David Mumford

Tata Lectureson Theta II

Birkhauser

Almost periodic solution of K-dV given by the genus 2

p-function D2log-& (z, St) with 92 = 10 22 10

An infinite train of fast solitons crosses an infinitetrain of slower solitons (see Ch. II2a,§10,IIIb,§4).

Two slow waves appear in the pictures: Note that eachis shifted backward at every collision with a fastwave.

David MumfordWith the collaboration of C. Musili, M. Nori,E. Previato, M. Stillman, and H. Umemura

Tata Lectureson Theta 11Jacobian theta functions anddifferential equations

1984 BirkhauserBoston Basel Stuttgart

Author:David MumfordDepartment of MathematicsHarvard UniversityCambridge, Massachusetts 02138

Library of Congress Cataloging in Publication Data(Revised for volume 2)

Mumford, David.Tata lectures on theta.

(Progress in mathematics ; v. 28, )Vol. 2 has title: Tata lectures on theta."Contains ... lectures given at the Tata Institute

of Fundamental Research in the period October 1978 toMarch 1979" - v. 1, p. ix.

Includes bibliographical references.Contents: 1. Introduction and motivation : theta

functions in one variable ; basic results on theta functionsin several variables - 2. Jacobian theta functions anddifferential equations.

1. Functions, Theta. I. Tata Institute of FundamentalResearch. II. Title. III. Series: Progress in mathe-matics (Cambridge, Mass.) ; 28, etc.QA345.M85 1982 515.9'84 82-22619ISBN 3-7643-3109-7 (Switzerland : v. 1)

CIP-Kurztitelaufnahme der Deutschen Bibliothek

Mumford, David:Tata lectures on theta / David Mumford. With thecollab. of C. Musili... - Boston ; Basel ; Stuttgart :Birkhauser

Progress in mathematics ; ...)

2. - Mumford, David: Jacobian theta functions anddifferential equations

Mumford, David:Jacobian theta functions and differential equations /David Mumford. With the collab. of C. Musili... -Boston ; Basel ; Stuttgart : Birkhauser, 1984.

(Tata lectures on theta / David Mumford ; 2) (Pro=gress in mathematics; Vol. 43)ISBN 3-7643-3110-0 (Basel, Stuttgart)ISBN 0-8176-3110-0 (Boston)

NE: 2. GT

All rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without prior permission ofthe copyright owner.

o Birkhauser Boston, Inc., 1984ISBN 0-8176-3110-0ISBN 3-7643-3110-0Printed in USA987654321

CHAPTER III

Jacobian theta functions and

Differential E uatiohs

Introduction

IIIa: An Elementary Construction of H erelli tic Jacobians

ix

§0. Review of background in algebraic geometry 3.1§1. Divisors on hyperelliptic curves 3.12

("92) Algebraic construction of the Jacobian of ahyperelliptic curve 3.28

§3. The translation-invariant vector fields 3.40§4. Neumann's dynamical system

Tying together the analytic Jacobian and algebraic3.51

Jacobian 3.75§6. Theta characteristics and the fundamental

Vanishing Property 3.95§7. Frobenius' theta formula 3.106§8. Thomae's formula and moduli of hyperelliptic curves 3.120§9. Characterization of hyperelliptic period matrices 3.137§10. The hyperelliptic fir-function 3.155§11. The Korteweg-deVries dynamical system

IIib: Fay's Trisecant Identity for Jacobian theta functions

3.177

§1. The Prime Form E(x,y) 3.207§2. Fay's Trisecant Identity 3.214§3. Corollaries of the Identity 3.223§4. Applications to solutions of differential equations 3;.239

§5. The generalized Jacobian of a singular curve and solutions

IIIc: Resolutions of Algebraic F uations b Theta Constants,

3.243

by Hiroshi Umemura 3.261

Bibliography 3.271

Introduction to Chapter III

In the first chapter of this book, we analyzed the classical

analytic function

(z T) = I e7T in2T+21T inz

of 2 variables, explained its functional equations and their

geometric significance and gave some idea of its arithmetic

applications. In the second chapter, we indicated how 17

generalizes when the scalar z is replaced by a vector variable

z E Cg and the scalar T by a gxg symmetric period matrix 0.

The geometry was more elaborate, and it led us to the concept of

abelian-varieties: complex tori embeddable in complex projective

space. we also saw how these functions arise naturally if we

start from a compact Riemann surface X of genus g and attempt

to construct meromorphic functions on X by the same methods used

when g = 1.

However, a very fundamental fact is that as soon as g > 4,

the set of gxg symmetric matrices n which arise as period

matrices of Riemann.surfaces C depends on fewer parameters than

g(g+l)/2, the number of variables in 0. Therefore, one expects

that the Q's coming from Riemann surfaces C, and the correspond-

ing tori XX, also known as the Jacobian variety Jac(C) of C,

will have special properties. Surprisingly, these special

properties are rather subtle. I have given elsewhere.

(Curves and their Jacobians, Univ. of Mich. Press, 1975), a

survey of some of these special properties. What I want to

x

explain in this chapter are some of the special function-

theoretic properties that * possesses when 0 comes from a

Riemann surface. One of the most striking properties is that

from these special 4's one can produce solutions of many

important non-linear partial differential equations that have

arisen in applied mathematics. For an arbitrary 0, general

considerations of functional dependence say that 4 (zSl) must

always satisfy many non-linear PDE's: but if g > 4, these

equations are not known explicitly. Describing them is a very

interesting problem. But in contrast when Sl comes from a?

Riemann surface, and especially when the Riemann surface is hyper-

elliptic, -S satisfies quite simple non-linear PDE's of fairly

low degree. The best known examples are the Korteweg-de Vries

(or KdV) equation and the Sine-Gordan equation in the hyperelliptic

case, and somewhat more complicated Kadomstev-Petriashvili (or KP)

equation for general Riemann surfaces. We wish to explain these

facts in this chapter.

The structure of the chapter was dictated by a second goal,

however. As background, let me recall that for all g > 2, the

natural projective embeddings of the general tori X. lie in

very high-dimensional projective space, e.g., IP3g--1) or IP45_1)

and their image in these projective spaces is given by an even

larger set of polynomials equations derived from Riemann's theta

relation. The complexity of this set of equations has long been

a major obstacle in the theory of abelian varieties. It forced

mathematicians, notably A. Weil, to develop the theory of these

varieties purely abstractly without the possibility of

motivating or illustrating results with explicit projective

examples of dimension greater than 1. I was really delighted,

therefore, when I found that J. Moser's use of hyperelliptic

theta functions to solve certain non-linear ordinary differential

equations leads directly to a very simple projective model of the

corresponding tori XQ. It turned out that the ideas behind this

model in fact go back to early work of Jacobi himself (Crelle, 32,

1846). It therefore seemed that these elementary models, and

their applications to ODE's and PDE's are a very good introduction

to the general algebro-geometric theory of abelian varieties,

and this Chapter attempts to provide such an introduction.

In the same spirit, one can also use hyperelliptic theta

functions to solve explicitly algebraic equations of arbitrary

degree. It was shown by Hermite and Kronecker that algebraic

equations of degree 5 can be solved by elliptic modular functions

and elliptic integrals. H. Umemura, developing ideas of Jordan,

has, shown how a simple expression involving hyperelliptic theta

functions and hyperelliptic integrals can be used to write down

the roots of any algebraic equation. He has kindly written up

his theory as a continuation of the exposition below.

The outline of the book is as follows. The first part

deals entirely with hyperelliptic theta functions and hyperelliptic

jacobians:

§0 reviews the basic definitions of algebraic

geometry, making the book self-contained for

analysts without geometric background.

xii

§§1-4 present the basic projective model of hyperelliptic

jacobians and Moser's use of this model to solve the

Neumann system of ODE's.

§5 links the present theory with that of Ch. 2, §92-3.

§§6-9 shows how this theory can be used to solve the

problem of characterizing hyperelliptic period matrices

0 among all matrices 2. This result is new, but it

is such a natural application of the theory that we

include it here rather than in a paper.

§§10-11 discuss the theory of McKean-vanMoerbeke, which

describes "all" the differential identities satisfied

theta functions, and especially the

Matveev-Iits formula giving a solution of Kd V. We

present the Adler-Gel'fand-Manin-et-al description of

Kd V as a completely-integrable dynamical system in the

space of pseudo-differential operators.

The second part of the chapter takes up general jacobian theta

functions (i.e., ,9 ,0) for 0 the period matrix of an arbitrary

Riemann surface). The fundamental special property that all such

.D's have is expressed by the "trisecant" identity, due to John Fay

(Theta functions on Riemann Surface, Springer Lecture Notes 352),

and the Chapter is organized around this identity:

§1 is a preliminary discussion of the "Prime form" E(x,y)

- a gadget defined on a compact Riemann surface X which

vanishes iff x = Y.

§2 presents the identity.

§§3--4 specialize the identity and derive the formulae for

solutions of the KP equation (in general) and KdV,

Sine-Gordan (in the hyperelliptic case).

§5 is only loosely related, but I felt it was a mistake

not to include a discussion of how algebraic geometry

describes and explains the soliton solutions to KdV

as limits of the theta-function solutions when g of

the 2g cycles on X are "pinched".

The third part of the chapter by Hiroshi Umemura derives the

formula mentioned above for the roots of an arbitrary algebraic

equation in terms of hyperelliptic theta functions and

hyperelliptic integrals.

There are two striking unsolved problems in this area:

the first, already mentioned, is to find the differential

identities in z satisfied by (z,SZ) for general U. The

second is called the "Schottky problem": to characterize the

jacobians X2 among all abelian varieties, or to characterize

the period matrices U2 of Riemann surfaces among all U. The

problem can be understood in many ways; (a) one can seek

geometric properties of XS and especially of the divisor e

of zeroes of ,9-(Z,Q) to characterize jacobians or (b) one can

seek a set of modular forms in U whose vanishing implies

comes from a Riemann surface. One can also simplify the

problem by (a) seeking only a generic characterization:

conditions that define the jacobians plus possibly some other

irritating components, or (b) seeking identities involving

xiv

auxiliary variables: the characterization then says that X0

is a jacobian iff 3 choices of the auxiliary variables such

that the identities hold. In any case, as this book goes to

press substantial progress is being made on this exciting

problem. I refer the reader to forthcoming papers:

E. Arbarello, C. De Concini, on a set of equationscharacterizing Riemann matrices,

T. Shiota, Soliton equations and the Schottky problem,

B. van Geemen, Siegel modular forms vanishing on themoduli space of curves,

G. Welters, On flexes of the Kummer varieties.

The material for this book dates from lectures at the

Tata Institute of Fundamental Research (Spring 1979), Harvard

University (fall 1979) and University of Montreal (Summer 1980).

Unfortunately, my purgatory as Chairman at Harvard has delayed

their final preparation for 3 years. I want to thank many

people for help and permissions, especially Emma Previato for

taking notes that are the basis of Ch. Iila, Mike Stillman for

taking notes that are the basis of Ch. Ilib, Gert Sabidusi for

giving permission to include the Montreal section here rather

than in their publications, and S. Ramanathan for giving per-

mission to include the T.I.F.R. section here. Finally, I would

like to thank Birkhauser-Boston for their continuing encouragement

and meticulous care.

3.1

§0. Review of background in algebraic geometry.

We shall work over the complex field C.

Definition 0.1. An affine variety is a subset X c (Cn, defined

as the set of zeroes of a prime ideal 4) a (C[Xl,...,Xn]; X

{x E cnlf(x) = 0 for all f EP }1). X will sometimes be denoted by

V(P) or by V(f,...;fk) if generate D.

A morphism between two affine varieties X,Y is a polynomial map

f: X ->Y, i.e., if (X1'...,Xn)E X, then the point f (X1,..., X) has coordinates Yi =fi(Xl, ...fi(n),

where fi E cE[Xl,.... Xn]; following this definition, we will identify

isomorphic varieties, possibly lying in different (dimensional) Cn's.

A variety is endowed with several structures:

a) 2 topologies; the "complex topology", induced as a subspace

of (Cn, with a basis for the open sets given by

{(xl,...,xn)Ilxi-ail< s, all i}, and the "Zariski topology" with

basis {(x1,...,xn)lf(x) 0}, f E (E[Xl,...,Xn].

b) the affine ring R(X)= CE[Xl,..,Xn]/p, which can be viewed as

a subring of the ring of (C-valued functions on X since D is the

kernel of the restriction homomorphism defined on CE-valued polynomial

functions on Cn, by the Nullstellensatz.

c) the function field (C(X), which is the field of fractions of

R(X); the local rings and &Y,X, where x is a point, Y a

subvariety of X, defined by a x = {f/gIf,g ER(X) and g(x) 01, with

maximal ideal m{ = {f/g EV I f('x) = 0 } , C5`Y,X = { f/gl f,g E R(X) ,g t 0 on y}2) if Y = Y(q) ;= R(X)Q

1) If a polynomial f ECE[X is zero at every point of V thenf E3 ; this is Hilbert 's Nullnstellensatz.

2) We denote by A the localization of a domain A with respect to itsprime ideal Q,q Aq = {a/bja,b EA, b q}.

3.2

(notice: if x E Y, R(X) c C. c

Xc T(X)the structure sheaf (Y,

subsheaf of the constant sheaf U i --- T (X) , which assigns to anyZariski-open subset U of X the ring 0U ax = r (u,cX) ; and a dimension

given by dim X = tr.dtC(X). dim X is related to the Krull dimension of Oy,X

(maximum length of a chain of prime ideals), by:

Q Proposition 0.2. dim X - dim Y = Krull dim. V xX

d) the Zariski tangent-space at x E X, which can be defined

in a number of equivalent ways:

TX,x = vector space of derivations d: R(X) - T centered at x

(i.e., satisfying the product rule d(fg) = f(x)dg+g(x)df); or

2TX, x = (rft / fl2) V , the space of linear functions on 1v/m ; or

TX,x = the space of n-tuples such that for all

f f 0 mod e2

where from a derivation d a linear function £(Xi-xi) = dXi and an

n-tuple (xl,...,xn) with dXi = xi are obtained; this sets up the

bijection. This vector is also written customarily as (xi)a/3x;i=1

Proposition O.D3 3 a non-empty Zariski open subset U c X

such that tr.d.T T(X) = dim TX,x for all x E U; if x 0 U, then

dim TX x > dim X.

O U is called the set of "smooth" points of X, X-U the "singular

locus". It can be shown from this proposition that U (with the complex

topology) is locally homeomorphic to T, where d is tr.deg.TCC(X).d

3.3

Lemma 0.4 For any x E X, .3 a fundamental system of Zariski

neighborhoods U of x such that U is isomorphic to an affine variety.

In fact, for any f E R(X) such that f(x) 0, Uf = {y E x(f(y) 3g 0}

is a neighborhood of x and if RX = T[X1,...,X ] /', then Uf is isomorphic to the sub-

variety ofTn+l

is defined by the ideal the isomorphism is

realized by

-1(x1,...rxn) (xl,...,xn, .f(xl,...,xn))-

But we need a more subtle definition of morphism from an open set

to an affine variety.

Definition 0.5. f: U Y is a morphism if (equivalently):flopenx

(1) for any g E R(Y), thought of as a complex-valued function

on Y, gof E r(U,O )

(2) 3 gikhk E (C[X1,---,X AI such that for any (x1,---,xn) E U

there is a suitable k such that hk(x) 34 0, and the i-th coordinate of

is given by gik(x1,-.-,xn) whenever hk(x) 0.

hk(xl,...,xn

(n.b.. there may not exist a single expression )ti= h(X ,--,X '

with h--1 E r(u,O'x) ., 1 n

Theorem 0.6 (Weak Zariski's Main Theorem). If f: X - Y is-el

an injective morphism between affine varieties of the same dimension

and Y is smooth, then f is an isomorphism of X with an open subset of Y.

3.4

QThe product of affine varieties is categorical, i.e., given

X (,n and Y c Cmaffine varieties, i) X x Y is an affine variety.

(in (Cn+m), ii) the projections are morphisms, iii) if Z is an affine

variety and morphisms Z -> X, Z -- ;;,Y are given, then there is a

unique morphism Z --> X x Y making a commutative diagram

Definition 0.7 A variety in general is obtained by an atlasn

of affine varieties: X =a S Xa, S a finite set, Xa c ( a, glued

by isomorphisms

(where Ua,s is a nonempty Zariski-o en subset of Xa), such that one

of the equivalent (se aration) conditions holds:

(1) X is Eausdorff in the "complex topology" (a subset of ,Xbeing open in the complex topology if and only if its intersections

with Xa are open for all a's)

(2) the graph of a$r,, c Xa X Xa is Zariski-closed.

(3) for any valuation ring R c T(X) = C(Xa) (any a, for the

function field of Uf c Uas coincides with that of Xa, hence has

identifies E(Xa) and VX)) there is at most one point x E X

3.5

such that R > (YX (R "dominates" 0 x, or R is "centered" at x,

i.e., R D(Y and inR = rrX) .

(4) for all affine varieties Y and morphisms f,g: Y --.X, the

set {y E YIf(y) = g(y)} is (Zariski) closed in It.

Such an X carries:

(a')) 2 topologies (the complex and the Zariski; as with the

complex topology, a subset of X is Zariski-open if and only if its

intersection with all the Xa's is Zariski-open)

(c') the function field Q(X); the local rings)

{f/gIf,g E R(Xa), g(x) 0} if x E Xa

; the structure sheaf

Ui noxxEU

(d') the Zariski tangent space Tx,X =Tx,X

if x E Xa.

0 f: X --- Y is a morphism between two varieties if the, restriction

res f: Ua f f-l(Va) ---ov is a morphism for all a,$'s or, equivalently,

if. for any open set U c Y and g E r (U,0 ) , gof E r (f-lu, ) issatisfied.

Key example. Projective varieties, defined by homogeneous

ideals P c cE(X0,...,Xn1, as

V(P) _ {(xo,..-,xn) E 7Pn If(x) = 0 for all f E}7 };

an atlas is given by V(). = f p E v (p) Xi (p) 4 0} .

variety X can even be defined as a set of local rings {Cr) withthe same fraction field M(X). Then the topology on X is defined asfollows - for each f ET(X), let Uf be the set of the local ringscontaining f.

3.6

The product of varieties is again a variety; we take (UxV)Zfg

to be a basis for the open sets in XxY, where U,V are open subsets

of X,Y isomorphic to affine varieties, fi E r(U,OX), giE F(V,( ) and

(UXV)Efigl is the set of points (x,y) E UxV such that

Efi (x) gi (y) 0 -

The product of projective varieties is again a projective variety,

for instance the map (xi,yj) i (xiyj) embeds 7Pn x3Pm into

g, (n+1) (m+l)-1 and the image is given on the affine pieces

(7Pn+m+nm)Xhk

by the equations sij = sihskj for all i h and j # k,

where sij = Xij/Xhk'

Definition 0.8. A variety X is complete (or proper if one of

the following equivalent condition holds:

(1) X is compact in the complex topology

(2) 3 a surjective birational morphism f: X' --> X, X' projective

(3) for all valuation rings R c OW, 3 x E X such that R y ax(4) for all varieties Y, Z c XxY closed, pr2Z is closed in Y.

A subvariety of a variety X is an irreducible locally closed

subset Y of X; the variety structure is given by the sheaf (7Y which

assigns to any open subset V of Y the ring

r(v,OY) = T-valued functions f on V b x E V, 3 a neighborhood U

of x in X and a functionf1E r(U,OX) such that

f = restriction to UOV of fl

So, any open subset of X is a subvariety; but a subvariety which

is a complete variety must be closed.

3.7

Divisors and linear systems.

The theory of divisors is based on a fundamental result of Krull.

if R is a noetherian integrally closed integral domain, then

a) for all p c R, minimal prime ideal, R is a discrete

valuation ring.

b) R= fl R4D

min.

. P

prime

Thus if Ordd = valuation attached to RP, and K is the fraction field of R,we get an exact sequence:

1 -> R* > K* ---D free abed.. groupon min. prime ideals

v1 Wf d> ord f- [P ) = (f)

Let be the primes occurring positively in (f),

P, ... , }

mto negatively in (f), then

Corollary 0.10. For all prime ideals p in R,

f E R any i

f-1 E R 3)any i, hence

neither f or f-1 are in Rl? + for some i,j"f is indeterminate at

(in particular, if f is indeterminate at , then 3) is not a

minimal prime ideal).

3.8

We will apply Krull's result to the following geometrical

situation:

Theorem 0.11: if X = J Xa is a smooth variety, then RX isa

integrally closed, the minimal primesJ

in RX are the codimension

one (closed) subvarieties Y of X which meet Xa and (RX ) P _ ay X.

Idea of the proof: for all points P E X,,, the hypothesis of

being smooth means dim /M2 = dim X = Krull-dim. ,, i.e., ( p is'regular" (this can be taken as a definition). One proves that a

regular local ring is integrally closed, hence C is integrally

closed. Since, for any affine variety, RX = f,

0 P5), RX

a PEXa a

is integrally closed. The rest of the statement follows from the:

Lemma 0,12 A (closed) subvariety Y of Z is maximal

dim Y = dim Z-l.

'(This follows from (o;%), or else can be used to prove (o.2.) . )

Thus the map f {. >(f) defines a homomorphismfree abel. group

T (X) * >Div X =on codim. 1 subvar.

Elements of Div X are called divisors on X and 2 divisors D1,D2 are

called linearly equivalent (written Dl='D 2) if D1-D2 = (f), some fET(X)*.

0 The corollary 0.10 has the following geometrical meaning: for

any f EQZX)*, set (f) = (f)O-(f). with M0 (zero-divisor) and (.f).,(pole-divisor) both positive divisors, and let, for any divisor

D = IniYi, supp D = U Yi; then

5) If x/yE f, e,, consider the ideal A = {zERX since x/yE 0j,PEX al y a

x/y can be written w/z, with wERX , zERX MP, so P MP. Therefore A is not

contained in any maximal ideal, s8 A = Ra P. This means that lEA, i.e., 2ERx .

a y a

3.9

fEt9' 4= P supp (f)

f c c, 4. P / supp (f) o

f is indeterminate At P 44 P E supp (f) o n Supp (f).'0 Moreover, if X is a smooth, affine variety of dimension 1 with affine

ring R, then R is a Dedekind domain, so all its ideals are products

of prime ideals. If fER, let:

(f) _ En.Y. where Yi corresponds to the prime ideal Vi in R.

Then:

Corollaryniif-R =T7 :P

i

We define Div+(X) to be the semi-group in Div(X) of divisors with

only positive coefficients.

We define Pic(X) as the cokernel:

T (X) * r Div X Pic (X) } 0

i.e., as the obstruction to finding rational functions with given

zeroes and poles. Elements of Pic(X) are called divisor classes.

Example. Pic (En) = Z. In fact, any hypersurface is given by

the zeroes of a homogeneous polynomial. The degree of a divisor

D = EniYi is defined by deg D = Enideg Yi where deg Yi is the

degree of the irreducible homogeneous polynomial defining it. Then

any divisor of degree zero comes from a rational function, and

degree gives an isomorphism Pic(1Pn) -.>2Z

3.10

Suppose D is 4 positive divisor; we define the vector space

Y_(D) = {fEu(X')*i (f)+D > 0} u {0},

Note; The condition (f)+D > 0 is equivalent to M. < 0 '(the

poles of f are bounded by D). Note that f(D) is a sub-vector space

of T (X) .Lemma 0.15. If X is proper, dim t (D)<c and for all f.E T(X)*

(f) = 0 if and only if fE(C'

0 In this case, we form the associated projective space 2 (s..,(D))

of one-dimensional subspaces of Y-(D) and note:

W W

Divfibre through D ofF (x(D) ) !-' [,-1 (irD) n Div+ (X)l _ +

(X)) Pic(X)

line{a.flaek} + )divisor (f)+D

These projective spaces and their linear subspaces are the so-called

"linear systems" of divisors. 1P (.,x (D)) is denoted D J.

If L c ,DI is a linear subspace of dimension k, set

B(L) = () Supp E, the "base locus" of L.

The fundamental construction associated to linear systems is the map

PL : (X-B(L)) * 1111

where Lv is the projective space of hyperplanes in L, given by

x i > [hyperplane in L consisting of the EEL s.t. x E Supp EJ

cpL isa morphism. To prove this and to describe cpL explicitly,

let's choose a

3.11

projective basis of L, i.e., k+1 points which are not contained in a

hyperplane:

E, E+(fl) ,E+(f2) ,... ,E+(fk) .

Set fo = 1; the map

x'-- (f o(x)...,fk(x))

is defined on the open set X-Supp E since the poles of fi are all

contained in Supp E; it coincides with q7L on X-Supp E, as we

see if we let coordinates on L be and note: for x Supp E,

kx E Supp(E + ( I c.f.)) c. f

1. (x) = 0.

i=0 i=O

hence f.L(x) = hyperplane in L with coefficients fo(x),--, fk(x)

= pt. of LO with homogeneous coordinates - - - fk (X).

3.12

§1. Divisors on hyperelliptic curves.

Given a finite number of distinct elements ai E T, i E S, letf (t) = (t-ai) . We form the plane curve C1 defined by the equation

iES

s2 = f (t) .

The polynomial s2_f(t) is irreducible, so (s2-f(t)) is a prime ideal, and C1 is

a 1-dimensional affine variety in C2. in fact, C1 is smooth. To

prove this, we will calculate the dimension of the Zariski-tangent

space at each point, i.e., the space of solutions (s,t) ET2

to

the equation

(s+e s)2 fl(t+e t-ai ) mod e2

for (s,t) E C1.

That is equivalent to the equation

2ss = t . I T (t-a ) ;j ES i#j

if s 0, the solutions are all linearly dependent since

s = 2s( I T (t-a )); if s'= 0, we get from the equation of thej ES i34j 1

curve Ti (t-ai) = 0, hence t = ai for some i; thusiES

0 = t.TT(a.-a.), so 0. Thus at all points, the Zariski

tangent space is one-dimensional.

We add points at infinity by introducing a second chart:

C2 : s 2 = i T(l-ait') if #S 2kiES

s' 2 = to .TT (1-ait') if #S = 2k--1,iES

3.13

glued by the isomorphism t' = 1t

sstk

between the open sets t 0 of C1 and t' 0 of C2.

The points at of C1 are:

Col 11002

given by t' = 0, s' _ +1 if #S even

00 w to t' = 0 = s' if #S odd.

On the resulting variety C we can define a morphism

n: C 3P1. Let t and t' = 1/t be affine coordinates in 7P 1

then define u by

(s,t)'+- t, on the chart Cl.(s' , t') 6 > t', on the chart C2 .

n is 2:1 except over the set B of the "branch points" consisting

in the ai's, and o in the case #S odd. The number of branch points

is therefore an even number 2k'in both cases. Topologically C is a surface with

k-l handles, so we say that it is of genus g = k-l;- this is called the genus of

the curve. This is usually visualized by defining 2 continuous functions

+,,T(t) , - f t for t E JP1-(k"cuts") and reconstructing C by glueing the 2 open

pieces of C defined by s = + f t) and s = - f t):

k disjoint cuts on each

copy of 1PI glueing ai's, ai's

3.14

Since C is smooth, the af fine rings of C1 and C2 are Dedekind domainsl),

and their local rings V are discrete valuation rings,

1: (s,t) t> (-s,t)

is an automorphism of C, that flips the sheets of the covering, hence

is an involution, with the set of orbits C/{±l} ]P1. Tr(B) is the

k (o1 + 00 2D =

2koo

Lemma 1.1. is a basis for the vector space

(D).

1) We already know that the tangent space to the curve at each pointhas the right dimension, in each of the two affine pieces; but it's alsoeasy to see directly that T[t,s]/(s2-II(t-a i))=R is integrally closed,

the reason being that f(t-ai) is a square-free discriminant over the

U.F.D. cC[t]. If we let a be the automorphism which sends

(s,t) to (-s,t), then the general element of the quotient field

of R is a+bs, with a,bE(C(t), and for all a+b.s integral over R,

(a+bs) +r; (a+b s) = 2a and (a+b s) a (a+b s) = a2-b2d are in V t) and areintegral over U[tl, which is integrally closed. Thus 2aEC[t],

a2-db2ECC[tl, so db2ET[tl; since d is square-free and M[t] is

U.F.D. we conclude b E T ['tl , hence a+bs ER.

if #S is even

if #S is odd.

3.15

Proof: The function field of X, (a (t) [ II (] , has aninvolution over (C(t), that interchanges1'002, or fixes the

point , hence sends 2 (D) into itself. Thus V(D) splits into

the sum of the +1 and --1 eigenspaces of i

,F_ (D) = [i(D) n T (t) ] ® [d° (D) n SC (t) ] .

If h(t) E Y (D) n r(t), since it has no poles for finite values of t,

then it must be a polynomial in t, h E (C[t]. On the other hand, in

the case #S even the maximum ideal of } = R(C2) is

generated by t' since the equation of the curve givess'-l = ((s'+1)-1 (TI(l--t'ai)-1) E(t')R(C2) (t',s'_l); in thethe max. ideal of C = R (C 2)t' = S,2 (II (1-t' ai)) -1; thus

case #S odd

is generated by s', since

or) = -v (t) =1 (similarly v (t' )= 1) ,

Av (t') = -v. (t) = 2,v O P

( t ' ll 2

2°° , #S odd

So in order for (h), to be < D we must have deg h < k.

Now consider h (t) E X (D) n s(C (t) , h = sg (t) with g (t) E Q(t);g may only have poles in C1 where s has zeroes, i.e., in the set

{Pi ... (O, ai) } .

CX) 1 + 00 2, #S eveni.e., (t)

The order of vanishing of s at Pi is 1 and that of

(t-a1) is 2, since the max. ideal in_ (gyp

1

is generated by s and

and (t-a1) = S2 ( 11 (t-a.)) l. That prevents g (t) ET(t)#i

3.16

from having a pole at Pi, because the product sg(t) would still have

a pole at Pi. Thus the only poles of g(t) must be at -., i.e., g(t)

must be a polynomial in t; but now

k k (°°l+°°2) #S even(s't _ (2k--1)- #S odd

hence g(t) must be constant in order to have D+(sg(t)) > 0.

This proves the lemma. Ci

Now a projective base for IDI is D+(s) ;

since W. = we have D+(tk) = either k(O1+02) or 2k02=

where 0 is a branch point in the 2nd case, and where 01,02 are the

two points in the fiber over the point t = 0 of IP1 in the 1st case. IHence 1DI contains D and whose supports are disjoint,

hence 4IDI) is empty. F]

Thus explicitly, corresponding to IDI, we get (PIDI: C 7pk+l

by (s,t) F > (l,t,t2,-..,tk,s) on `IC Supp D = Ci,

and (s',t')I --. (t'k...... ,t',l,s') on C2.

Note that these 2 mapsdo agree on the overlap: (l,t,t2,,---tk,s) t

This map, which is an isomorphism of C with its image, makes C

into a projective curve.`

3.17

Remark. If : 1P1 P1P1 is a linear fractional transformation

( at+b a bli"e'' fi(t) = ct+d' (c d)

E SL(2,(D)), then it is not hard to check

that the two hyperelliptic curves whose sets of branch points are,

respectively, B and OB) are isomorphic. So we can henceforth

assume that #S is always odd by sending one branch po:int,to Co.

Our aim is to describe a variety of divisors on C, and from

this the Jacobian variety of C; the idea of this construction is due

originally to Jacobi and appeared in

"Uber eine neue Methode zur Integration der hyperelliptischen

Differential leichun en and fiber die rationale Form ihrer vollstandigen

algebraischen Integralgleichungen" Crelle, 32, 1846.

Let's consider the subset Divv(C) of Div(C) given by all the

divisors of degree v : Div(C) = LL Divv(C), and inside the setvE7,L

+ v j j those with the following property:of ositivelones Div `f CV(

IDiv+,v (C) = Div+,v (C) = D E Div+'v (C) I if D = Pi r then P all i

o i=1and Pi 34 t (P all ij

invdl

Our basic idea is to associate to D E Div+'v(C) three polynomials:

V(a) U(t) _ fl (t-t(Pi)), monic of degree v (t(Pi) is the value of

i=lat Pi)

(b) If the Pi's are distinct, let

Y (J (1,a'(

3.18

V IT (t-t(Pj))

V (t) _ s (Pi)li=1 TT (t (Pi ) -t (P

J))

jEi

V(t) is the unique polynomial of degree < v-1 such that

V (t (Pi) ) = s (Pi) , 1 < i. < v

If P. has positive multiplicity in D, then we want to "approximate

the function V-f--(t) up to the order mD(Pi)", and in order to do

that we let

V (t) = the unique polynomial of degree < v-l such that, if

mD (Pi) = ni,

)'[V(t) - it (t-a2)]LES

= 0 for 0 < j < n -1- it=t(Pt)

By construction f(t)-V(t)2 is divisible by U(t), hence

(c) Define W (t) by: f(t)-V(t) 2 = U (t) .W (t) .

Let's assume v < g+1; then deg V(t)2 < deg f(t) and since U(t) is

manic in t of degree v , W is manic of degree 2g+l -v.

Conversely, given any U,V,W such that f-V2 = UW, U and W manic,

having degrees as above, we get the divisor (U)o of v points on the

t-line; over each zero of U, the corresponding value of V gives a

square root of f(t), i.e., a value of s (either one of ±4f-(t) );

thus the divisor of the points on the curve so obtained isin Div'v.0

3.19

Remark. Given U,V,W satisfying the above equation, then in

ct[s,t];s2_f (t) _ (s -V (t)) (s+V (t) ) -U (t) W (t) , hence

(s2-f(t)) C (U (t) , s-V (t)),

and the bigger ideal defines a zero-dimensional subset of C, which

is in fact supp D, or a zero-dimensional subscheme, which is D.

We have now proven:

'Proposition 1.2. There is a bijection between

Diva' andtriples ofpolynomials

U,V,W

f-V2=UW, U, W are monic,

deg 'V < v-l, deg U = v, deg W =2g+1- v

Notice how the bijection gives us a way to introduce

coordinates into Div +'y (x): let0

U(t) = tv + Ulty-1 +... + Uv

V(t) = Vlty-1 +... + Vv

W(t) =t2g+l-v

+ W0t2g-v +... + W29-v

be 3 polynomials with indeterminant coefficients, and expand:

f - V22g

UW = I a a(Ui,Vj,WQ)taa=0

Then, taking Ui,Vj,Wk as coordinates;

[the set of triples (U,V,W) as above] = c (r2g+l+v_

3.20

Or else, since U and V determine W whenever the division is

possible, we can write using the Euclidean algorithm:

f (t) -V (t) 2 = U (t) . [t2g+l-v+B0 (Ui,V)t2g_v+..

]+Rl( 1,V.)tv-l+...+R (U1,V)

remainder

Using only Ui,Vj as coordinates, we find:

2v[the set of triples (U,V,W) as above] ~ c .

The structure of affine variety is the same in both cases because the

morphisms:

(Ui), vi , k)projection > (Ui,V and

(Ui,Vj'Bk (U. 1v ) ) =* (Ui,Vj )

are inverse of one another.

On the other hand, we can parametrize Div+'v(C) by points of C,

in the following way: we have a surjective map

Cv ) Div+'v(C)

now let (C9)o c CV be the Zariski open set defined as follows:

3.21

V _CV - (Cv)o = [ U pi1(°°) Jut U pij ()i=1 O<iaj<i

'A- i<j.V

where pi:CV

> C is the i-th projection, pij: CV - ;:C2 the

(i, j) -th projection, r = [locus of points (P,1 P)1 1 the Zariskiclosed subset of C2 given by the equations sl = -s

2,t1 = t2 if

(si,ti,s2,t2) are coordinates. Then everything is tied together in:

Proposition 1.3. The equations a prime ideal

in (C[Ui,Vj,WzI, the variety is smooth and the composite map

CV)o ---->-> Divo`v (C) =( V(a0, ,a2g) is a surjective morphism making(orbit space for the group of permutations

V($, -.,a2g) = I

Sv acting on (Cv) o 1

iJ\ ,_.. The proof of proposition 1.3 will consist of 2 steps.

1. In order to prove that V(aa) is smooth, let's consider a

small perturbation of the coordinates (U1,...,Uy,V1,...,Vv,WO,...,2g`v)'

Starting with any solution U,V,W to the equation f-V2 = UW (with

prescribed degrees) we will show that the vector space of triples

U,V,W (deg U,V <,v-l, deg W < 2g-v) such that

f- (V+EV) 2= (U+sU) (W+cW) mod s2 (*)

has dimension v .

The dimension must be > v since in general k equations in

n-dimensional affine space define a closed set whose irreducible

components are varieties of dimension > n-k; which in our case means

> (2g+l+v) - (2g+l) = v.

3.22

On the other hand, the condition (*) is equivalent to the equation

(*) Uw + Uw + 2VV = 0.

If we can prove that any polynomial of degree < 2g can be written

in the form TjW + UW + 2VV, then the number of linear conditions

imposed by (*) equals the dimension of the space of polynomials

in t of degree < 2g, which is 2g4 l, and we conclude that the dimension

of the space of solutions of (*) equals (2g+l+v)-(2g+l) = vV. But

notice:

WU gives all the polynomials that are multiples of U

2VV assumes any given values at the points t where U = 0, V 0

UW tt of it If to tt It n it U = 0 , W ..F 0

These make a vector space of dimension (2g-v+1)+v = 2g+l, since

U = V = W = 0 never happens, or f_= V2+UW would have a double zero.

2. (CV)o - V(aa) is a morphism. First observe that the map is a

morphism on the smaller Zariski-open set (C)) = (Cv)o - U pi1(6),ooi<j

7

where o c CXC is the diagonal. This is because the coefficients

of U,V (hence W) were given above by an explicit formula as

rational functions in the coordinates s(Pi),t(Pi) with denominators

products of t(Pi)-t(Pj). These denominators are zero only if Pi = Pa

or P. = I(P (i j), i.e., only on pg(/ UT").

To see that the map is a morphism elsewhere, we use Newton's

Interpolation formula. This is expressed in

3.23

Theorem 1.4 (Newton): Let f be a Cm function on an open set

U c kt (resp. an analytic function on an open set U c (L). Define

by induction on n

f(xl,...Ixn) =f (xl . ' . , xn-1) -- f (x2 , ' , xn)

x1 _ xn

Then f is a em function (resp. an analytic function) on Un,

symmetric in its n arguments xi, and for all

f (x) - [f(a1)+(x-)f(a11)++(x-a1) -f (av... ,a )] +--a fn i=l

Note that the expression in brackets is therefore the unique

polynomial V(x) of degree < n-l such that:

-)k( [f(x)-V(x)l Orof a. \

0, 0 < k < ( ) -- 1 .equal to a

Ix=a

To apply this to our problem, we define by induction on n rational

functions on Cn by:

S(P1,..,Pn) =t (P1) -t (Pn)

As in Newton's theorem,. it is an easy calculation that

is symmetric in P1,.-,Pn. I claim

3.24

For n = 2, note that

s (P1) -s (P2)s (P'P)12 t (P1 -t (P1 +s (P2) ]

f (t (P1)) -f (t (P2)) 1 - 1

t (P1)-t(P2) 1 s(P)+s(P2)

polynomial in

t(P1) ,t(P2)

Thus s(P1,P2) has no poles in the open set t(P1) 76 t(P2) nor in the

open set s(P1) -s(P2) The union of these 2 open sets is C2-r

since t(P1) = t(P2) and s(P1) -s(P2) implies P2 = L(P1). Thus

s(P1,P2) has no poles on (C2)o, hence is in r((C2)o, 0 2). ForC

n > 3, by induction and, the expression for s(P1' 'Pn),

has poles only if t(P1) = t(Pn). But by symmetry, it has poles only

if t(P2) = t(Pn) too. The subset t(P1) = t(P2) = t(Pn) has

codimension 2 in (Cn)o, so has no poles at all in (Cn)o.

Finally, by Newton's theorem, the interpolating polynomial V(t)

can be expressed by:

v-1V(t) = s(P1)+(t-t(P1)).s(P1,P 2)+...+ TT

i=1

Thus the coefficients Vi of V(t) are polynomials in t(Pi) and

hence are functions in r((Cv)0, 0 ) . This provesC

that (C') 0 -- V(a) is a morphism.

s2 (P1)-S 2 (P2)

3.25

A consequence is that the set V(aa).is irreducible since (CV)o

maps onto V(a and (Cu)o is irreducible. To complete the proof,

use the elementary:

Lemma 1.5: If V c (Cn is an affine variety,

f1'...'fk E T[X1,...'Xn are polynomials such that

V = x E (Cn fi (x) = 0 all I

p x EV, TV,x = ;x E n fi(x+ek) = 0 mod e2, all i}

then is the prime ideal of all 2olynomials zero on V.

(Proof omitted).

We want to emphasize at this point the rather unorthodox use

that we are making of the polynomials U,V,W:

we have a bijection

divisors D on C three polynomials(of a certain type (U (t) ,V (t) ,W (t) ofa certain type

Thus

b) these divisors D become the points of a variety for

which the coefficients of U,V,W are coordinates.

To take the coefficients of certain auxiliary polynomials as

coordinates for a new variety is quite typical of moduli

constructions, although it is usually not.so explicitly carried out.

In all of this Chapter, U,V,W will play the main role, and we will

talk of (U,V,W) as representing a point of the variety Divo'v(C).

3.26

Actually, for any smooth projective curve X, it's possible to

describe Div+,V (X) as a projective variety, although not as

explicitly asinthe above construction. we outline this without

giving details, as it will not be used later. we use the bijection

Div +'v(X) - Symmv(X), the orbit space of Xv under the action of the

symmetric group permuting the factors.

A) Given an embedding X r > 3P n, we have the associated

Segre embedding:

j X' > 2P(n+l)v -1

given by:

V Pa = (Xpa),...,Xna))EX

v (a)then (P1,...,Pv) r--- (..., TT Xa(Wa=l

(one coordinate for every map a ;

B) j is equivariant under the action of the symmetric group on Xv

von the homogeneous coordinate ring of X, R =

S

Sv acts preserving the grading; the ring of invariants R v

is finitely generated by homogeneous polynomials and OM

such that if are a basis of R. in degree M, then:

Sd S"h R V

I

(r[g0r...rgN a

Ctt mehfS w,oqree.ivisible by M

3.27

C) Via gi we have the embedding:

Symm VX C ]PN

Xv C 3P (n+1)V -1

D) The smoothness of Symm V X follows from local analytic

description:

SymmV(z-disc.) {open set in a }

biholomorphically

via -> Celem. symm. functions of z(Pi)l.

The explicit coordinates given by prop. 1.2 are particular to

the case of hyperelliptic curves.

3.28

§2. Algebraic construction of the Jacobian of a hyperelliptic curve.

Let's recall that a hyperelliptic curve C is determined by an

equation s2 = f(t), where f is a polynomial of degree 2g+l; C has

one point at infinity, and (t). = 2--

(s) . = (2g+1) --.

We shall study the structure of Pic C = {group of divisors modulo

linear equivalence}.

Since the degree of the divisor (f) of a rational function is

zero, there is a homomorphism

deg: P

i

Definition 2.1. The Jacobian variet of C is given by:

Jac C = Ker[deg: Pic C ` 2Z]

we wish to endow Jac(C) with the structure of an algebraic variety.dO-Cl

The possibility of doing this by purely algebraic constructions was dis-

cussedsby A. Weil. In the hyperelliptic case, his construction becomes

quite explicit. For the general case, see Serre [ ].

Step I. Given any g+l points on the curve P1.....Tg+1, such

that Pi # - and P i # 1Pj if i # j, the function

gs+4(t) where (t(Pi)) = s(Pdeg c ¢ g,

tr (t-t(Pi))i=l

3.29

has simple poles at all of the Pi's and no poles anywhere else, forqhd

numerator and denominator are both zero at LPi,Athe numerator is 54 0

at Pil) . At (s)ue (2g+l) -°°. (t) )co < 2g.cog+l(n (t-t(Pi)) (2g+2) co,i=l

g+lPi

i=1

so the function is -z r-o.-. Thus :C", -c

Consider also the function

an equality of divisor classes:

00 + Q1 ., suitable Q's.v'i=l 1

t-afor any numbers a,b; it gives

P + tP - t-a t-a _ +a a t-b) o Qt-b) Pb t Pb

Similarly, using the function t-a, we see:

Pa + tPa = 2= 9)Let's define ]G to be the divisor class of degree 2 that

-

coI

nntains

P+tP, all P E C.2)

The above remarks show that for every divisor D of degree zerog

SPl,...,P such that D P. . - g-03. In fact for any D ofg i=1

Q Qdegree 0, D = R. -- S. _

i=1 i=1 1 GRi + IlSi - P1Si+Si) =- JRi +Jt i-22-co;

also write whenever a pair R+tR occurs in JRi+ ItSi; now we can

use the construction above in order to decrease the number of points

Or if Pi is a branch point, then the numerator vanishes to 1st order

and the denominator to 2nd order at P..2)Also calledAfundamental "pencil" on

th1

e h perelliptic curve; "pencil"because the projective dimension of IP+tPI= W(d(P+i P) ) is 1. In theaffine part of the curve C1 c STS, IP+tPI is cut out on C1 by the pencilof lines through the point at infinity of the curve.

3.30

in R; + iSi to < g. Therefore the map:

I: SymmgC ` Jac C

Ipi

iIpi-g..'

is surjective.

(This in fact is true for every curve.)

Step II. Given a divisor Pi, Pi -, Pi ZP. if i > j,i=1

then X a non constant rational function on C whose poles are

bounded by IP i.

gProof. Let h be such a function; then h II (t-t(Pi)) has poles

i=lonly at a polynomial in the affine coordinates s,t,

i.e., it has the form where q and iy are polynomials;wlhrcti ''

now vco(s) = 2g+lAodd, vo,(flt)) is even, hence v,(std) # so

0 = v,,, (h) = v. (fl (pi))/ > v,,. (sip) - 2g = 1+v (V) > 1,

which is a contradictionunless (t) = 0, i.e., h is a function of t only, hot = h; this

implies that the poles of h are bounded by I1Pi also, thus h cannot

have poles, i.e., is a constant.

Definition 2.2. 0 = [subset of Jac C of divisor classes of the

g-lform P.-(g-l)°' ].

i=1 1

Steps I and II imply that a suitable restriction of the map I is

injective:

3.31

divisors Pi such thati=l

z " Jac C - 0Pi co if i `j

SymmgC

res I

->a Jac C

in OL4

,by Step II, I(D) = I(D') for-D E Z implies D = D' because a function

such that D'-D = (h) would have poles only on D = 7 P., hence bei=l i

a constant; in particular z n 1-10 = ¢ since 0 is the image of,

g-lPi + CO. Now if we represent any divisor class in Jac C--S as

Pi-g-- by Step I, then P. is in Z, because if P _ ori=1 1

Pi = 1Pi, i.e., Pi + Pi = 2--, then ( Pi-q--) E S.1

By the previous section, Z is a smooth g-dimensional variety;

by translation, we will cover Jac C by affine pieces isomorphic to Z.

Step III. Recall that B c C is the set of branch points P,

defined by P = iP: thus 2P = L, for all P E B.

Definition 2.3. Let T c B be a subset of even cardinality;

define

eT = ( P)( LT-) EJac CPET

3.32

Lemma 2.4. a) 2eT = 0

b) eT1+eT2 = eT16T2 where T1oT2=(T1UT2)-(T1fT2),

(symmetric. difference)

c) e = e, if and only if T = T2T1 121 or T1 = CT2

,,

the complement of T2 in

Thus,the set of the eT's forms a group isomorphic to (a/2Z2)2'.

Proof. a) 2eT = 12P - (#T)L E 0.PET

#T1+#T2b)

eTl+eT2 PET P + P

_(

2the P's that

1PET

2

occur once in I P + I P are those in T1OT2; the others can bePET1 PET2

cancelled against L's because 2P = L, and the multiple k of L in

e +e is determined by deg( P) = 2k.T T1 2 PET1oT2

c) eT + eCT = eg = 1 P - (g+1)L: the function s hasPEB

a simple zero at each of the branch points except' , and

-- =)- (2g+1)' eB.(S)CO = (2g+l) - -, so 0 = (s)cEB

To prove the converse, it is enough to check that if T Sb or B

then eT # 0. By replacing if necessary T by CT, we may assume

#T < g+l and, in the case #T = g+1, - is in T. eT = 0 means

I P = 1 L = Therefore there must be a function h withPET

(h)w _ I P; by putting - on the right if it occurs, we bound thePET

poles of h with at most g distinct branch points, none of which is

since two distinct such P's cannot be conjugate, by step II f must

be a constant and T = 95.

3.33

Lemma 2.5.* U (Jac c - 0) + era = Jac CT

or (1 (e+eT)T

rProof. Write any D = as = Q.-r= with

i=1 1 i=1

Q Qi IQ (by replacing P+1P with 2- if it occurs).

Now choose g-r branch points distinct from the Qi's

and Then

D + (g--r) co) _ IQi + IRi -goo E [Jac C-0]

(because it's the image of a point in Z). If g-r is even, then

D E (Jac C-0) + e{R ,,,,R } if g-r is odd1, g-r

D E (Jac C-0) + eJR ,,..,R}.

1 g-r"00QED

So, we take one copy of Z for each T, and we glue them together

according to their identification as subsets of the Jacobian; we have

to see that this glueing satisfies the conditions to give the atlas

of a variety.

*Here - in Jac C-0 is a difference of sets, but + in (Jac C-0)+eTmeans translation of a set by a point using the group law on Jac C.

3.34

Step IV.

Lemma 2.6. Given any {Bi}iET,

1 {Bi}iET2c B, with T1,T2

of even cardinality, let rT .T c ZxZ be the set of pairs1 2

P + e = + ei=1 1 T1

i1Q.

T2 i

Then rT,T

is Zariski closed and ro'ects isomorphically to1 2

Zariski open subsets of each factor.

Proof.) Rewrite the definition of I'T

JPi' YQi

1r T

as2

Pi + tiQi + I Bi + I Bi =-

i=1 i=1 iET1 iE T2

Consider the vector space V of functions whose poles are bounded by

N = 2g+#T1+#T2; as we saw .before (lemma 1.1)

polynomials in t polynomials in t

g-1]V

of degree < I N+ s

of degree < {N_2_l])

is

Say f1, ... , fM a basis of V, where M = N-g+l.

Among these functions, those which have zeroes at

I Pi + I tiQi + I Bi + I B. for fixed Pi, lQi, are just 3)i=1 i=l iET1 iET2

the elements of the ideal in 34ve. 6J ftie troducf

use the fact that a(s,t]/(s2-f(t)) is a Dedekind domain (91, Note 1)plus Corollary 0.14.

3.35

(*) IEP.,EQ.`u(1) (t),s-V(1) (t)). (U(2) (t),s+v(2) (t)JO

e 1T (s,t-a.) iT (s,t-a.iETI iET2

where the divisor P. <---> (U

and (U(2) ,V(2) ,W(2))

e

Note 1Lntif h E V n IEPa.EQi then ( Ipi, IQi) E rT T since h has1' 2

exactly N zeroes, and poles only at - .

Note also that membership in I imposes N linear conditions on a

function, so codim I = 2g + #T1+ #T2 = N, independent ofIpi' 'Qi-

Let RZ =(affine ring of Z)- (E[U.,V.,Wk}/(aa). Then we get a"universal" I

I c R(1) ® RZ(2) [s,t(s2-f (t) )

defined by the same formula (*) with U.1),V(1) E R(1), U(2),V(2)E R(2)1 1 Z 1 1 Z

being variables. Consider

A = R(1) ® R(2) '[s,t)/(s2-f (t))+I;

A is an algebra over RZ1)®R Z2), finitely generated and integrally

dependent (I contains,a monic polynomial in t) such that for all

homomorphisms RZ1)® R(2) -> T (evaluation of the coordinates) it

3.36

becomes a c-vector space of fixed dimension N. It follows`) that A

is "locally free", i.e.,

3ha,ga E RU) ® R(2) such that 1 =jhaga

and Va 3 e(a),...,e(a) basis of A as (R(l)®R(2) --module.1 N ha Z Z ha

Now let the map

be given by

V >A

fi

14 rk (c ( ) <N--g+lu defines rT

Tin the open set ha # 0 of Z X Z1,2

4)This follows from theProposition. If R is the affine ring of an affine variety, S a

finitely generated R-module, and

dime S ®R R/,.

is constant as fl varies among the maximal ideals of R, then S isa locally free R-module.

Proof: If m is a maximal ideal in R, let E S

a basis for the vector space S ®R RM /MR m = S ®R R/tn; byNakayama's lemma, e11 .. ,eN generate S. as R. -module; we claim

they are a free set of generators. Since S is finite we can express

its generators as combinations i)ei,

f it follows that generate 1 , for any

max. ideal 11. of R which doesn't contain f. Now if there were a

relation among the e1,...,eN, jXiei = 0, Xi E R,tt,F and say X 31 0,

let's express the X.'s as g'/f', g',f'E R. Then g'Em., sinceI 1 1 Ie1 , ... ,eN is a basis for S./MS.. There is a maximal ideal IS suchthat glff' !Z f, since R is the ring of an affine variety. But if

g1ff''f then al is not zero in R/t. R and dimcc St 0R R/4R < N,

nwhich contradicts the assumption. (See Hartshorne, Algebraic

Geometry, Ex. 5.8, p. 125.)

3.37

(such open sets cover Z x Z because Jhaga = 1). This proves that

rT T is Zariski-closed.l,

2

Note: the rank of (ci7a)) is never less than N-g since

2 functions in the kernel of V -o A must be linearly dependent,

having the same zeroes and poles. Therefore r is defined locally

by g equations. This follows from:

Proposition 2.7. Given a matrix

0

all (x) ... aln (x) al,n+l (x) .. alm(x)

N

and (x) ... ahnN) an,n+l (x) .. anm(x)

where aij(x) are polynomial functions on an affine variety, let

MIJ = det(ai.), #I = #J,iEI 3

jEJ

and suppose M (O) # 0; then 3 a Zariski

open neighborhood U of 0 such that for all x E UftM(x)-0or n < i < mu if and only if 'irk M (x) = n-l`.

Therefore all components of rT FT have dimension > g. But eachl 2

projection rT1,T2

---> Z is injective and dim Z = g. Therefore

rTT must be irreducible and of dimension g too. By Zariski's1' 2

Main Theorem (0.6), rT ,T is isomorphic to an open subset of Z

1 2

under each projection.

3.38

This proves that rT,,T

can be used to.glue the T1 and T2 copies1 2

of Z. This procedure therefore constructs Jac C as a variety;, we will

see later that it is projective. Note that it is complete, because

there is a surjective map Cg -> Jac C, and Cg is compact in the complex

topology as C is complete.

In fact, Jac C is an abelian variety:

Definition 2.8. An abelian variety X is a complete variety with

a commutative grou2 law such that addition X x X - X and inverse

X -b X are morphisms.

We know that Jac(C) is a complete variety and a commutative group.

To see that the group law is a morphism we need:

Lemma 2.9.

{JPi,JQi,JRi

For all T1, T2, T3 even sets of branch points

IPi+IQi+G Bi+ GBi = I Ri+1

Bi+(g+#T1+#T2-#T3)-}

1 1 T1 T2 i=1 T3

is Zariski closed in ZxZxZ, and projects isomorphically via p12 to Zx2.

(This can be proved in the same way as Lemma 2.6, so we omit the

details.)

Proposition 2.10. As a complex manifold, every abelian variety

X of dimension n is a complex torus Cn/L.

Proof: We use the Lie group structure of X. The exponential mapping

Cn = (Lie algebra of X) --- X is a homomorphism because X is commutative;

thus, being a diffeomorphism in a neighborhood of the identity, it is open,

and since the image is connected exp is surjective. Again by bijectivity

in a neighborhood of 0, the kernel is a discrete subgroup of Cn, and

the only discrete subgroups L such thatTn/L

is compact are lattices.

QED.

3.39

In fact, we already showed in Chapter lI that Jac C was

a complex torus (Abel's theorem 11.2.5). we have thus a 2nd

proof of this based on the chain of reasoning:

Jac C is a complete variety --> Jac C is an abelianvariety -> Jac C is a complex torus.

Corollary 2.11. Every 2-torsion element of Jac C is of the0

form eT, some T c B.

Proof: The 2-torsion subgroup of the abstract group

Tg/L = Jft2g/ZZ2g is 2L/L, which is isomorphic to (72/ZZ) 2g. But

it contains the group of the eT's, which has the same order, sothey coincide. QED.

3.40

§3. The translation-invariant vector fields

Let X be a variety. Then a vector field D on X is given

equivalently by:

a) a family of tangent vectors D(x) E TX x, all x E X

such that in local charts

nX a U c CC

a

rd.D (x) _ ai (x). a/aXi, ai E (C [X1, ... ,Xn ] .

b) a derivation D: O -> (.

In fact, given D(x), f E r(u,OX), define Of by

Df (x) = D (x) (f) .

When X is an abelian variety, then translations on X define

isomorphisms

TX'O TX'x

for all x E X (0 = identity), so we may speak of translation-

invariant vector fields. It is easy to see that for all D(O)E TX'O,

there is a unique translation-invariant vector field with this value

at O. In general, the vector fields on X form a Lie algebra under

commutators:

[D1,D2](f) = D1D2f-D2D1f.

For translation-invariant vector fields, the commutativity of X

implies that bracket is zero (see Abelian Varieties, D. "Mumford,

Oxford Univ. Press, p. 100,

3.41

The purpose of this section is to give explicit formulas for

the invariant vector fields in the chart Z in Jac C. Our method is

this. Let P E C and choose a non-zero 6P E TC P. Let

E > P(e) E C be analytic coordinates in a small neighborhood

of P with P(O) = P, so that 6P is the image of the unit tangent vector

a/ae at 0 in this coordinate. Then we get

DP (O) E TJac C,O

defined as the image of a/ae for the map

c-disc -----> Jac CS 1 [divisor class P(0)-P(&)]

at e = 0, i.e., the tangent vector to this little analytic curve

in Jac C at 0. Note that dp and DP(O) are determined by P only

up to a scalar.

Starting with any divisor D = ( Pi - g-=) E Z, leti=l

D - P(e) + P Pi(e) g--

and let

lPi (e) - _ (Ue (t) ,Ve (t) ,We (t)) .

Since Z is open, choosing IcI small enough, we can suppose

(D - P(e) + P) E Z.

3.42

Then

(due

dE

dVsC=O

'de

dW

E=Ode

/ E=0 ) E TZD TJac C,D

and this represents the translate of-DP(O) toTyac C ,D'

Note that

for this to be an invariant vector field, it is possible to use

different uniformizations a ti--. P(e) for each D, so long as the

tangent vector SP to this ma is independent of D.

The result is:

Theorem 3.1. For any P E C, P 7( -, for suitable SP the

above tangent vector is given at (U,V,W) E Z by

U(t) V(t(P)) -U(t)--U(t(P)) V(t)t - t(P)

V(t) U(t(P))).W(t(Pw(t(P)) U (t) _ U(t(P))'U(t)

W(t) = W(t(P))V(t)-V(t(P))-W(t)t t (P) L*(t(P))'V(t).-

Note. Equivalently, this means we have a derivation

DP: cr[Ui,Vj,Wk]/(a(,)O given by

DP(Ui) = [coeff. of tg-i in V (t (P)) U (t) U (t (P)) -V (t)

DP(Vi), DP(Wi) = (coeff. of tg-i in the other expressions) .

3.43

Note. Corresponding to P = w , we get the vector field

U (t) = V (t)

V (t) =z!I-W(t) + (t-U1+w0) U (t) ]

W (t) = - (t-u1+w0) v (t) ,

obtained by letting t(P) go o - , and replacing 6P in the Theorem

by dP/t (P) g-l. To c keck 4 L ve Calculkf e :

lim U (t)1 = lim[_t(P)Lv(t)+lowerordertennsin t (P)]

= V (t)P->-t(P)te (-t(P)+'t) t(P)g-1

lower terms}

lim V (t) = lim t (P) gw(t)-(t (P)1+Wt (P )g) U (t)+ in t (P) /-(t (P))U (t)

P->00 t (p)g-1 2 (-t (P) +t) t (P) 9-1

lower termslim 2 t(P)q(t)-w0 t(P)gU(t)+Ult(P)gU(t)-t(P)gtu(t)+ in t(P)

2-t(P)g + lower order terms in t(P)

W (t) + (t-U+W) U (t1D Jl)lower termsJ)

W(t) (t(?)9+U1t(P)9 )V(t)+ in t(P)lim = lim

P->.t(P)T-1 (-t(P)+t)t(P)g-1

= -(t-U 1+W0).V(t).

Note. DP(O) E TJacC,O will depend on P and on the chosen

uniformization; as P varies, we should only have g independent

vector fields. To see this, it suffices to expand the above

expressions in powers of t(P). As before let:

3.44

then

u (t) _

V (t) =

gI

i=OUi

IVi tg-ii0

W (t) _ Wtg-i.

i=-1 1

U0 = 1

V0 = 0

W-1 = 1;

t (pp-y-j-t (P) g-jtg-iU (t) =i,j=0

V1Uj t - t (P)

V. U t (P) g-jtg-j t (P) -i-tJ-a +

tlV0<.<3<g

1 3 t-t(P) g>i>j>0 i0<. t t(PP)']

so g-1'g-j so g-i<g-j

1(iUj V3Ui) t (P) g-itg-i /ti-j-1 + .... + t(P)3_)i >j ll

t (P) gktg-R,

k+R;-.A.+j+11 < j+1<k,R, <i<g

_ I t(P)g-k[ tg-P,

I (VU. -VU. )k=1

1..=1i+j= (k+i) -1

iix (k, R,)3 <min (k, !,) -1

3.45

2V(t) _ U.W.t(P)g-itg-j-t(P)g-itg-i

- U.U.t(P)g-1tg JO<i<g 1 7 t-t(P) O<i, j<g I J

I (U1Wj-UjWi) t (P) g-ltg-i (ti -j -1+.:.+ t (P) i-i -EUiUjt (P)g`at9 7

7

I t(P)g-k 1 I tg-k( I (UiW. i-Ui Wi U U >k=1 k=1 i+j=k+k-1 k ki>max (k, k)

j <min (k, k) -1

-60 rVy%

(k and k are allowed from 0 to g, but for k = O,k = 0

j(UWj-UjWi)-UkUk = UW-1-Uk, =UkW_i-Uk(resp.) = 0).

t(P)g-ltg-,-t(P)g 3tg-1 g-i g-jW(t) _ 1Wivj t - t(P) + IUiVjt(P) t

_ 1t(P)g-k

1tg-k

(WiVj-WjVi) + UkUkk=1 k=0 i+j=k+k-1

i>max (k, k)j <min (k, k) -1

(k is allowed to be zero but for k = 0 7 I(WiVj-WjVi)+UkV = 0,

while for k = 0 we get - I t(P)g-ktgVk).k=1

3.46

So if

DkUZ

_ (V.U --V.U )i

j <min (k, Q) -l

i+j=k+i-li>max(k,k)

3 3 ].

DkVQ = ZC (UiW.-UjW.) -UkURJsame

as above

DkWQ I (WiV.-W.Vi) + UkVQsameas above

. then we find DP = I t(P)q-kDk.IProof of the theorem 3.1. For the proof, we also assume

P 0 Supp P1., and that neither P nor any P is a branch point.i=l a.

The result will follow by continuity for all P and IPi. Let

Pi correspond to (U,V,W) as usual and note that as no Pi is ai=l

branch point, U,V have no common zeroes.

We consider the function

q (s, t) = U (P).(s+V (t)) +U (t).(s (P) -V (P) )U (t), (t--t (P) )

the denominator is zero at Pi + Iti(Pi)+P+r,(P), but the numerator

is zero at L(P1.)+ t(P) so q has poles at P. and at P. Its

1 i=1 1principal 2s (P)part at P is

t-t(P))independent of IPi. At infinity,

q is like so q has a zero at

3.47

So the equation q(s,t)-i

or U (t) (t-t (P) i [U (P) (s+V (t))+U (t) (s (P) -V (P)) ]

has solutions( Pi(e) near Pi=1

P(E) near P,

and f Pi (E)+P (e) ] - [ Pi+P] = I q - t E 0111 Ll

Unfortunately, this analytic family eI--a P(E) of points near P

depends on the choice of U,V,W, hence on the divisor IPi. But

since the principal part of q at P is independent of 1Pi, the

tangent vector SP to the family E F P(e) at E = 0 is independent

of "jPi. In fact, this gives

zu (P) s = U (t).(t-t (P)) -z

[U (P) V (t) -U (t)v (P)+U (t) s (P) ]

squaring both sides

U(P)2f(t) = U(t)2[t-t(P)]2-U(t).(t-t(P)).E[U(P)V(t)-U(t)V(P) +

2+ U(t)s(P)I+ 4 [U(P)2V(t)2+2U(P)V(t)U(t) (s (P)-V (P)) +U (Q (s (P) -V (p))

or (substituting f(t) = V(t)2+U(t)W(t) and dividing by U(t)(t-t(P)) )

U (t).(t--t (P)) -c [U (P) V (t) -U (t) V (P) +U (t) s (P) ] +

E2 {-u(P)2w(t)+2u(P)v(t) (s (P) -V (P))+U (t) (s (P) -V (P)) 2t - t (P) I

3.48

or O = [u(t) + g U(t)V(P)-U(P)V(t) + s2( )+...1[t-t(P)-es(P)+e2( )+...]t-t(P)L

JJJ

degree g in t degree 1 in t;

defines P(s)

Thus the 1st factor is U5(t), hence differentiating, we find:

U (t) = U (t) V (P) -U (P) V (t)t-t (P F

Now from the relation

f - (V+eV) 2 =- (U+&U) (W+sW) mod. E 2,

-2VV = UW + WU

UV(P) -U(P)V W + WUt -t(P)

we have

Therefore

or

O = V(2V - t(t(P)) U1 W + t-t`P) )

2V - U(P)W = - a(t)Ut-t P

W + V (P) W = + a (t) V where a is a rational functiont--t(P)

U (P) -W - a (t) U2t-t P) I

W = a(t)-V - V(P)Wt-t P)

3.49

Set a (t) = t-W (t P)(P) + a (t) . Then

U(P)W-W(P)UV = 1 - a(t)U2 t-t (P) 1

W= W (P) V-V (P) W

t-t (P) a (t)V

so aU, aV are polynomials in t; it follows that a is a polynomial

(since U,V are relatively prime), and since deg V < g and

V = [UJP)tg-a(t)tg + (lower order terms in t)], then a(t) = U(P).

If U,V have common zeroes, the formula follows by continuity.

QED

In fact, we have something more here than an expression for the

invariant vector fields on Jac C. Suppose we let the curve

s2 = f(t) vary too. We see that we have a morphism

C3g+1 AMspace of all polynomials U,V,W

s.t. U monic, deg. g

V deg.Eg-1W monic, deg. g+l

, coord. Ui,Vj,Wk

Tr

I2g+l . (space of polynomials f1 coord. as

s.t. f monic, deg 2g+1 f(t) t2g+1+ a1 2g+...+a2g+l

where r is defined by:

= V2 + UW.

3.50

The fibre of 7r over any f with distinct roots is the affine

piece Z of the Jacobian of s2 = f(t). Thus all Z's fit together

into a fibre system. The formulae above define vector fields Dk,

1 < k < g, on all of T3g+1, which are tangent to the subvarieties

7r-1(f) (and generate their tangent spaces at each point). Thus

E D F D = 0 (because the Jacobians are commutativel 2

groups, hence [Dk Dk

I is zero onDk(aa)

= 0 each Z) 1 2

To summarize we have found an explicit set.of.g commuting vector

fields on Tag+1, with 2.g+l polynomial invariants aa, and integral

manifolds Jac C - 0, where C varies over all hyperelliptic curves.

3.51

§4. Neumann's dynamical system.

In classical mechanics, one encounters the class of problems:

M = real 2n-dimensional manifold, with a closed non-degenerate

differential 2-form w

w = dual skew-symmetric form on TM

H = 00-function on M, called the Hamiltonian.

XV

H

the vector field on M defined by

w(XH,Y) = <Y,dH> for all vectors Y

or

<XH,a> for all 1-forms a

Recall that we define the Poisson bracket by:

{f,g} = +<Xf,dg> = -<Xg,df> = w(df,dg),

and that the compatibility condition

[XfFXgl X{f,g}

holds.

Moreover, {f,g} = 0 means that dg is perpendicular to Xf, or that g

is constant on the orbits of the integral flow of Xf. The main

problems of classical mechanics were all to integrate various vector

fields XH.

Unfortunately, it never happens except in trivial.cases that

there exist 2n-1 functions defining M 1R 2n-1 such that

tt-1(x) = orbits of XH. However, what does happen occasionally and

3.52

to date unpredictably is that there exists. aCco

-map h: M ---)U,

U open in ]Rn . (a) if Xi are the coordinates on ]Rn{Xioh, Xioh} = 0, (b) h is submersive, (c) h is proper, and

(d) H = foh, f a Coo function on U. In this case, if Hi = Xioh,

it follows from {Hi,H.} = 0 that the XH commute,

hence the fibres of h are

n-dimensional compact submanifolds whose connected components areYES

orbits of{xH

1,.. ,XH

n} hence are isomorphic to,tori: 1n/lattice).

It can be proven that near each one of these tori M has coordinates

xi determined mod 2Z, Hi = Hi(yl,...,yn)

independent of (xl , ... , xn) , w = Idxi A dyi and (xl r,'- , xn )

coordinates on the torus ]Rn/2,n; such (canonical) x,y are called

the action-angle variables.

But the orbits of H by itself are almost all dense 1-parameter

subgroups (as soon as the

ay

's are rationally independent, fork

instance, in action-angle coordinates); in this case the closure

of 5in1eorbit of XH is already an n-dimensional torus, and that's whyA

we cannot find any more rational continuous invariants for the flow,wQ CA"Kef f,hj

In particular 7: H > ]R2n-l which would give 2n-1 functions

constant on the orbit.

A Hamiltonian vector field XH with properties (a), (b) , (c), and (d) icalled a completely integrable system.

Given a completely integrable system, suppose M is the set of

real points on an algebraic variety and that w,Hi are rational

differentials and functions without poles on M. Then the Lori

3.53

Mc = 7r-1(c) are the real points on complex algebraic varieties

M. It may then happen, although this is a strong further

assumption, that the vector fields XH still have no poles on

a compactification of M. (Typically M c I and is the set

of real points of an affine variety MC, leaving plenty of room

for poles at infinity: e.g., take M = IR2, w = dx A dy,

H = x4+y4.) If this does happen, then for a suitable complex-

ification of the system, each Mc will be the group of real

points on an abelian variety (or a degenerate limit which is a

group formed as an extension of (C*)k by an abelian variety).

We call such systems algebraically completely- integrable. More

precisely:

Definition: (M2n,w,H) is an algebraically completely integrable

system if there exists a smooth algebraic variety R, a co-symplectic

structure w on Il, i.e. , w E A2TR. and a morphism

h : in ---B U

U a Zariski open subset of ci , all defined over the real field,

such that

a) {Xi o h, X. o h} (d(X3 oh),d(Xi oh) ) = 0

b) h is submersive

c) h is proper

d) M is a component of am, the w on M is the w on

RI along M, and H is a Cco-function of Xi ohIM,

3.54

In such a situation, it is easy to prove that the fibres

of h are abelian varieties or extensions of these by (T*)k.

These remarkable cases give us methods of describing families

of abelian varieties by dynamical systems.

Neumann discovered a remarkable example of this in:

C. Neumann, De problemate quodam mechanico, quod ad primam

integralium ultraellipticorum classem revocatur, Crelle, 56 (1859).

To describe this we start with n particles in simple harmonic

oscillation, whose position is given by x1,...,xn. The equations

of motion are

-a.x.1 1 1

or equivalently a system of 1st order differential equations

xi = yi

yi = -aixi.

We assume that a1 < a2 < < an, and we want to

constrain the position to lie on the sphere jx2 = 1; then

Jxiyi = 0, too, and the equation of motion is given by

adding a force normal to Sn-l that keeps the particle on

Sn-1.

3.55

We get

(4.1)xi yi

2yi = -aixi xi (lakxk-- yk)

(In fact,2 `these imply: (Zxkyk) = lyk + :xk'(.-akxk+xk(yaQxP. yk))

2ayk

x2+ (jxk),( aixQ- YZ

which is zero if jxk = 1. Hence if we start with a point such thatIxk = 1, Ixkyk = 0, these will continue to hold if we integrate

these equations.)

Let T(Sn-1) _ (locus of points s.t. Ixk = 1, Ixkyk = 0), i.e., the

tangent bundle to Sn-1.

(4.1) gives a vector field on T(Sn-l)s

D = lyk axk jak xk ayk + ayk .

If we put a symplectic structure on T(Sn-1) in the usual way by

restriction of the 2-form jdxiA dyi, then (4.1) is Hamiltonian

with

(4.2)1H =2

{jakx 2k + ly2k} (= potential + kinetic energy).

To check this, it is convenient to develop formulae which

express Hamiltonian flows on symplectic submanifolds in general.

Thus say

3.56

M C IR 2n

is defined by f = g = 0. Let w = Jdxindyi define a symplectic

structure on IR2n

and let resMw define one on M. We assume resMw

is non-degenerate, so w gives us a splitting

_ TM ® TM

TM is generated by the vector fields Xf,Xg, and for all functions

h on )R2n the vector field Xres(h) gotten from the Hamiltonian

structure on M is the projection of Xh to TM:

w(XhrXf) w(xh,xXres (h) = xh _ w (x ,X ) }Xg \w (X f,Xg

g f

where, as usual, on ]R 2n

Xh = v ah a _ v ah ayi axi axi ayi

Now, consider the special case f = (Ix2 - 1), g = 1xiyi., Then

X = -2 Ix a, xif ayi g

a a

i axi - yi ayi

w(Xf,Xg) = 2 Ix2 = 2

hence

3.57

Xah - x ( x. ah a

resW ayk k ayi ] axk

ah r ah ah 1 / ah 1 1 a+ k _ axk + xk xi ax yi ayi +yk ayi/ ayk

Substituting Z(Jakx2+Iyk) for h, we get (4.1.).

Following Moser, we can link these equations with Jacobians as

follows: let n = g+l; define a map:

7r: T(Sg)

by (x,y) t

a (c3g+l

(Ux,Y"Vx,Y'Wx,Y)

where we let

nf1 (t) _ (t-ai)

i=l

2

Ux,y(t) = t-amonic, deg=n-1 = g

k k

Vx,Y (t) fl (t). I Xkyk deg < n-2 = g-1k k

2YWx,y (t) = fl W. t_k + 1) monic, deg=n = g+l

k k

and the coefficients of Ux"y,VX'y,Wx,y are taken as the coordinates

3g+1in a:

3.58

Then

222

x_ {kr

kyR

U xrYwxrY+Vx,Y fl(t) l,Q t-ak t-aQ)

xk2

xkYkXP,

t-ak - k-ak t-aR

22+x22 2

f (t)2xkYiQyk-2xkxtykyi + xk

1 k<Qt-ak)(t-aQ) k t-ak

2(xkY -x

yk= f1 (t) L k<lQ (t-ak t-aQ)

because the second factor has only simple poles atak

(with

Q

J"singular part" 1 [x2 + (x kyz-x k )2

/ and is 0 at o0t--ak k Qk ak-aQ

we can re-expand by partial fractions

2 2 (xkYQ-xQf1(t)

fkt-ak Lxk + Jk ak-as

If we set F = x2 + I (xkyz-xQyk)2k k

ilk ak-ak

then

and finally:

f2(t) =

k

f (t-ai)Fk is manic, of degree n-l,

2

Ux,y x,y + vx,Y fl, f2

so that x,y defines a point of the affine point of the Jacobian of the

algebraic curve s2 = embedded in T3g+l by the method of §2!

3.59

The map 7: (x,y) (U,V,W) extends to a map wT on the

complexification T(Sg)T of T(Sg), i.e., the complex variety given by

equations Irk = 1, lxkyk = 0, and the image if contained in the set

of complex polynomials tJ,V,W such that f1IV2+UW; or equivalently the

set of affine parts of the Jacobians of the curves s2 = f(t) for which

filf. The situation is summarized in the diagram (4.4) below.

Lemma 4.3: 7r(C is surjective; TrCC (x, Y) = rr(C (x' ,y') if and only if(.x',y') is the image of (x,y) under one of the transformations

(xk'yk) (Ekxk'Ekyk), sk +1, which form a group of order 2g+1

hence 7rT,

is unramified outside the subvariety of the (U,V,W) such that

U(ak) = V(ak) = W(ak) = 0 for some k.

Proof: Given f with the property f11f, and polynomials (U,V,W)

such that f = UW+V2, then we make partial fraction expansions

u _ I Xkff 1 k t-alt

V ukf

kt-ak

w Ivk

+ 1f1 k t-ak

it follows that Irk = l because U Monic, and it follows that

uk = 0 because deg V < g-1, and it follows that XV i = p1 becausei

ii2

at each ai, UW+V2 has a zero, hence UWZV has a simple pole. Now wefl

can solve for (xi,yi) E T (Sg)T

Xi _ xi2

2vi = yi4i = xiyi, uniquely up to a single sign

for each i. QED

3.60

(4.4)

3g+1

U Zariski-closed

cx.dim 2g

UT (Sg)

real dim. 2g

7r

union of the Jatobianf-V2=U W,all f such that f 1!

U real dim. 2gsubset where f,U have

res RCreal roots as below, Wis real and V pure imaginary

g+l g . g

f1(t) _ 1T(t-ai), f2(t) = (t-bi), U(t) = fl (t-ci)i=l i=1 i=l

Lemma 4.5: If (U,V,W) satisfy f1IUW+V2, and (U,V,W)

then x and y are real if and only if U,W are real, V is pure

imaginary and f(t),U(t) have real roots separated as in (4.4).

Proof: If x and y are real, then

u (t) ._ I T (t-ai) xk2

k ifkis a real polynomial and sign U(ak) _ (-1)g-k+l, so U must have a

zero in each of the intervals (ak,ak+l),k < g. Also U is monic

cx.dim. 2g

3.61

so U(t) > 0 for t >> 0. Thus U(t) has signs like this:

0 + 0 - p +

at C, a7- c1 A5 5 a ft l

Next, f-V2 = U-W and V(a) is pure imaginary, all a E ]R, hence

f(t) is negative at all zeroes of U(t), hence f2(t) is alternately

+ and - at these zeroes. Thus all zeroes of f(t) are real with one

zero of f1 and one zero of f2 in each interval (-co,cl) , (Cl,c2) ,---,(c c9-1! g

as shown in (4.4). (In all of this we have assumed the zeroes of f

are distinct, but limiting cases ai = bi and bi = bi+1 are possible.)

Conversely, if the zeroes of U and f are real, and interweave like

in the diagram, then in the partial fraction expansion above

ak > 0 and Riuk is imaginary, so the equations

x 2

2

have real solution (x,y). QED

If we fix f with real zeroes, the curve s2 =f{¢)is a double covering

of the t-line; it has a real structure given by coordinates2 g+l g

S, t) . Since sI = - iT (t--ai) TI (t-bi), the real pointsi= l i=l

3.62

on C are givens by :

f<0 f > 0 f>O . - - - - - f>0 f>0

By complex conjugation (s;,t) t----- (Set), we get an antiholomorphicinvolution

Jac C - > Jac C

(Jac C)IR is defined as the set of fixed points of this involution:

it is a subgroup of Jac C which must consist in a g-dimensional real subtorus plus

a finite number of cosets. Since oo is fixed under this involution, we can determine

the real points in Jac C-0 as follows:

` 0 = { pi -- g- I Pi oo,P lpj , if i j, and Pi = JPiJ.(Jac C)3R 3'i=1

Note that Pi = Ipi means that Ipi consists in some real points and, some

pairs of conjugate complex points. If (Io,---,lg) are the intervals in

IR where f < 0 (see diagram (4.6)), then any subset S of {0,1,---,g}

whose cardanality is <g and S g mod 2 defines a connected

component KS of (Jac. C) IR - 0, namely the set of the

2-sheeted 1covering map JJ

C---> JP

V

f<0

3.63

divisor classes

ICs

Pi-goo

i=1such that, if S = fill ... is),

A) t (Pk) E Ii'`

01

ah

Ps+l ps+2

Clnd

's+3 P s + 4

5o Pk = Pk , far 1 .

k. - S

or t(Ps+l),t(Ps+2) E (same TP,1

)

or t (Ps+3) I t (Ps+4) E (same IQ2

) .

etc.

Example:

10

21 12

the real (affine) part of the Jacobian breaks up into 4 components;2 2

in order for P to be equal to P., the only possibilities are:i=1 i=1 1i

K{01) P1 over 20, P2 over I1

P1 over T0, P2 over 12

K{tP1 over T1, P2 over 22

K (Pl,P2 over the same 1K) or (P2

Note that this last set K0 is connected because we can continuously

move P+P on the complex curve C until P = P Elk and then move the two

points independently on the real loop over Ik; KO also contains t-

in the limit, hence it's the connected component of the origin.

Thus we verify that when g = 2,

(Jac C), is a real 2-dimensional closed subgroup,

isomorphic to 3R /lattice x (7L/2?Z).2 2

3.64

Returning to Neumann's dynamical system, the following theorem

will show that the functions Fk(x,y) are integrals of the vector field XH give

ahaw fl.acommuting flows on T(Sg) T and their image under v is tangent to all

Jacobians, and gives the translation-invariant flows on them.

Theorem 4.7 (Moser-Uhlenbeck*). On T(Sg)

{Fk,FQ} = 0

l1 -j

I akFk = Hb) kElFk =xk2

= 1,2 k=l

c) Tr* (XF ckDa ck = 4/ T fl (ak-aZ)k k

Tr* (XH) = CD., c = -2v "-- T

d) the XF span a g-dimensional s ]Race, except over thek

Zariski closed subset of triples (U,V,W) which have a common root.

Proof of a).

First one checks that on (C 2g+2, with coordinates xi,yi, {Fk

with respect to the symplectic form I dx1Adyi = w; if we let

zkZ _ xkyZ-ykxZ, then

azkfZ azi{zkP'zij} - oaxa

ayaaykZ Ox

13 hence

{zkV'zij} = 0 if {k,2.} !1 {i,j} =

{zkZ,'zkj}=-YQ(_xj)+(-xz)yj

=-ZQj

=0

*J. Moser, Various aspects of hamiltonian systems, C.I.M.E. conferencetalks, Bressone, 1978; K. Uhlenbeck, Equivariant harmonic maps intospheres, Proc. Tulane Conf. on Harmonic Maps.

3.65

Thus if2

k.

J kak-a. and k 34 k

{ 1 1 2 2 1 1zj#k ak-aj aCa {zkj' ki} = 4 3fk ak-a. ak-ai zkjzki{zkj'zki}i `k ilk

- q 7 1 1

r z z zk

zkk zkjzk j

jk ak-aj ak-ai I zkk zk i zkikj kk jk

1 1=-4 zkki;k ak-ai aCa

izki zki +

or k

+zki zki + 4z 1 zk7,.,,k4 zk. akak i L ak -ai k k ak -ak a -a .jqdk k

;..4z

1kk ilk (ak-ai} ak -ai) - (ak-ak) (ak-ai -(aak) (ak ior k

(ak-ak) - (ak-ai) + (ak-ai)(ak-ai) (ak-ai (ak -ak) zkizki

Finally

0

{Fk,FL} _ {xk2+qk,

{xk2,x2k} + {xk2,Ok} + {Ok,xk2}

=-2xk ay 4. + 2xk aykk

3.66

= 2x2(xkyi-xiyk)(-xi)

- 2x2(xiyk-xkyi)(-xk)

k ak-ai lC aQ-a.k

= 0 .

To conclude use the following

Lemma 4.8: Let M2 =,N2n-2be symplectic with basis 2-forms w, res w; let

f,g,h be functions on M such that df is nowhere zero and f = 0 on N.

Then if the Poisson brackets on M satisfy {f,g}M = {g,h}M = 0, the

Poisson bracket on N satisfies {g,h}N = 0.

Proof: Via w, we can split the cotangent bundle to M into the

cotangent bundle to N and its orthogonal complement

TM* _ TN* a C*

Now write dg = d(res g) + a

dh = d(res h) + Sdf = y 76 0 a,a.Y e C* .

From w((X,y) = w(S,Y) = 0 it follows that a = by are

linearly dependent, because C* is 2-dimensional, so w(a,s) = 0,

so 0 = w(dg,dh) = w(d res g, d res h) = {res g, res h}.

2(xkyR-xQyk)2

Proof of b). Fk = xk + a -ak,k k Z

kQ

QED

2 2--!-+ 1_ Xk + kZk(xkyi-xiyk)

(ak-at ai-ak

= 1

3.67

and21 j1 2 k k k k

2 akFk 2 akxk +2 ak ak-ak

k?`k

a a2

2

akxk + 1 kk

(ak-k ak + at-ak) (xkyk-xkyk) 2

_ akxkk +k 2

(xkyk-xkyk)2

akxk + 412 E xk'1 yk - 2(Q xkyk)2lLL k k k

X2 + 2 Yyk= H.

Proof of c): Under the Fk flow on T(Sg)(,, by the formulae for

Hamiltonian flows on a submanifold:

arkpp

aFk

ayk- xk 1 p xp ayP)

Since

xaFk x 2(xkyp--xpyk)xk +

x 2(xky2-xpyk)(`xp) - 0p p ayP P39k

P ak-apk

P54k ak-ap

then

k a -a kx =2(xkyk-xkyk)

--- xk k

or 2(xkyp-x pyk )_ (-xp) if k = k.

p34k ak-a p

3.68

Under the action of it, this vector field becomes

x x (x y ""x X )k 9 k Q Q k2Q = (xk) = 2xz P, 4 ak-ak

4xkuQ-x Z"k

ak-aL if

-4xpxk(xkYp-xpYk)p#k ak-apor I if R=k

= aku xpuk-4pfk akap

So U (t) = f1(t) I t

f (t)[4Xkpk-XZuk 1 _ 4 1

akup-Apukl

X #k ak--a t-axp 34k

ak-ap t-ak

SOP.-xkukl= i k (t-ai)(t-ak J

Xk uQ Ilk xQ4 f1(t) t-ak k t-az t-ak R t-a2

I AkV(t)/vT-UkU(t)= - 4 t ak

ButU (t) = I 1 r (t-ak) Ak

k £ k

V(t) = I iT (t--a2)ukk ,34k

3.69

so U (ak) _ (ak-aQ) . lk, V (ak) = 'T -fffGk Qk

V (ak) U (t) -U (ak) V (t)and finally U(t)

r_

t - ak 1-cr(ak-a

=I 4 - fl (ak-aR) -1)- Dak (U (t)) .kft

The argument at the end of the proof of theorem 3.1 shows that for

any vector X on the space of (U,V,W)`s which is tangent to the fibre

UW+VZ = f at a point where U,V have no common zeroes, the equality

X (U (t)) = c- U (t) V (P) -U (P) V (t) , c a constant, impliest-t(P)

X (V (t) ) = 2. IU (P)

t(t

(P)(P) U (t) _ U (P) U (t)

J

X(W(t) ) = c. IW(P)V(t)-V(P)W(t) _ U(P)V(t)t-t(P) .

So in order to finish the proof of c r, where the constant is

c = 4/T -'TT (ak-aQ)-1, we notice that by a) the collection of

vector fields XF are tangent to the loci (Fi = constant, all Q.),k

hence their images via r* are tangent to the fibres sitting over

f = fl Fk - l( (t_a,). Thus we get 7r*XF = cDa on the part ofk Zk k k

the fibre where U,V are relatively prime; the result holds everywhere

by continuity.

The formula for w*(XH) is proven similarly, the calculation

being much simpler.

*Alternatively, one may calculate 1 (t) ,CV' (t) directly as we did U W.

3.70

Proof of d). Assume that at a particular triple (U,V,W), there

are constants d such that d.D = 0. Theni i aV(a.)U(t)--U(a.)V(t)z 1 -d 0

i=1i t -- ai

or

or

(i=V(a ) U(a )

di t_a. )U (t) _ (1 1d V (t)

di V (ai) I f (t-a ] U (t) _ di U (ai) I[ (t-aj ) V (t)jai jai

V* U

since deg U = g, and deg U* < g-1, this implies that U and V have

a common root a where U*(a) 0 (or at least the multiplicity of a

as a root of u* is less than its multiplicity for U).

Likewise

or

g rU(ai)W(t)-W(ai)U(t)i=ldi L t _ a i

U(a )

(I d, t-ai) W(t) =

- U(ai)U(t) I = 0

W(a. )1

t-ai + 1 di U (ai) IU (t)

implies

3.71

I diU(ai)jITl(t-ai)+W(t) = {jd.W(ai) IT (t-ai )+Id U(ai)II(t-aiU(t)

I j54i 7JJ

U* W*

hence U(a) = 0, U*(a) 74 0 implies W(a) = 0 too. Thus U,V,W have a

common root.

Corollary .9: Almost all orbits of {XF } (defined byk

Fk = const., all k) are compact real tori, isomorphic to connected

components of the real points ona 29+1--order covering of the Jacobian

of a hyperelliptic curve.

The covering that occurs here will be described analytically in §5.

Finally, Moser discovered a beautiful link between the dynamical

system T(Sn-l),{Fk} and the problem of finding the geodesics on an

ellipsoid. The result is so elegant that we want to reproduce it here:x2

Theorem 4.10 (Moser). Let E be the ellipsoid k= 1 nak ..'

g+l if x,y EIRg+l satisfy j = 1, <x,y> = 0 then:

/the line LxrY {y+tx t E 7Rk(x,y) = 0) if and only if (

\ is tangent to E

and if this holds:

(Exkyk

ak= y - x = L n E.

C ak2x,Y

k

If x (t) 'Y (t)1

is an integral curve for the vector field a Fk'ak

then t(t) is a geodesic on E, up to reparametrization.

3.72

Proof: First we calculate out

L- l (x2 + (xkyR, -xkyk)

ak Fk(x,Y) _ ak

kt7k

ak-at

xk +1

1 (xkyQ-xQyk)2

ak 2>k ak aQ) ak-a91

2xk

- 1 (xkyRTxQyk)2ak 2 z,k akaZ2 2 2x xk YR

\I

xkyk

/ak k ak k a \ ak

a(1 _ I ak) + xayk! 2

k k k

or if B(u,v) is the bilinear form I ukvk/akr

Fk (x, Y) = B (X, X) (1 - B (y, y) ) + B (X, y) 2a

Call this function F. We calculate the flow associated to F.

Ya)

xk ay

__ B(x,x) I(- 2xk)ak + 2B(x,y) I xk(xk/ak)

k k k k

= -2B(x,x)B(x,y) + 2B(x,y)B(x,x) = 0 .

b) k xka3F =xk 2F because F is homogeneous in x of degree 2.

c) Likewise I yk ax picks out the quadratic y-terms in F, i.e.,k

aF =l-2B (x, x) B (Y,Y) + 2B (x,y)2,Ykayk

so

Xk 8x Yk aF 2B(x,x)k yk

3.73

d) The flow therefore is

Y. xxk = aE 2 B (x,x) + 2 B (x,y)

ykakk

ak

aF + x (2B(x,x))Yk axk k

Y- 2 ak(l - B(Y,Y)) -- 2 k B(x,Y) + 2 xk B(x,x).

k ak

Let E be the ellipsoid B(x,x) = 1. Note that B(y+tx,y+tx) has a

minimum at t = -B (x,y) /B (x,x) , i.e., at B (t, ') if = y - X(X,y) x.

But

B(E,) = B(y,Y) - B(x,Y)2B x,x

So Lxry is tangent to E if and only if B(,) = 1, which holds if

and only if l-B(y,y) + B(x,y)2/B(x,x) = 0, i.e., if and only if

F(x,y) = 0. Now differentiating along a flow line:

B (X, y) B (xrY)k k B(x,x)xk B(x,x ) xk

y) + 2 xk B(x,x)

+ 2 yk - 2 xk B( )2 _ /B(x0` xY) a x,x 1B(x,x)/ k

Now define a function T(t) by setting

T(W ' = 2B(x,x) - (B (X, y)yB (x,x)

3.74

along this flow line. Then

dt

or

Therefore

dTdt*xk

xk.d9kdT

d2

9k ddt Xk

dT dt xk

- B(x,Y)yk B(x,x) xk

lkdT/dt[ a

2B(x,x) . EkdT/dt ak

(-2 B(x,x))

This simply says that the acceleration of t(T)6 E is always

normal to the ellipsoid E, i.e., that 1(T) is a geodesic. QED

3.75

§5. Tying together the analytic Jacobian and algebraic Jacobian

So far in this Chapter, we have definetan algebraic variety

Jac C and studied its invariant flows. In Chapter II, we associated

to any compact Riemann Surface C a complex torus Jac C. If C is

hyperelliptic so that both constructions apply, they are isomorphic

by Abel's theorem. We would now like to make this isomorphism

explicitly, i.e., express the algebraic coordinates on Jac C-S as theta functions.

To study C as we did in Chapter II, the first thing we must do

is to choose a homology basis A±,Bi. There is a traditional way

to do this in the hyperelliptic case. One first chooses on

7P1 = (C U (co) , a simple closed curve P through the set of branch points B.One then chooses paths in 1P1-B as in the diagram below. Noting

that each of them circles an even number of branch points, these

paths can be lifted to the double cover C. see Figure on next page.

On C, the paths Ai are disjoint from each other as are the paths Bi1

and Ai,Bj meet only if i = j, and then in one point so that

i(AA) = i(Bi.Bj) = 0

i(Ai,Bj) = Sij.

Thus Ai,Bj are a symplectic basis of HI(C,2Z). To make this picture

clearly homeomorphic to the figure in Ch. II, §2, we can also add

disjoint tails to all Ai,Bi, connecting them to the base point P1.

Widening each tail into 4 parallel paths, we can lengthen Ai,Bi to

disjoint simple closed loops AiBi' all beginning and ending at P1,

which is exactly as in §11.2.

3.76

top layer

curves onbottom layer

crosscuts where(5.1) layers join

3.77

4,111

Next, on C we can describe the g-dimensional vector space of

holomorphic 1-forms:

Proposition 5.2: r(C,& ) consists in the 1-forms:

w = P (t) dts

P a polynomial of degree < c_1,

Proof: Because s2 = f(t), we have

2s ds =

so

P (t) dt = 2P (t) dss f' (t)

On Cl (the affine piece of C with coordinates s,t), s = 0 implies

f(t) = 0 which implies fl(t) 0 because f has no double roots.

Thus at every P E C1,either s(p) 34 0 or f'(t(P)) 74 0, so using

one of the above expressions it follows that w has no poles on C1.

Now at t' =

t

and s' =t}a

are coordinates. Then

3.73

ds - (g+l) s-dtds' =

tg+ltg+T

(f' (t) - +1) s2 dt

2.t g1 tg+2 ) s

tf' (t) - (251+2) f (t) . dt2tg+2 s

_t2g+1+(lower terms in t) dt

2tg+2s

Now we saw in §1 that s',t' have respectively a simple and a double

zero at hence s' is a local coordinate near - and ds' is a

1-form with neither zero nor pole at So the above equation

shows that

st = (-2(t') g-l + higher order terms in

i.e.,dt has a zero of order 2g-2 at Thus if deg P < g-1,s

P(t) has pole at - of order < 2g-2 and w has no poles at all.

Thus we have found a g-dimensional space of 1-forms without poles

and as dim F(C,21) = g, this must be all of them. (We could also

start with an arbitrary rational 1-form n = and show

directly that if n has no poles, then = 0,

(t) = (polyn. of deg < g-1)/f(t).)

The next step is to choose a normalized basis

QED

P. (t) dtw. _

s

3.79

of holomorphic 1-forms such that

A.I

The period matrix of the curve C is then

Wi

and the analytic Jacobian is by definition:

(lattice 2Zg + S2.7Z g) .

By means of the indefinite abelian integrals we have holomorphic maps

I

(P1, ,Pk)

> Og/LS2

P.

mod LP .

Abel's theorem (I1, 52) states that these induce an isomorphism

Jac C > Cg/Ln

k kif we map a divisor class Pi -- Qi to 1(P1,...,Pk) -

Taking k = g, we compare this with our algebraic

description of an affine piece of Jac C:

3.80

CgU

(Cg) o11

open set of

g )PiO°°,Pi01PJ

if i 0 .7 f

I a* CTg/LQ

Z c CCZg

11

variety of polyn. U,V

such that UIf_V2 )

We have seen that z is the open piece Jac C-0 of the Jacobian,g-l

where 0 = (locus of divisor classes P.-(g-l).=). Our goali=1 1

now is to prove:

Theorem 5.3:

1) There are E ZZ,g such that for all z E Ego,

lg ] (z. St) = 0P I...FP E C such that1 g-1 1

z W mod Lg-1

i,i-1

2) Thus Jac C-0 can be described analytically as

and algebraically as the above variety Z,

whose coordinates are the coefficients of U(t),V(t). Thus

the coefficients of U(t),V(t) are meromorphic functions on

CTg/LQ with poles where'' S = 0.

3.81

3) For all branch points akE B, there are r1' (k) (k) E 12Z9

and a constant ck such that for all divisors

D = p. in (Cg)., if UD(t) _9

(t-t(Pi)) is thei=l

corresponding polynomial, then

UD(ak) = ck.

P .

+q 9 4.

'LS +n \iI1 JAW)k CO

2

l (pi->.

jd°I \i l f )

CO IThis determines the coefficients of U(t) as meromor hic

functions on CCg/Ln.

In the course of proving this, we shall determine d,rl(k) explicitly.

In fact, ck can also be determined, but this will not be done until

the next section.

We, first prove (1). Note that (1) is exactly Corollary 3.6 of

Riemann's Theorem, (Ch. I1,§3), except that we assert that

E 2LQ and we want to compute Q too. (Also, we have used

the fact that z) = t4' (z) ) WeAdetermine 4 by arguingbackwards from some of the Corollaries of Riemann's theorem, or else

we can go back to the proof of 11.3.1 and work a little harder on

the integrals there. We shall do this although the reader should

be warned that the details are such that it is almost impossible

not to make mistakes of sign, orientation conventions, etc. The

result is that for hyperelliptic C,

3.82

(5.4)

where

I = S11' + " mod L

_ 1 1 ._. 1l 1 g8 2 2 2 E 2

2 ........ 1 2) E

Proof: Recall the expression for Z mod L2: let gk be the

indefinite integral of wk on C - UA! - UBi, normalized so that

gk(-) = 0 (we are extending Ai and Bi by "tails" to get a figure

homeomorphic to the one in §12.2). Then

P1kk

A k - 2 - f k +R, 1 f +gkwk

A

P1

(In the term f wk, the path should be taken in C-UA'-UB from com

to P considered as the beginning of B'.) Firstly, wQ = dgp,

1 f 2

Ak

d (gk) . This is

1Ig2(end of Ak,+)-g2(beginning of A + the

of the beginning of

at the end becauseJ

Wk = +1. So this term is

1t

Ak

3.83

f

P 1 2rPl

2

z[(J wk) - (1

Wk -- 1 ) I

00

Secondly, if k k, then

J

has the same value at the beginning

and endofA+

because wk = 0. So the contribution of the

tails onA'+

in

Ak

is zero and we may as well integrate around Ak.

on AQ by paths as follows, missing all Ai'P

gkis evaluated

-R = access road to A-,4

3.84

here we have chosen Ai exactly along the cut between a2X_1,a2Q,

so that it consists in a path az from a2Q to a2Q_1 and a return

along i(cL). Now i*Wz = -WQ , and t(ak) is traversed backwards:

so

J+gkWQ= J (gk + l*gk)WQ

AR aQ

But gk+l*gk is constant because d(gk+l*gk) = Wk+l*Wk = 0. Thus

J (gk+l*gk) WQ 2gk (a22, ) J WR

a21 aQ

gk (a22)

JU)k

pQ

(since

1

2 WkRoop pQ.i*pz]

= 2 J(since pQ-l* pi is homologous

(Al+... +At).-BZ to (A1+...+Ai) _BQ

0 if k > i+ {1/2 if k < i

3.85

Altogether, this shows

1

1

0 if g-k+l even

k '2 Q=1 k£ + .1/2 if g-k+l odd

which proves (5.4).

Part (2) is just a restatement of Part (1). Before proving (3),

we need to tie together the different descriptions we have introduced

for the 2-torsion in Pic C. In fact, in §2, we showed that

(Pic C)2

where

group of divisor classes eT

(T c B, #T even, mod eT = eCT

err = I P - #T.-

We also know

PET

(Pic C)2 = 2-torsion in Tg/L

The link between these is given by

P

eT I > I (eT) w

P T

3.86

We can calculate I(eT): th place.r

Lemma 5.6: a) I(e ) _ tt0,...,2, ...0){a 2i-I a 2il

'GC3l)mod I-"I"

2li

'. _

' 2...I (e{a2i a2i+l' , 2g+1}

b) I (e ){a2i-l"'

I (e {a2i , 00} )

1(2: 12'2

t 1 1 ..(2,2,

ith place

1...r2'0, 0)

ith p lace

2,

Ln

+ mod L2 2

Proof: The path Ai in the diagram (5.1) above may be moved so

that it follows p from a2i to a2i_1 on one sheet of C, and then

goes back on the other sheet:

C.

N

TI

3.87

But each Cok reverses its sign when you switch sheets. As the

direction in which Ai is traversed also changes:

w

a2i-lam a2if

= w -J

w

0* 00

z

e{a2i-l'a2i} mod LS2

a(2i(Note: 2.J w E L because I(2eT) = 1(0) = 0 E Cg/LQ.)

CO

The same argument with Bi shows

= I(e{a2i,...a2g+l

This proves (a). (b) follows because of

} ) mod Ln .

{al,a 2}o...o{a 2a.-3,a2i-2}o{a2i,...,a2g+1} = C{a 2i--1,Go}

and

{al,a2}o ..o{a2i-l'a2i}o{a2i,...,a2g+l} = C{a2i,o}

and lemma (2.4). QED

3.88

.th1Definition 5.7*: place

2 tn2i-1r = the 2xg matrix2i-1 0

2 2 n2i-1

ilace0...0 2 0...0

_ fl2in2i = the 2xg matrix ( ) ( )

1 if

2..2

2 0...0 2i

nT I 'nk for all T c BakET

a k#o

Then lemma 5.6 says that

(5.8) I (eT) = SZn + TIT

and the more precise version of part (3) of the theorem states that

UD (ak ) = ck CO

To prove this, note that both sides are meromorphic functions

on Jac C with poles only on the irreducible divisor 0 . Suppose we

prove that both sides are zero precisely on the translate of 0

* There is an unfortunate conflict here between conventions for row and column vectors.

In Ch. 2, {n was defined for n',n" columns of height g. In the 19th century,

W,n" were rows of length g (easier to write!). To make these compatible, we must

put a transpose in this definition.

3.89

ak

by J w and vanish to 2nd order there. It follows that the ratioCO

of the LHS and RHS is finite and non-zero on J'ac-0, hence it is

either 1) a constant, or 2) has a zero on 0 and no poles, or

3) has a pole on 0 and no zeroes. But using the fact that a

bounded analytic function on a compact analytic space is constant,

applied to the ratio or its inverse, we see that 2) and 3) are

impossible.

Consider therefore the zeroes. The RHS has a double zero*

on the translate of V(9[6]) by sink + ?Ik*

By our remarks above,

this is the translate of 0 by I (e,

)' i.e.,a

{k, }

k

f w. As for the LHS, as D =

CO

Pi'

UD(ak) = 0 Pi = ak for some i

( Pi)-ak (effective divisor of

1 degree g-1)

(divisor class Pi-ak- (g-l) -) E 0

P.ak

J

W E (translate of 0 by 1 w)

Note that if '& [S] vanished to some higher order r > 2 on 0, thiswould contradict Riemann's theorem: because for a general choice

of P1, ' - , pg E C, f (P) w + 1w) vanishes to first00 00

order at P = P1'...'Pg.

3.90

To check the order of vanishing, go back to the covering:

(Cg)00res ir

Z0

g-tuples Pl,---,Pg open set of Z of U,V

( s.t. P.P.,#t(p), (such that U has g1 distinct rootsif j O i,# o0

The group of permutations acts freely on (Cg)00,

so 7T is an

unramified covering map between g-dimensional complex manifolds,

i.e., they are locally biholornorphic. Now

i h place1 zeroes of g

(res rr } = U [Cx...x{a

UD (ak) i=l k

I gThe pull-back of the function UD (ak) is f (Pl, ... ,Pk) _ 11 (t (Pi) -ak) -

i=l

But the function t-ak on C vanishes to order 2 at the point s = 0,

t = ak, i.e., at the point we are calling ak. So f vanishes tog

order 2 on i,,;prll(ak) as required. QEDi=l

An interesting restatement of part (3) of the Theorem is

3.91

Corollary'5.9: The 2g+2-meromorphic functions

#[-Ik] (z) 2k E B, on Jac C span a vector space V of dimension

*[Olw )f

only g+1. In the projective space 1P(V), the individual functions

lie on a rational curve D of degree ,g and on this curve, give a

finite set projectively equivalent to B in IPI. In this way, we

can reconstruct the hyperelliptic curve C from Jac C and i9'

Proof: Part (3) says that, up to a translation in z, these

functions are D i--.. UD (ak) . But

UD(ak) _ I U?-ag-' , UDy = coefficientsi=0 of UD(t).

So the 2g+2 function UD(ak) are all constant linear combinations

of the g+1 functions D t----> UD (including UD which is theconstant function 1).

Taking these U? as a basis of V, the individual functions

UD(ak) have coordinates in V

(ak,ak-1,...,ak,l)

The rational curve D in the theorem is just the locus of points

in IP(VJ whose homogeneous coordinates in V are:

some b E C.

Thus b is a coordinate on D and the individual functions UD(ak.)

have coordinates b = ak. Thus the Corollary is just a geometric

restatement of Part (3). QED

3.92

In 93, we described algebraically the translation invariantvector fields on the variety Z of polynomials U,V,W such that

f-V2 = U-W. In analytic coordinates on (Cg, the translation

invariant vector fields are just Ic1 8z , c-, E. We can tie these

together too. The result is:

Pro position 5.10. Let Wi = (t)dt/s, (t) =

Then in the isomorphism

the vector field Da on Z corresponds to the vector field

- vi(a) az and the vector field D,, on Z corresponds to the

vector field - ei 8z1

Proof: Let D(E) _ Pi(s) represent an integral curvei=1

of the vector field Da Let ci(E) = t(Pi(e)),

U E(t) = IT

g(t-ci (E) ) = UD (E) (t), and Vs (t) = VD (E) (t), so that

i=l( P = V E(ci (E)) . Then

U F- E (t) = VE (a) UE (t) --UE (a) VE (t)C t , a

D(E)

The corresponding curve in (Cg-space isJ

w and we want to

prove

=

E=0

3.93

Letting ci(0) = ci, we calculate aEUE(c in 2 ways:

(aE UE) (ci) _ _ UE (a) VE (ci)ci-a

and aE (U,: (ci) ) = II (ci-ck) (- as ci (E)) .ki

UTherefore ae (ci (E) ) = 1

(ci-a) II (ci-ck)

k#i

Letting t=a, s=b be the point on C over a, we recall the rational

function

U (a) . (s+V (t))+U (t) . (b-V (a) )U(t) (t-a)

on C used in §3, which has poles at P = (a,b) and at

Take its product with w7

and use the fact that the sum of its

residues at all poles is zero:

0 resQ (U(a). (s+V (t)) +U (t) (b-V (a)) (t) dt\= U (t) (t-a s

2U(a)V(c) .(ci) dt(

;(a)dt l((t= resP U(a) (t-a) b l + resP-ci) 11 (ci ck)(ci-a) V(ci))

-71i

(using s(Pi) = V(ci))

W(ci20.(a) + 2 ) V(c)E=0

3.94

But

E=0

j (Ci) ai V(ci ae (ci(e))E=0

The proof for the vector field D. is similar. QED

ra Jwj

g.CoC=0

C. (C)(t)dt

jj

s0*

3.95

§6. Theta characteristics and the fundamental

Vanishing Property

The appearance of in the main theorem of '§5 looks quite

mysterious. It appeared as a result of an involved evaluation of

the integrals in Riemann's derivation. As in the Appendix to §3,

Ch. II, we would like to introduce the concept of theta characteristics

in order to give a more intrinsic formulation of (5.3) and clarify

the reason for the peculiar looking constant It cannot be

eliminated but it can be made to look more natural in this setting.

Recall that theta characteristics on a curve C are divisor

classes D such that 2D = KC. For hyperelliptic curves, we can

describe them as follows:

Proposition 6.1:

i) KC = (g - l)L

ii) Every theta characteristic is of the form

f7def I P + ( T)L

PET

for some subset T a B with #T ; (g+1)(mod 2).

,iii) fT = fT if and only if T1 = T2 or CT2, hence the1 2

set I of theta characteristics is described by:

set of subsets T c B modulo

#T -= (g+l) mod 2 1/ T - CT

3.94

But

3.96

g-liv) For all such T, E C such that P.

if and only if #T 34 g+1, and if #T < g+l

g-1dim (f T) = dim oG ( Pi)

1

g+l--#T2

The pi

(if #T > g+1, replace T by CT to compute dim t(f,V);

Proof: In the proof of (5.2), we saw that the divisor of the

differential dt/s was just (2g-2)-, which belongs to the divisor

class (g-l)L. This proves (i). As for (ii) and (iii), note that

hence fT

2fT = I 2P + (g-l-#T) L = (g-l) LPET

But all 2-torsion is representable as divisor

classes e,,, and it's immediate that:

(6.2) fT + eS = fTOS .

Since any 2 theta characteristics differ by 2-torsion, they are all

of the form fT for some T. Moreover

fT=

fT eTT = 0 by (6.2)

1 2 1o

2

T1oT2 = or B

T1 = T2 or T1 = CT2.

3.97

Finally, to calculate 9(fT), use

(g-l+#T)c

hence

I P,

PET

_ space of fcns. f with (g-l+#T)-fold

fT) (pole at - and zeroes at all PE T )

We assume #T < g+l, so (g-l+#T) < 2g. Now functions with 2g-fold

poles at are polynomials in t of degree < g (s has a 2g+l-foa.d

pole at ). So

polynomials in t of degree < (5-1+#T)

Tzero at all P E T

The dimension of the latter space is g+2-#T, hence (iv). QED

Comparing Prop. 6.1 with 11.3.10, we come up with a set of

canonical isomorphisms as follows:

T= B: #Tsg+l(2)1>

fsymmetric translates} Iand 'T - CT > of 0 in ,Yac C f 2

locus of div. classesT t fT -- p +...+P f }1

g-l-T

zeroes of '$[nl n

3.98

Thus the symmetric translates of 0 in Jac C can be described

combinatorially in 2 ways: by subsets T of B and by p E2

2g/m2g.

Riemann's theorem tells us how to link these up. The result can be

better phrased like this:

Proposition 6.2: Let U c B be the set of g+l branch points

al,a3,,a2g+l. In the above correspondences, the.followin objects

correspond to each other:

(a)

¢i if g oddif g even

0 itself ,i.e.,locus of

< (g-l)00 4-; (P1+...+Pg-ll-(g-1)co

(b) For all T c B such that #T E (g+l)mod 2:

T <

especially:

TIToU

U C 0

Proof: (a) is a rephrasing of (5.3) part 1, except for the

first description. For this note that

g odd # 0 = (g+l)mod 2 and f, = (g--l)-,

g even -=> # ({ }) = 1 = (g+l)mod 2 and f{-} _ (g-l)

3.99

To check (b), build from (a) as follows: for all S c B, #S even,

if g odd rtranslateISoS [eSI

L

of0 by [6+if g even I(eS)

S1.

If g is odd, #U is even and one checks

1

:iii:)(

1:II) inU(...)+( ... )6

z 2 2 z z

while if g is even #(Uo{0o}) is even and

0...0l (0.. -02,a4,...,a2 } (1 0...OJ+\1 L.

0)+....

g 2 2 2

0 0...2

1 1...3.) = S

2 2 2

Letting T = S if g is even, T = So{oo} if g is odd, part (b) follows.

QED

4.This gives the following "explanation" for A and 6: the

symmetric translates of 0 are - without any unnecessary choices -

naturally parametrized by the divisor classes Z ,. hence by subsets

T c B, #T 1_,q+1(2). The points of order 2 on Jac C are naturally

parametrized by subsets T c B, #T even. The theta function, after

a lot of non-canonical choices, picks out a particular symmetric 0,

i.e., 't9 (z) = 0. (6.2) shows that in effect all these choices just

boil down to fixing. a "base point" in the set I which is the set U

of odd-numbered branch points.

3.100

In Ch. II, §3, Appendix, we also noted that I came with a

natural division into even and odd subsets." We can identify this

division in the hyperelliptic case:

Proposition 6.3:

a)

e2(n5rnS (-1)#(S1ns 2)

for Sic B, #Si even,1 2 (#T-g-1)

b) e* (nToU) 2 for T c B, #T (g+l) (2) ,

hence:

c) If T c B satisfies #T = (g+l)(mod 2), fT is an even

element of I if and only if #T - (g+1)(mod 4),

odd if and only if #T = (g--1) (mod 4) .

Proof: Check (a) as follows:

Note that

# (Sl n (s2Os3)) - # (S1 n s2) + # (Sl n s3) (mod 2)

(see figure 6.4), hence #(S1n S2) mod 2 is a symmetric bilinear

2Z/22Z-valued form on the group of sUJ,sets of B. When S1 and S2

are generators {ak ,°°},{ak ,=} of this group, one checks the result1 2

directly. This proves (6.3a).

A = S1 n (S2oS3)

(6.4)B = points occurring in

both S1ns2 and S1ns3.

3.101

To check (b), recall that

e (a+13)e*' a)e* s) = e2(a, ).

#T--g-1Let e* (T) _ (-1) We check that

all a R EI 2Z2g

e (ToS1QS2) e* (T)e,(To51)e*(ToS2) (--1

#slns2)

for all S1,S2,T C B, #Si even, #T = g+1(2). This is equivalent to:

(6.5) #(ToS1OS2) + #T - #ToS1 -- #ToS2 = 2#(S1ns2)(mod 4)

Proof by Venn diagram:

7

(6.6)

(+ for membership in ToS10S2 or T; - for membership in ToS1 or TOS2).

Thus in (6.5):

3.102

LES = 2# (TfS1fS2) - 2# (S1f,S2fCT)

2# (Sill S2) (mod 4) = RHS.

Putting together part (a) and this equality, we find that

T I e* (ToU) /e* (r1T

is a homomorphism from the group of even subsets of B to (+1).

Next check

e* (Uo{ak,co +1 if k is odd

if k is even

while e*(pk) = +1 if k is odd

-1 if k is even.

This proves (b). (c) is a restatement of (b). QED

Note that (6.1 iv) and (6.3b) together confirm the formula:

(-1)dim t (fT) = e,ti (nToU)

asserted without proof in 11.3 for all corresponding divisor classes

D with 2D = KC and theta functions &[n].

Putting together (6.1) and (6.2), we obtain the following

vezy important Corollary:

Corollary 6.7: Let C be a hyperelliptic curve, with branch

points B. Describing the topology_of_ C as:above, let U c B be the

(g+1) odd branch points and let 92 be its period matrix. Then for

3.103

all S a B, with #S even, let I(eS) E Jac C be the corresponding

2-division'point. Then

1 5 1 [nS) ( 0 , n ) = 0 '&(I (eS) ,n) = 0 4r` (SOU) # (g+ 1) .

Proof: Combine Car. 3.12. of Ch. II with (6.2) to find:

(4[nsl(0,n)'= 0) (fSOU Pl+...+Pg_l for some Pi).

Then apply 6.1 iv. QED

The importance of this Corollary is that it provides a lot

of pairs n',n" E 22Z2g such that for hyperelliptic period matrices

SZ,

I 0nE2Zg

We know (II.3.14). that for all odd n',n", i.e., 4tn'. n" odd, this

vanishes for all n because in fact the series vanishes identically.

But Cor. 6.7 applies to many even n',n" as well.

We shall see, in fact, that these identities characterize

hyperelliptic period matrices. To get some idea of the strength

of this vanishing property, it is useful to look a) at low genus

and b) to estimate by Stirling's formula, what fraction of the

2-division points are covered by this Corollary for very large genus.

3.104

g = 2: = {S c {1,2,3,4,5,6}I#S = 1, 3, or 5}/(S ^ CS)

[the 6 odd characteristics

U the 10 even characteristics

{1,2,3},{1,2,4}, --,{1,5,6}

[normaizing S by assuming 1 E S)

g = 3: 1 _ {S c {1,2,.--,8} #S = 0,2,4,6 or 8}/(S - CS)

= [the one even characteristic S = with g(fS) .(0)l

U [the 28 odd characteristics S = {i,j}l

U [the 35 even characteristics S = {1,i,j,k}, J(f5)=(0)l

g = 4: = {S c {1,2,...,10}I

#S = 1,3,5,7 or 9}/(S - CS)

[the 10 even characteristics S = {i}, with dim X(fS)=2l

U [the 120 odd theta characteristics S = {i,j,k}l

U [the 126 even theta char. S = £(f)= (0) l .

3.105

Fraction of 2_ ;s.ovt Fraction ofpts.wh,ck are odd 2-division pts. a

(so that &(a,f)=0 where i3 (a,St)=0_,.9 all S) S2 hyperelliptic

in dimension 2,

2 6/16 6/16hyperelliptic Q'sare an open,dense set

3 28/64 29/64

4 120/256 130/256

5 496/1024 562/1024

large g ^- 1/2 "- (1- 2 l)

-IT vg-+1

The last estimate comes from:

# ({S c B I#S=g+l}/S _ CS) - 1 (2g+2) !2 (g+l )! 2

1[(2ge2)2«+2 2Tr ( .C(9+1) 9+1 2Tr )-2

22g (_2 __L_]-

JTr /g-+l

3.106

§7. Frobenius' theta formula

In this section we want to combine Reemann's theta formula

(11.6) with the Vanishing Property (6.7) of the last section. An

amazing cancellation takes place and we can prove that for

hyperelliptic SZ , 4(z,S2) satisfies a much simpler identity

discovered in essence by Frobenius*. We shall make many

applications of Frobenius' formula. The first of these is to make

more explicit the link between the analytic and algebraic theory of

the Jacobian by evaluating the constants ck of Theorem 5.3. The

second will be to give explicitly via thetas the solutions of

Neumann's dynamical system discussed in §4. Other applications will

be given in later sections. Because one of these is to the Theorem

characterizing hyperelliptic Q by the Vanishing Property (6.7), we

want to derive Frobenius' theta formula using only this Vanishing

and no further aspects of the hyperelliptic situation. Therefore,

we assume we are working in the following situation:

1. B = fixed set with 2g+2 elements

2. U B, a fixed subset with g+l elements

3. - E B-U a fixed element

4. T j' 71 T an isomorphism:

even subsets of B 12Z

2g/ 2Z2g

(modulo S ^- CS ) 2

such that

a) ns O5' = n + fS1 2 1 2

b) e2 (n5 rns ) _ (-1) #S11 52

1 2

W-Uber die constanten Factoren der Thetareihen, Crelle, 98 (1885);see top formula, p. 249, Collected Works, vol. 11.

3.107

# (ToU) -g-l

c) e*(nT) _ (-1) 2

5. St E j satisfies n9jnT1(O,SZ) = 0 if #ToU g+l.

6. We fix ni E 2 ZZ2g

for all i E B-= such that ni mod 7L2g

equals n{i,.}, and also let no, = 0. (This choice affects

nothing essentially.)

We shall use the notation

eS (k) = +1 if k E S

-1 if k % S

for all k E B, subsets S c B.

Theorem 7.1 (Generalized Frobenius' theta formula). In the

above situation, for all zi E Cg, 1 < i < 4 such that

z1+z2+z3+z4 = 0, and for all ai E(p2g, 1 < i < 4, such that

al+a2+a3+a4 = 0, then

(Fch)4

e (j) 11 &(a+nj] (z. ) = 0jEB

Ui=l i

or equivalently:

(F)4

e (j)exp(4Tritfl cfl ) IT z+Pn+T1 = 0J 3jEB U

3 i=1 i

3.108

Proof: By (Rch), for every w E I 2G2g

2-g 1 (z4)XE22Z2g/IL2g

a1+2w+a2--a3-a4_ '1} [wl (0) l C 2

l (z1+z2-z3-z4) ( ...) (.... )

or 19'[w1 (0) = 0

since

40 = I exp(4Tri [ai+al (zi)

XE2Z2gtZ2g i=l

-e (a1 l

2Trit (a'+a') 2w" i,9,[a1+x1 (zl) .

Therefore, V T c B, #T even, #ToU # (g+l),

#snT 40 = (-1) '&[ai+nsl (zi) .ScB,#S even i=1

SMCS

Thus, for any coefficients cT,

(7.2) 0 = I I

I

S B#S even

mod S^+CS

T c B#T even

# (ToU) 34g+1

c (_l) #SnTT

41 z9'[ai+nsl (zi) .i=1

3.109

What we must do is to choose the cT's so that "most" but not

all of the terms in brackets vanish! For this, we resort to a

combinatorial lemma:

Lemma 7.3. For all S c B, #S even,

Y

TcBET

#T w (g+1)mod 222g-2 if S - B,B-,k} and g is even.

Proof: We note first the following points:

a) for all finite non-empty sets R,

# (subsets T c R) # /subsets TcRR = 2#R-1skt #T even J #T odd

In fact, the subsets T c R form a group under o and the even subsets

are a subgroup of index 2.

b) for all finite sets R with at least 2 elements,

rr

I (#T) = I (#T) = (#R) .2#R-2

TTeven) #T odd)

In fact, the first sum here is the cardinality of the set of

pairs (i,S), where i E R and S is an odd subset of R-{i}, and

we count this by (a). The second sum is the same except that S is

an even subset of R-{i}.

0 if S # ,{-,kj,B-{-,k},B

#T- (q+l) . (-l) #Sf1T =2 g2 if either S 0, {-,k } Or

2 S=B-{- k},B and gis odd

3.110

Given these facts, we can easily work out the sum of the lemma.

Note that it is invariant, up to the sign (-1)#T = (-1)g+1, under

S H CS = B-S, so we may assume

2:TcB00 ET

#T =(g+1)mod 2

If #CS > 2 and #S > 3, then

1[#T2+(#T1-(g+1))) ]

T2cCS

#T2(g+l-#T1)

#T

(-12)#T1

#CS-2#CS-2 + [#Tl- (g+1)12 #CS-11

coET1cS

= 2#CS-3{

I #CS+2[#T1-(g+l)]- I #CS+2[#T1-(g+1)]}-ET1c S -ET 1cS

#T1 even #T1 odd

w E S. We then have

#T- (g+1) . (-1) #SnT _I

#T1+#T2- (g+l)(--1)

#T

2

_® ET11S 2

T2cCS

#T1+#T2=g+l

= 0, using b) again.

3.111

If either #CS < 1 or #S < 2, we must have S = {-,k} or S = B;

in the first case we compute directly:

2#CS-3f E

co ET1CS

#T1 even

#CS + 2[#T1_(g+1)] -- I #CS+2[#T1-(g+1)]} _ET1CS

#T, odd

= 22g-3 (2g+2 [2- (g+1) ] - 2g - 2[1-(g+1)}) = 22g-2 ;

in the second case,_ g+1#T1 2g+1) (-1) #T1

= (-l) [ (2g+l) 2g+1-3 - (g+l) 22g+1-2ET1CS 2

#T12g+1

= (-1)g+1 22g-2.

To apply the lemma, note that

#S n (ToU) = # (snT) + # (SAU) mod 2.

In formula (7.2), set

cT

Then by the lemma,

# (ToU) - (g+1)2

0

if E ToU

if not

0 = # (ToU) - (g+1) . (-1)#s n (ToU), (_1)#(snrJ)

S CB TC B#S even #T evenmod S - CS

QED

4

(zi)i=1

ti22g-2

(-1) #Snu (zi) . QEDS={}o{k}

k E B

3.112

Corollary 7.4. Let T c B have g+2 elements and let

S = ToUo{oo}, so #S even. Then

e.(j)exp(4Tri tT1 Snj) 9 [ns+nj] (O)2.9,rn (z)2 = 0.jET

Then

Proof. In Foh, take

zl = z2 = 0, z3 = z, z4 = -z

a1 = ns, a2 = -ns, a3 = a4 = 0.

o = I eU(j) 4 4 [ns+nj] (0) ' [-n$+njl (0)'9'[n.] (z) 't9'[nj) (-z)jEB

now for any A E 2Z2g

so

/&[a+Xl (z) = exp(27Tita' A')'S'[al (z),

45'[-ns+n (0) = 4T[ns+n2nsl (0) = exp(-4Trit(r1;+n ) ns) z9'[6 S+n (0),

and

But

'&(nj l (-z) = e* (nj) 49'[nj l (z) .

(wo{'}o{= -1')

so putting all this together, we have the formula (7.4). QED

3.113

$[n3 J (O) *[in l (z) 2Corollary 7.5:(

0 O O z ) 1jEU

Proof: In (7.4), set T = U U {-}, hence S = .

We now apply Frobenius' identity to refine (5.3) above:

Theorem 7.6: As in (5.3) consider the map

Jac C - 0

W

space of monic polyn.

U(t) of degree g

W

D t , UD (t)

Then for all finite branch points ak, 1 < k < 2g+l, and for all

V c such that #V = g+1, k E V

2D [nu,V+nk1 (0) 19 [+fk] (z)

'TTU (ak) _ - (ak`ai)i9 [nUeVl (O) 'c9[SliEVi'k

where z = Jw

and the sign is given by

D

Proof:

(-1) (-'1)

We make a partial fraction expansion:

DUD(t) = C(t)- k,

TT(t-ak) kEV t-ak

kEV

Then UD(t), is monic but otherwise arbitrary, so I AD = 1 butk k,V

otherwise the ak,V are arbitrary. In particular, if IdkXk V = 1,P

3.114

for all D, then dk = 1, all k. ',dow by (5.3),

2D UD (ak) ck (19-[6+nkl (z)

1k,V iUV(ak-ai) II (ak-ai (z) )vitilk

i34k

On the other hand, by (7.4), with T = V U (-), S = UoV,

2

1 exp (4t 9'[n oV+nkl (0) ,9'[nkl (z)

)U(*) = ri nU

kEV 15)_

[nUoV[ (O) - &101(z)

Using the definition of 19'-functions with characteristic, it

follows:

2 2[d+nk) (z) t ' k)

exp Wri d nk)9[d] (z) (z+Std'+d") '

Since z is arbitrary in (*), we can replace it by z+Sld'+d"

and find:

(+flkd1 =(0)

2

l(z)/kEV [nUovl (0) [dl (z)

p, 2 TT (a -a )exp (47ri.... ) . ((fl01(0)U°Vkl O1 iEV,i k D

kEV J ck k,V

for all D. Thus the coefficients of lkV are all 1, hence

ck = exp (41ri ) . f (ak-ai) t9[nuov+nk1 (0)2

iEV,i#k ( & [nUoVl (0) )QED

3.115

A second application is to the explicit solutions (xk(t),yk(t))

of Neumann's system of differential equations,

xk Yk

Yk = -akxk + xk(jaix - iyi)

where a1< --- < ag+1 are fixed real numbers and jxk = 1 and

lxkYk = 0. We saw that

2

(xkyk-xkyk)2

k k ak k

are integrals of this motion and set up the following maps:

subvarietyTT (Sg+1 F1 (x,Y)=cl

(Fg;;(,;og+i)

n

space of polyn. zeroes of3g+l U(t) , V (t) , W (t) (Jac C--O) gs.t. FZ - (z,St)

f--V2 = U-W

Here TT(Sg+l) is the affine variety of x,y such that 2 = 1,

1xiyi = 0, rr(x,Y) _ (U,V, W) where2

U (t) = fl (t) tXkk

V (t) = fl (t) tkakk2

W(t) = fl (t) - ( Yk + lt-ak

3.116

g+l ckfl (t) = TT (t-a) f2 (t) fl (t) ' It-ak=1 k

, f (t) = f1 (t) f2 (t). We shall

assume for simplicity that the constants ck are all chosen to be

positive. The other cases may be treated quite similarly. This

means that sign f2(ak) _ (--1)g+l+k, hence the zeroes

b a2

< b2 < ..... < age bg< ag+1

Graph of f:

We assume that the cycles Ai,Bi on the curve C given by s2 = f(t)

are chosen as in §5, with respect to the linear ordering of the

branch points on the real axis.

Neumann's equations are the equations given by the Hamiltonian

vector field XH on TT(Sg+l), which is tangent to the above subvariety.

We have seen that

Tr*XH = -2/ -D.

and that the vector field D. on Jac C is given by

- ei(a/azi)

on (Cg, where if wi are the normalized 1-forms on C, wi = 4i(t)dt/s

and fi(t) = eitg-1 Therefore the solutions (xk(t),yk(t)), t E 7R,

3.117

of Neumann's equations project to curves on Jac C-0, which lift to

the straight lines:

z0 + 2/T t e, t E ]R, e = (e1, . eg) .

in (Cg. Moreover:

xk = TI (ak-aQ) -l U (ak)Qk

+ / y (0) (Z)

f\ -&101 (0) t4'[ 6 } (Z)

2

by (7.6), where in (7.6) we choose V = U =

(i.e., corresponding to the branch points al,-,a a+1 ), andD

z = J W, D = (divisor defined by U(t) = 0, s = V(t)'). The n2k-1goo

appears because ak is the (2k-1) st branch point in the linear

ordering. The sign becomes +1 if we put the characteristic S

back into a translation by I:

2 $[n2k-l] (O) . [n2k-l] (z-1) 2xk = +

X9'[0] (0) . 19'[0]

(see proof of (7.6)). Now note that whereas (t' [n2k-1] (z) /YO] (z) ) 2

is periodic with respect to L0, 'i92k1 ] (z)/9'[O] (z) is not.In fact,

19[n2k-1] (z+On+m) mk+nl+ +nk-l 19'[n2k_lI (z)

'r[O] (z+Stn+m) 't9' [0] (z)

(Ch. II, 1 and Def. (5.7)). Thus let L be the sublattice in L.

3.118

of index 2g+1 defined by

L = SZn+m n,m E ag and ml,m,+n1,....

,m +n +...+n ,n eveng 1 g--1 1 g

These ratios are La-periodic. So if we consider the torus CCg/L

which covers CCg/L,, it follows that we can complete the previous

diagram as follows:

subvariety

T (Sg+1)F1(x'y) cq

CCg/L (zeroes of

Fg+1(x , y) = Cg+1

space of polyn.3g+1 U(t),V(t),W(t) 4

F14

Tg/L_ (zeroes ofs.t. - St [Sl (z,)

f-V2 = Uw

if we define the upper arrow by

t' ['12k_11 (O) . h2k-il (zxk & [ol (0) . 'i9' [d] (-_J)

3.119

Note that the action of the group Ln/L = (?L/22Z)g+l on

Cg/L -(zeroes of 1l9[S1) corresponds to the action of the elementary

2-group (xk,yk , (Ekxk,Ekyk), El,...,Eg+1E{+1} on T(Sg+1).

Now the liftings of straight lines in (rg/Ln are straight lines in

(rg/L so the solution to Neumann's equations are:

&1n2k-11 (O) " &[n2k-l3 (zo-4+2 t e)xk (t) _ , t E JR

'l4'[O] (0) 19'[0] (z0--1+2/T t e)

Finally, if we want xk(t) to be real, we have seen that this means

that the divisor b given by U(t) = 0, s = V(t) should consist in

g points (P1,...,Pg) with bi < t(Pi) < ai+l' s(Pi) pure imaginary.

In this case

P. (bi,0) Pri

z O-L _ I j ' - j w=J

w

Coi-

00 i(bi0)

3.120

§8. Thomae's formula and moduli of hyperelli tic curves

As a consequence of the formula expressing the polynomial UU(t)

in terms of theta functions, we can directly relate the cross-ratios

of the branch points ai to the "theta-constants" '&[n1(0,Q). This

result goes back to Thomae: Beitrag zur bestimmun von 19'(0,---,0)

durch die Klassenmoduln algebraischer Funktionen, Crelle, 71 (1870).

We claim;

Theorem 8.1. For all sets of branch points B w {a11...ra2g+l,'},

there is a constant c such that for all S c B--, #S even,

17'[nSl (C)4

=0 if #SoU 4 g+1

c-(-1)#$SU . 'iT(ai-aj)-1

if #SoU = g+1iESoU

jEB-SoU-o

The result looks more natural if we don't distinguish one branch

point by putting it at infinity. Let B = {al,.-.,a2g+2} be all

finite, put the ai's on a simple closed curve in this order and

choose Ai,Bi as before. Then for all S c B, #S even:

(8.2) 1g'[ns1 (o) 40 if #SoU g+l

(- 1) #Sf1U "ff (a1-a]) -1 if #SoU = g+liESoUj%SoU

3.121

In fact Thomae evaluated c too. The answer is:

(8.3) q' [ns] (C) 4 = +(deta)-2 . TI (ai-a ) i I (ai-a )

i<j i<j ji,jES0U i,jgSoU

where (I2iV'-_1 wi = j=1 vJ

For a proof of this, see Fay, op. cit., p. 46.

We can deduce (8.2) from (8.1) by making a substitution

a =

Aa'+B

---' 1 < i < 2 +li +DCa g

CO

= Aa2g+2+Bor

- - Da

Ca2g+2+D' 2g+2C

.

The numbers &[-n s](0) are not affected but the RHS changes to

I r -1a - a.c- {_11 #Sf1U

iES U 1 (Ca ) (Ca+D jjEB-SoU-=

2g+2 ^ _l(C. tT (Ca! +D) 9+1) (-l) #Sf10

1 ! (a'-a') iT (Ca'+D)-1

i=1 iESoU i 3 iESoU ijEB-SoU-0O

.'TT(Ca"+D) g+11 (-1) #Sf1U T T (a' -a') -1.lJ iESoU a

j0oU

3.122

To prove (8.1), we substitute D = a. - g into 7.6.1<i<2g+1

Then

uD(t) = TI (t-a.)iEV'

Y

where V' = and

UD(ak) _ UTiEV'

But D B eV if g is odd, D B eVU{-} if g is even. So

eU g odd

D ; eUOV +

eUo{oo} g even

I (D) = I (e ) +rrov f

I (eu) g odd

)I (e g evenUo {-}

(211 UOV+nU0V) + (S261+611).

Therefore replacing the argument by a theta characteristic:

I" +nkl(I(D) 2

l (1 (D) ) )t [n oV+nk1 (o) 2

exp (s'*nUoV) -rill) ( ..S [nl (0)UOV

hence (7.6) reads:

3.123

(8.4)iT, (ak-at) 4 (tn oVnip

-ktnto

UoVkn' ) [nUoV+nk] (4) 4(-, U

IT (a-a.)iEV-{k} k Y

By (6.3), the sign is

(_1) # (UoV) fl{k,ao}

qKnU,V) (0) 4

Since k E V, this is +1 if k E U, -1 if k 0 U. Now in (8.1), we

may choose c to make the formula correct for S = 4,, and then prove

it for any S by decreasing induction on the number of elements. in

(Sou)r, U: if (SoU) = U, then S = 4 and we are done. Formula (8..4)

is just the ratio of the 2 cases of (8.1):

S = UoV and S = C(Uo(V-k+-)).

This is straightforward although somewhat painstaking to verify.

Therefore applying .(8.4) twice, we obtain the ratio of (8.1) for

S = UoV, and S = Uo (V_k+k) , if k E V, k % V.

For these SoU is respectively V and V-k+k, hence, step by step,

we can move from the formula in the case SoU = U to S0U = (any V

with #V = q+l). QED

We now introduce a moduli space in order to formulate our

results more geometrically:

3.124

set of pairs C a hyperellinticcurve, {1,2,---,2g+l,°}=a B a mod

(8.5) °Q, 52 ) bijection where B c C are the branch isomorphism.points of n : C ..-> 3P 1

We are merely defining dVg2) as a set here, the set of isomorphism

classes of hyperelliptic curves with "marked" branch points.

However because the image of the branch points in ]PI determine

d1othe curve, d'K.g2) can be described equivalently as:

(2)set of sequences Pw mod projective

(8.6) g {of distinct points of IP1 equivalencePGL(2,IC)

Since we can normalize P,, to be co , we can also say:

(8.7) dt

(2) _ set of sequences a1, a2, --,a2g+1

g { of distinct complex numbers }

mod affineequivalenceai i , Aal+u

hence further normalizing, e.g., P1 to be 0, P2 to be 1:

(2)(open subset of C2g-1 of points (a3'a4'...,a2g+1))

(8.8) JrVg such that ai aj, ai # 0,1

This makes °Qg2d),Dinto an affine variety. In terms of the 2nd

description of di(g), if t is the coordinate on ]Pthe affine

ring of d'[.(2) is generated by the cross-ratios9

3.125

t (Pi) -t (Pk) ( P

t(P-t(Pk) t(Pi)-t(PQ)

In terms of the 3rd description (8.7) of {i(92), with one point

normalized, the affine ring is generated by the functions

a. --ia. --

Jak

Now consider the universal covering space g of Q 2)

Letting Aij c CCn be the diagonal zi = zj, and let b Edg(2) be

the base point B = we can describe it concretely

as follows via (8.8):

(8.9)

space of maps c: [0,11- [((-(O,1))2g_l_iuj"ij]

gsuch that QU(O) = b = (2,3,'-',2g)

modulo homotopy: o - Oj if

3(p: [0,112 - [((r-(O,1))2g-l- UM..]

4)(0,t) = 0 (t) , (D (l,t) _ 1 (t)

The projection

0(s,0) = b, c(s,l) indep. of s.

(2)

is the map q s cb (1) , and the covering group r = Sri (°g (

92)is the group of loops in (C-(0,l))2g-1- UAij mod homotopy. This is

essentially what E. Artin called braids with 2g+l-strands except

3.126

that 2 strands are normalized at 0,1 and each strand comes back to

its starting place ("pure braids"). Here is an example:

(In fact 2Z is easily shown to be 7r1 (G2g+1_L'oij), the

group of all pure braids.) We call r the group of normalizedpure braids. We can describe r a bit differently as follows:

let G = group of all orientation preserving

homeomorphisms 0: ]P1 > ]Pl such that

fl O) = 0, c (1) = 1, $ (00) = -, topologized inthe compact-open topology.

let g = subgroup of cp such that c(i) = i,

'i = 0,1,...,2g,-

Then we have a map Tr : G (2) , Tr $ _ { cp (0)

inducing a bijection:

3.127

(8.10) G/Kg (2)

The following lemma is easy:

Lemma 8.11: For all E ((r-(0,1))2g-l- UDij, there

are disjoint discs Di about Pi and a map

: D2x...xD2g G

such that xi. Thus iP is a local section

of TF

Cr

hence IffW l (IIDi ) N = HDi x Kg by the group structure.

The lemma can be proven by use of suitable families of

homeomorphisms of ]Pl which are different from the identity only

near one point P and move P a little bit in any desired direction.

The lemma implies that rr has the homotopy lifting property and

hence the following map p is bijective:

3.128

Equiv. relation O evof if

space of gaps 30: (0,112 - G,[0,1)----a G,) 0(0,t)0 (t) r(D (t)tit0)=iden y (D(s,0)=id., Tr(D(s,l)indep. of s

p

f space of maps

Ahomotopy of paths

[011)--a(a!_(0,1))28-1--k/Aij with fixed initial00) = b and end points

oy.,

g

(In fact, the surjectivity of p is just the lifting of a path:

i V(2)g

and the injectivity of p is a lifting of the type

G

a10(2)di9

3.129

Starting with 4: [0,1] --- > (a-(0,1))2g-1- UAij, lifting to

[0,1] G and taking p(1), we get a map

a : ° g ----> G/K0

where Kg is the path-connected component of the identity in Kg

i.e., { E Kg j (0,11 -- Kg such that (0) = e, i)(1) = }.

From the homotopy lifting property of 7 , it follows immediately

that a is continuous. c relates to our other constructions by

a commutative diagram

gG/Kg

(2) a G/Kg

g

equivariant for the homomorphism:

a*: r Kg/Kg

(Kg/Kgo, acts by right multiplication on G/K°g) given as follows:

Vy: [0,1] M°0,1)) 2g-1_UAi. , with y (O) = y (l) = b, lifty via p to cp: [0,11 -- G. Then Tr (1) = b, i.e., 0(l) E Kg

Ltd T,jyl = d()All this follows formally from the definition of a . in fact,

it can further be shown, but this is not merely formal,that g

is homeomorphic to G/Kg, hence r Kg/K;; but we omit this because

we don't need it. The reason we have defined a so carefully is

3.130

that we wish to use a to define the global period map:

2- dg 4i

In fact, given a point of °9g, we have 2g+2 branch points

B = {0,1,a2,....a2g'Col C 1Pl and a homeomorphism : A'> P

such that p (O) = 0, $ (1) = 1, 4 (i) = ai, 2 < i < 2g and (°°) _

given up to replacing by 4o , i fixing the i's and isotopic

to the identity. Let C be the hyperelliptic curve with branch

points B . Then induces a homeomorphism of the standard

hyperelliptic curve C0 with branch points with C.

Taking the standard homology basis A.,Bi on CO, we obtain a

homology basis on C, hence normalized1-forms wi

such thatf

W7 = Sid, hence finally S2aj =J

w3. This

(Ai) f(Bi

defines the map 0

It should be mentioned that all the topology on the last

00,

3 pages was traditionally compressed in the following few sentences:

to each choice of branch points P, we associate a period matrix S2(B).

As B varies, we move the paths Ai,Bi continuously. S2ij(B) is locally

in this way a single-valued holomorphic function on the space of B's.

Globally, if we replace the space of B's by its universal covering

space SZi.(B) is still a holomorphic function by analytic continuation.

I'm not sure whether this "sloppy" way of talking isn't clearer!

3.131

Note that the map 0 is equivariant with respect to a

homomorphism of discrete groups:

S2* : r Sp (2g, ) / (+I)

To define S2*, let

[0,1) --- M- (011)) 2g-1" UAi7

(0) = 0 (1) = b

be a braid. Lift 0 to

D: (0,11 > G.

Then c(1) is a homeomorphism of 1P1 carrying

to itself. Lift c(l) to a homeomorphism `' of C0 itself. Thentke

`Y acts on H (C,2L) , in its basis {Ai,Bi}, by4a 2gx2g integral

symplectic matrix The equivariance of n is clear (see Ch. 21, §4).

An interesting side-remark in this connection is:

Lemma 8.12. The image of S2* is the level two subgroup r

of y E Sp(2g,2Z)s'.t. Y = I2g(mod 2).2

Proof: Note that if Xi E Hl(C0-B,7L/2ZZ) is the loop around

the ith branch point, then the image of ai in 1P1-B goes twice

around the ith branch point, hence is zero in H1(1P1-B,2L/22G).

Therefore we have a diagram:

3.132

H1 (C0-B,2Z/22Z)/r - ,Xi, - > = HI (CO,2L/22Z )

K!

(]P1-B, 2Z/22Z)

and it's easy to check that K is injective*. '(1) carries each

point of B to itself, it maps a loop p i in ]P1 going around the ith

branch point to a homologous loop. Thus (D(1) acts by the identity

on Hl (]Pl-B, 2z/22Z) . Thus T acts by the identity on I~1 (CO, 2Z/2ZZ) .

SZ* c r2.Thus In

To prove the converse, recall from the Appendix to §4, Ch. II,

that r2, or rather its image in the group of automorphisms of

H1(C0,2Z), is generated by the maps

x i> x + 2(x,e)e

where e is one of the elements Ai,Bi, Ai+Aj, Ai+Bj or Bi+Bj.

To lift these generators in the braid group r , consider the

following simple closed curves in 7P1-B:

This is the purely topological version of the description of2-torsion on Jac C0 by even subsets of B. H1(]Pl -B,TL/22Z)is the

free group on loops ui around the branch points mod E}.ti- 0,21ji- 0.

One checks that K(Ai)P2i-l+u2i' K(Bi) _ "2i+...+u 2g+l' This

proves K injective and identifies H1(C0,2Z/22Z) with even subsets

S of B (mod S - B-S): let a,S correspond when

K (a> _ I pi,iES

3.133

P

A; -Z

loj MI 2; xy rl rz

Each of these lifts in C0 to 2 disjoint simple closed curves and a

little reflection will convince the reader that they lift as follows:

i) Cij lifts to C j U C a, Cij Ai+A3, Cij - -(Ai+Aj)

ii) Dij lifts to D. U Dij, D. Bi-B., Dij Bj-Bi

iii) E. lifts to E' U E', Ei Ai+Bi, E (Ai+Bi).

For every simple closed curve F in IP 1-B, there is a so-called

"Dehn twist" 8(F) E G: take a small collar Fx [-e,+e] around F.

Then 8(F) is the homeomorphism which is the identity outside the

collar and rotates the circles Fx{s), - c< s < e, through an angle

rr(s+e varying from 0 to 21T as s varies from -e to +e. Now the

Dehn twist S(Cij) lifts to a homeomorphism of C0 which is

3.134

6(C,j)o8(Cj). And the Dehn twist S(F), for a path F in CO, acts

on homology by

x i > x + (x.F) F.

Thus S (C i7 ) acts on H1 (CO, 2Z) by

x i > x + 2 (x,Ai+Aj) (Ai+Aj) .

Likewise:

S (D ) t b + 2 ( B -B ) (B -B )--- >ij ac s y x. i i ix x i6(A ) t b i ' + 2 ( A ) .Ai ac s y x x x. i

S (B ) ts b i + 2 ( B> ) Bi ac y x x x. i iS E ) bt : +B ); +B ) (AA+ 2 (( i s yac x ii i ix x.

all of which generate r2. Finally, all Dehn twists S(S) are

induced by braids,t.e.S(S) = cs*(y) mod Kg: making S the boundary

of a disc, one shrinks S to a point obtaining an isotopy of

homeomorphism S(S) with the identity. This may move 0, 1, and °,

but by a unique projectivity, one can keep putting them back. Thus

we find (D: [O,l) -k G with ' (O) = e, c (l) = S (S) . Then

Toy is a braid in r inducing S(S). QED

Finally, having set up the spaces g(2)

gand the map Q

we can reformulate Thomae's theorem more geometrically. In fact,

for all n E 2 2Z 2g, we have the holomorphic map:

3.135

-> T

W

g W 4 [n] (O,sZ(P))

Either by the functional equation for 19'(z,S2), or by Thomae's

formula, we see that the functions

j&[n] (O,s2)4

3.[O] (0,52) 4

on dig are r-invariant, hence are holomorphic functions on

depending only on S E 27Z2g/2Z23. Thomae's formula implies:

Corollary 8.13: The affine ring of °Q g2) is generated by the

nowhere zero functions:

+4i[ns] (O,SZ)1(7PI-0-1

S B such that #UQS = g+l.

Proof. Normalizing one branch point to oo , and letting

al,a2,...,a2g+l be the others, we must check that each

ratio ak-aR/ak--am is a polynomial in these 4th powers. We use

the identity:

2 2ak-aQ a,-a( ml

\ak-am - \ak-am) + 1 = 2Cak-a-)

ak-am

If we write {l,2,...,2g+l} = vy1.V2a{k}, #V1 = #V2 = g, then by

t

3.136

Thomae's formula

(8.14)

4

Tr(ak-al)

M)iEV

iT (ak-ai) [n(V1+k)

oUl (O, S2) 4

2

Write instead {1,2, ,2g+1} = V3 .u. V4 u{k,i,m}, #V3 = #V4 = g-1

and apply (8.14) to the pairs V1 = V3+i, V2 = V4+m and then to

V1 = V3+m, V2 = V4+Q. Dividing we find

ak-ai 2 &In114 n9-[n214

(ak_am) 19'[n3l4 14'[n414.

for suitable ni. QED.

The relations among these generators presumably may all be

derived from various specializations of Frobenius' identity.

3.137'

§9. Characterization of hyperelliptic period matrices

The goal of this section is to prove that the fundamental

Vanishing property of §6 characterizes hyperelliptic Jacobians.

The method will be to show that any abelian variety XQ which

has the Vanishing property must have a covering of degree 2g+1

which occurs as an orbit of the g commuting flows of the Neumann

dynamical system.

To state the result precisely, we fix, as above, the

following notation:

B = fixed set with 2g+2 elements in it

U c B, a subset of g+l elements in it.

n:

where n

group of subsets

T c B, #T even 1 292 /2L2g

mod T - Cr

is any isomorphism satisfying

# (T1nT2) (2n' (T1)) . (2n" (T2)) - t(2n' (T1)) . (2n' (T2))]mod 2

# (TOU) - (g+l) t(2n' (T)) . (2n" (T)) mod 22

where n (T) Cn'(T), n" (T)) . We shall subsequently abbreviate -n(T)by nT.

Theorem 9.1: Assume S2 E g satisfies

ITIT] (0, 2) = 0 E-- # (ToU) 54 g+l

Then S2 s the period matrix of a smooth hyperelliptic curve

of genus g.

3.138

Proof: First of all, to write our formulae with

unambiguous signs, we are forced to make a choice of lifting

of nT from 1 2Z2g/2Z2g to 1 2Z2g To do this, we choose a fixed2 2

element - E B-U, and choose

ni E2g ni ` nco}mod 2Z2`

for all i E B--. We also set n = 0. Then forAT define a

lifting nT E 2zz2g by

nT_

iITny-

The "standard" choice, if B = {l,--.,2g+l,co}, is

0...0 2 0...0

n 2i-l ... 0 0---02 2 _

ice` place

V.

0...0 2 0...0

n2i l 1 1 0...02 2 2

but there is no need to get that specific.)

The first part of the proof is to investigate the differential

of the theta function -9 [nT] (z,2) at z = 0. The tool at ourdisposal is Frobenius' formula, and we propose to differentiate

it, and substitute so that very few terms remain. In formula

3.139

(Fch) in theorem 7.1, replace xl by x1+y, x2

the differential with respect to y and set y

assuming Jai = Jzi = 0:

(F h) EsU(j) (d49[al+n] (zl). Q[a2+njl (z2) -jEB

by x2-y, take

0. We get,

d9[a2+nj] (z2) a[al+rljl (z1), ,*(a3+7111 (z3) 4[a4+f j1 (z4) 0.

Note first that ,9[n (z,S2) is an even function if

# (SoU) E g+l mod 4

and is odd if# (SoU) E g-l mod 4.

Therefore

d-9(nsl (0,52) = 0

if #(SoU) = g+l mod 4 and we may restrict our attention to the

case #(SoU) = g-l mod 4, and, replacing S by CS if necessary,

# (SoU) < g-l.

Lemma 9.2: d.- hS] (O,c2) = 0 if # (SoU) (g-1) mod 4 and

# (SoU) < (g-1).

Proof: Let T c B satisfy #T = g-5. In (Fch), let zi = 0,

all i. Moreover, take A,B,C c B-T 3 disjoint sets of 3 elements

each, let a E A be one of its elements and set

3.140

al =nTCU

+ na

a2 n(T+A+B)oU + na

a3 n(T+A+C)OU + na

a4 = -a1-a2-a3

2g= n(T+B+C)oU + na mod ZZ

Then all terms with a factor 9[a1+ni](O) are zero, so the d,%9

goes with the1st

factor. For the last 3 factors to be non-zero,

we need

#(T+A+C-a)0{i} = g+l,

#(T+A+C-a)o{i} _ g+l,

and #(T+B+C+a)o{i} = g+l.

This only happens if i = a. So the formula reduces to

d$[nToU] (0)'9[n(T+A+B)oTP (0) [n(T+A+C)°T. _4[n(T+B4C)°U}-(0) 0.

Since the last three are non-zero, the 1st is zero. A similar

argument shows that d [nTQU](0) = 0 if T c B satisfies#T = g-9, g-13, etc. QED

Lemma 9.3. For all R c B, #R = g-2, and elements a,b,c E B-R,

there is a relation

a d [ n (R+a) oU} (0) + (R+b) aU](0) + v -d°[n (R+C)

O U(0) = 0

where aiiv 34 0.

3.141

Proof: Let d,e be 2 elements of B-R-{a,b,c} and let

f E R. In (Fch), let zi = 0, all i, and let

a1 = n(R-f)oU + of

a2 n (R--f+d+e) oU + of

a3 = n (R--f+a+b+c+d) bU + of

a4 = -a1-a2-a3

n(R-f+a+b+c+e)oU + of .

Then all terms with a factor 9[a1+ni) (0) are zero, so in each

term the d9 goes with the first factor. For the last 3 factors

to be non-zero, we need

# (R+d+e) a{i} = g+l

#(R+a+b+c+d)o{i} = g+l

#(R+a+b+c+e)o{i} = g+l.

This happens if i = a, b, or c, giving 3 remaining terms and a

formula just as required. QED

Lemma 9.4: Let S,T c :B, #S = g, #T = g-1. Then in T '0,

dR [fi TOU] (0) E span of differentials 4[r1 (S-i) OU] (0)

for all i E S-SIT

3.142

Proof: Prove this by (n duction on # (SST) . If, to start with,S = R+a+b, T = R+C, (a,b,c E B-R distinct, and #R = g-2), thenthe result is precisely lemma 9.3. In general, choose

a,b E S-S S T and c E T-SST. By lemma 9.3,

1 : 1 = X d,9 fn (T-c+a) oU] (0) + V .d a & (T-c+b) oU1 (0) .

Now (T-c+a) and (T-c+b) both have one more element in common

with S than T did, so by induction d,9[n(T-c+a)oU1(0) and

d'9[n(T-c+b)oU](0)are both in the required span. Therefore,

so is d9 [nTQU I(0) . QED

Lemma 9.5: Let S c B, #S = g. Then the g differentials

Wa = d-5 fn (S-a) oU 1(0) , a E S, span TXS2' 0.

Proof: We use the fact that the abelian variety X0 is

embedded in projective space by the functions -9 [n] (2z,S2) , when n

runs over coset representatives of 2ZZ2g/2Z2g (see Ch. 2, § ).

It follows that the whole set of differentials d9[nS] (0,c2) must

span TX,0

, as S runs over all even subsets of B. By lemma 9.2

we may as well assume #(SOU) = g-1. By lemma 9.4, the whole

space is still spanned by d9[n5](0,Q)'s where S = (S0-{a})*U,

S0 is any one set of g elements of B and a runs over the elements

of S0. QED

Lemma 9.6. For all a E B, there is a unique vector Da E TX0'

S2',up to a scalar, such that for all T c B, #T = g-l

Da9 fnToU] (0) = 0 <=> a E T.

3.143

Proof: Fix a subset S c B with #S = g, a E S. Then the requirement

Da19[n(S-b)oU] (0) = 0, all bES-{a}determines Da up to a scalar by lemma 9.5. To see

that Da9[fTuU](0) = 0 if a E T, #T = g-l, use lemma 9.4.On the other hand, if Da9 [nTQU ] (0) = 0 when a T, #T = g-l,we would have that the differentials dd9 [ nT+a--b] (0) , all b E T+a,were linearly dependent, contradicting lemma 9.5.

We now concentrate on the vectors Dk, k E U, and D,,. By

9.5, no g of the vectors {Dklk E U) lie in a hyperplane, so

we may normalize the whole set up to multiplication of the

whole set by a single scalar by requiring

D = 0.kEU k

Then define scalars ak, k E U, by:

Da, = GakDk.

Note that for fixed D,, {Dk}, the ak are determined up to a

substitution ak f-- ak+u; and if D., {Dk} are changed byscalars, the ak change by an affine substitution ak I--a ak+'µ.

So far the proof is quite natural. We must, however, normalize

D.O,{Dk} a bit more and for this a rather ad hoc Corollary of

Frobenius's formula is needed.

Lemma 9.7: For all j,k,Q.E U distinct,

47rin n.e

R [njl(0).D,,,,41

n (0) -D

47ritn ,e nk

19(nk](0)";D-Sfn{Q,7}1(o)DQg[n{R,j}](o)

3.144

'Proof: Start with the formula

e WO [ni1(0)2.4[ni](z)2 = 0i ETJ+co

19

(Corollary 7.4). Replace z by z + O'n {j,k,Y,,oo} + n{j,k,k,°°}

and it becomes:

iE U + -

- 4 v r ] tt n {jr kr ZI 2 2E,,(i)e -9 [nil(0) 0.

Differentiate this first by D,,, second by DQ and set z = 0:

L.a47ri n{7rk,Z,°°}* ni,9

[nil (0)2-[D, [ni+n{j,k,i }] (0) D [fi+fl{akg,°°}](0)iEU+=

+9[ni+n{j,k,t,co}](0).D-D11'9[ni+n{jrk,i 4(0)1 = 0 .

Since the sets Uo{j,k,i}a{i} all have at most g-l elements,

i[ni+ n{j,kirk,°°}

](0) = 0 for all i E B. Moreover, if i E U+oo,

#(Uo{j,k,9}0{i}) = g-1 only if i = j,k,i or To get a non-zero

term in the above formula, we also need

CO,Q4 Uo {j,k,i}e{i}

which narrows down the possibilities to i = j or k. We get

t 1

4Trink9[nk ](0)2.Da°° p[n {j

Q} Q3(0)-D P [n{irQ}](0)e

e n{k,k}] 0](O) = 0J Q,

3.145

Using the fact that

and

-1 = (-])4Tri

j,00}0U)--14Tritn nrr

+1 = (-1)= e 3

# ({k,oo}oU)-g-1

+1 = (--1)2 e

4TrinQ-nk 4Tritthe two signs may be replaced by e and e j, respectively.

QED

Corollary 9.8. Replacing D00 by X E (C*, we can assume

2 4Tritnk-nQ29fol(o) e [nQl(0) S[nkl(0)2

for all Q,k E U.

Proof: If we choose D., suitably, this will be true for one

pair QO,ko. Now vary k. By lemma 9.7, the formula is true for

QO and all k. Now interchange QO and k. Since I D. = 0, we getjEU

o = I Dj8[,j{Q,k} ] (0) = (o) + DX& [n{Q,k} ] (d) .jEU

4Tritri n"

e

'k= (-l) # _l '

4Tri r nrr

e k

the formula is also true for k and Y.0. Now varying RU, it is

always true. QED

3.146

The next step in the proof is an elegant and quite important

consequence of Frobenius's formula:

Proposition 9.9. Let T c B satis #T = g-l and let a,b,c

3 distinct elements in B-T. Then

-9 [nTOU+ n{a,c}] (0) hToU+ n{b,c}] (0) [Dc na] (z) i [nb) (z) -D[ nb] (z) $[na] (z)]

= a. D c [nTQU](0) 9[fToU+n{a,b}] (o) 9 [nc) (z) -9[n{a,b,c,-} ] (z)

41Titna .n 4Trin'OU.nf

where 6= e b. e T c= +1.

Proof: In formula (Fch) , set zl = z4 0, z2 --= z, z3 = Z.

Moreover, seta1 = nc

a2 = 0

a3 = n{a,b}

a4 = -al-a2-a3 2 fToU+n ,,}mod 2L2g .{a,b,c,

Finally, evaluate the differential on the vector Dc.

The coefficients in the jth term of 2h) are, up to constants

Dc&[nToU+fc+nj ](0) v4[nToU+n{a,b,c,oo}+nj ](0)

and

9 [nTQU+nC+nj l(o) .9[nToU+n{a,b,c,-}+nj ](o) .

3.147

For the 1st to be non-zero we need

# (T+c-) o{j } = g-l, c t (T+c)o{j}

by lemma 9.6. This means j = c. For the 2na to be non--zero,

we need

# (T+c) a{j } = g+l

#(T+a+b+c)o{j} = g+l

which means j = b or c. Writing out the three non-zero terms

and evaluating the sign with some pain, we get the result. QED

We are now ready for the key point of the proof. We define

a 2g+l-sheeted covering X of X,, and a morphism

yS : X - V (9 f 0 ]) ------ > T2g+2

as follows:

X = (rgnS2 S2

r n

L =t,SLe+q, Ip,q E ZLg and tniq tnip E 72, all i E U}

H xi, i E U; y,, i E U} are coordinates on c2g+2

i defined by

-9[ni ](0) Ofni] (z)xi

,6f01

(0)

9i E U .

Note that 2L L' C L and 2g+1, and that L

is precisely the lattice with respect to which all the functions

3.148

-9 r r1 i l (z) J$ [ 0 l (z) , i E U, are periodic. Moreover, by Corollary 7.5,the image of 0 lies in the affine variety

1.2

hence by differentiating, the image lies in

(1x)-l = Ixiyi = 0

- called TT(Sg) in §4, the complexified tangent bundle to Sg.

What we shall prove now is that the vector fields Dk, k E U, on

X are mapped to half the Hamiltonian vector fields XF onk.T2g+2

defined in §4. It will follow that 0 is an isogeny of

the torus X onto one of the tori obtained by simultaneously

integrating the XF k, which by the theory of 94 are precisely

2g+1-fold covers of the hyperelliptic jacobians. It will then

follow easily that, in fact, XQ is isomorphic to the

corresponding jacobian.

Recall that Fk(X,Y), k E U, are the functions:

(x,y) = x 2 + Z (xkyk_xkyk)2Fk k

£Ok ak-akkEU

(the ak here are the same ak defined earlier in this proof).

The corresponding vector fields X , are given by:k

3.149

and

2(xjyp xQyk)

XF (xP, ) _ _a ."k, if . kk ak

X

XFk(yk)

^ 2( x= L, k

(_x

ppkk a.K _a p

PEU

2 (c aZYk)yk +

2

2(_ak)(

_Y,)+ 2xk (xk-1)

P PPEU

if 2 = k

if f, k

if Q=k

(See 94, Proof of Theorem 4.7.)

Note that IXFk

= X£Fk

= Xl = 0. Now let capital Xi be the

function on

X1 (z)

and let capital Yi by D,,Xi, again a function on What we

claim is that if we substitute Dk for Xf k, Xk forxk, Yk for yk,

then if k 74 k, bk(X9), Dk(Y9) are given by almost the same

formulae on X:

3.150

Lemma 9.10. If Q k, then on XI:

(XkYz-XP. Yk)

Dk (XQ) ak-as Xk

(X

k

Y Q-X

tY k)

Dk(YQ) =a k-a

k

Yk + XQX

But IDk = 0, so Dk(Xk),D k(Yk) are also given by the same

formulae as21XPk

(xk 2), 1XFk (yk ). Hence the lemma implies:

Corollary 9.11: The differential of 0 carries the vector

field Dk to the vector field 1X, ,k

Before proving lemma 9.10, we shall evaluate XkYi-XtYk

in simpler terms:

Lemma 9.11. If i k,

(0) [n{k,Z} ](z)XkYQ-XiYk = e

,9 [0](0) 4[0](z)

Proof: XkYP,XiYk = Xk -(XQ/Xk)

-i[nk] (0)'9[nX] (0) nk] (z) .ID J[n2] (z)--[nR] (z) (z)

4[o](0), $[ (0)](z)

Using Proposition 9.9, with T = U-{k,&}, c a = k, b = k,

the second term on the right equals

3.151

e4rrit nin D,1-4[f{k,Q(0) 4[2n{k,P.}] (0) t9[11{k,R}] (z)k nk+2fQ 0 nR+ nk 0 0 z

Simplifying the characteristics and working out the sign, this

gives Lemma 9.11. QED

Proof of 9.10: If k -/ k,

'[nk1(0) 9[0] (z) D [nt] (z)-W [fP] (z) D [0] (z)Dk (X&) ,s_` ]moo " 2

[ 07 (z)

Using Proposition 9.9 with T = U-{k,P}, c = k, a = Q, b = -,

the second term on the right equals

e47ritnk,Rnk DkLn{k,R}(0) -9[k2n1 (0) '9[C (z)9[n{k,Q}] (z)

9[2 +2n ](0) - [2n +n ](0'plc k ,q [0] (z)

hence

4rritnI ifDk$[n{k,Q}] (0) [rtkl (0) [nkl (z)

Q,}] (z)

Dk Q(X) e

-9g(0](0) $[01(2)2

On the other hand,

--a.) DpD =E

apDp - (ap pU PEU

hence

(0) =p

EU(ap--a.)Dp9[n{k,k}l (0) .

3.152

But DP-G [n{k,Q}] (O) = 0 if p 76 k, k (because p E Um{k, It}) , so

D Cn{k,0](0) =

Therefore,

D

(z) N Il (0) L[nk] (z)e4Tritfl .fl Dj[11{k,Q}] (0)(X) - ( .k Q ak Q \ 19 [0] (0) . -9 [0] (z) [0] (0) LO] (z)

XkYk-Xi Ykak-aZ Xk by Lemma 9.11.

Finally,

Dk(YQ) = DCO (Dk(XQ) )

D(Xk

XkYXQYk9k- z

XkYQ.TX£Yk +Xk

ak-ai ak-a!C

hence it remains to prove

D,(XkYi-XQYk) = (ak-aL)XQXk.

But

D00 (XkYQ-XQYk);

[0](z)47rit.n ..nZ D -9[71 {k,i}] (0) 9 (0l (z)D (n{k,i}] (z) [n{k,9}] 00

D00 (,Vk XRYk) - e-9[0](0) [01 (Z)

3.153

NOW in the proof of 9.11, we deduced as a special case of 9.9

that

.9 [rk] (z) DJ4[nk] (z) - .9[n.] (z) (z)

4Tritn T1 it * [ n{k, k) 1(0) -.&(o 1(o)e

[nk](0) (Tit 1(o)

Substituting

this gives

z+Qnk+nk for z and rewriting the theta functions,

4[o}(z)-D [n{k,Q}1(z) -00

D,-9[n{k,i}1(0) [o](0) Lnnko (o) R] (z) S[nk] (z)

(The minus sign comes from

41ritr .nn

eQ

This gives

4Rltr.01 to 11 )

fn{k,i}+nkl(z) = e k- nQ

47Ht

nQ D zX1 1 ( 0 ) [nk] (z) -t9[nz] (z)XQYk)

[ nk] (a)9(,)z 3 (o) t9 [01 (Z)

2

4Tr3.

(ak-ad e

which by corollary 9.8 is

29[] (o) D 9(0D° [n ki ), [0] (0){k. R.} n{k,L}]

'& [ nk] (0)2,9[n Z] (0) Z cxx

(ak-a.) XR,.QED

3.154

The rest of the proof is now simple. it follows that the

image of 0 is contained in one of the orbits of the g flows

XF , i.e., in one of the complex varietiesk

Fk = ck, k E U.

But these are affine pieces of 2k1-sheeted coverings Yc of

jacobians Jc of hyperelliptic curves (or of generalized jacobians

of singular limits of hyperelliptic curves). Since the

differential of 0 carries the invariant vector fields on X'

to the invariant vector fields of the algebraic groups Yc,

0 must extend to an everywhere-defined homomorphisms

0: X > Yc'

with finite kernel. In particular, Yc is also compact, hence

is a covering of a jacobian of a smooth hyperelliptic curve.

Next, the finite group

Ker (X > X

and the finite group

Ker (Yc > Jc )

both act on the coordinates xi,yi by sign changes

(xi,yi) 1 ),(sixi'siyi)' hence 0 descends to

00: XS c.

But by construction 00 `XS2-V (-9 [ O ])) c Jc-0 , hence 95-1 (9) = V (1 10 ]) ,

if 00 had a kernel, the divisor V09[01) would be invariant under

a non-trivial translation, which it is not. Therefore 00 is an

isomorphism of XR with the jacobian Jc and V(.9[0]) is isomorphic to 8.

QED

3.155

510. The hyperelliptic p-function

On any hyperelliptic jacobian Jac C, there is one meromorphic

function which is most important, playing a central role in the

function theory on Jac C. When g = 1, this function is

Weierstrass' p-function, so, at the risk of precipitating

some confusion in notation, we want to call this function p(a)

too.

We fix a hyperelliptic curve C, and let:

B = branch points of C

= E B

U c t-(* a set of g+l points

t = tangent vector to C at w

This defines for us

a) an invariant vector field D. on Jac C. Namely, if

{wi} is a basis of F(c ), zi ='Wi

are coordinates on Jac c,

and swi(=),t> = ei, then D, _ lei a/azi.

b) a definite theta divisor 0 a Jac C. Namely, 0 is

the locus of divisor classes

g-1ID

1. - (IQ-2°°)

1 QEU

c) the p-function. Namely, let 0 be given by .9(z) = 0,

then

P (z) = D log .9 (z) .Note that v is La -periodic, hence is a rational function on

the variety Jac C. More intrinsically, p(z) is characterized

3.156_

up to an additive constant as the unique rational function f'

on Jac C such that

For all U c Jac C open,

for all hoj.omorphic g on U, g vanishing to order 1

on e n U and nowhere else,

f = D2log g + holo.'fcn. on U.

(In fact, we can even construct p(z) in characteristic p!

Start with a Zariski-open covering {U } of Jac C and local

equations fa of 0 n Ua in Ua. Then fa/f is a unit in

Ua n US, hence

Dcofa D,, fs

f fa $

is a 1-cochain in 0Jac C. But DO: H1 (d J) ---a H1 ((Di) is zero,hence

CO fD

D

ga

f

gaa

How much does p depend on the given data? 1st, the

additive constant in p depends on the choice of .9 itself,

i.e., the choice of homology basis {Ai},{Bi}. if t is changed,

p will be replaced by c-p. If U is changed, p(z) is replaced

3.157

by p(z+a), a E Jac C2. Thus p really depends essentially only

on C and , though to get a definite p many further choices

must be made. Note that p(-z) = p(z). We can easily identify2g+l

p in our affine model. Let C be given by s2 = f(t) = fl (t-ai).1

Proposition 10.1. In the affine model of Jac C:

Jac C - 0 = {(U,V,W)

let

f= = U-W, degrees as before }

U (t) = tg+d1tg^1+- - .

W (t) = tg+1+WOtg+- - -

2g+1Note that U,+W0 = - ai. Assume t chosen so that D,,"U = V. Then

i=I

for some constant d.

Proof: Recall that f (t) = II (t-a.) and1 iEU

2 2U (t) _ xk V (t) xkyk W (t) 1

ykf1 kEU t-ak f1 (t) kEU t-ak kEU t-ak

If we expand

2U(t) _ I TT(t-aQ) -xkk Rk

3.158

we see that

Similarly,

U1 = jakx]c - jak

W(t) = II (t-a ) + I II (t-a ) y 22 k 2#k 2 k

=tg+l + tlyk - jaJ

t'+

hence

Therefore

WQ = lYk - jak

Ul-W0 = lakx2k - iy

2

k

which proves the lst equality.

Now D0U = V, so we find

or

D.(xk2 ) _ /T xkyk

Yk

i.e., D = 2 x (the derivative of Neumann's dynamical system).

Now in Neumann's system

:k = -akxk + xk Q a2x2 y22)

2

hence

2 2+a2x2 - y2, xk

Now in terms of theta functions

3.159

x = c-9 111 k7

kk

Consider the difference

DOG (log. [ 03) - ( a1A-

4 - ye2)

U j U

The first term equals

D2'3[0](

D.,9[0] 2

0

and the second equals

Working this out,

t+ 1 oxk

2 xk18 ak

(i 2 1 DAkI - D. %,' E0 D2

IX012) 1 V11dO '7 Do lkl 12 2_1Y a,SLo]) -8 y = z 09[1k7

which has at most simples poles at V('9101) U V (3 [rik 1). Since

this is true for every k, the difference has in fact only poles

at V(4(01). But the only functions with only simple poles at

V(-9[0]) are constants, and this proves the second equality in

the Proposition. QED

More generally, we can relate all the functions on Jac C

defined b the coefficients of U,V on the one hand, and by

derivatives-of log.9[0](z) on the other:

3.160

Proposition 10.2. (I) The two vector spaces of rational functions

on Jac C spanned by

a) D DO* log -9 [0] (z) , all invariant vector fields D,and 1

b) the coefficients Ui of U(t) including UO = 1

are equal. This space has dimension g+1, and consists of even

functions with at most double poles at 0.

(II) Likewise, the two vector spaces of rational functions

on Jac C spanned by

a) D D22log-9[0](z), all invariant vector fields D

b) the coefficients Vi of V(t)

are equal. This space has dimension g, and consists of odd

functions with poles of order exactly three on 0.

In fact, for suitable constants c,c' and dkr

D 0 log,-,9 [0 ] ( z ) = CX D + daCO k k

k

D D2 log-9 [0 ] (z) = c PD , for all k E U.ak

Proof: We calculate D D2 log-9 [01 as follows:a

k

Dak D2 log -9 [0 ] =8

Dak (U1-WO )

Dak(U1) - BDak(UI+WO

by Prop. 10.1

2g+lBut UI+W0 = - ai is constant on Jac C, and

1

3.161

hence

V (ak) U (t) -U (ak) V (t)ak t _ akD U =

DakU1 = V(ak) = cluk

for some constant c1 depending only on C. This proves

b D Z log -& [0 l= c 1akak

and hence proves (II). Moreover, as Dakak = Uk' it proves that

fk = is a function on Jac C killed by b..

But fk has poles only on 0 and either fk is a constant or

Dc, must be everywhere tangent to 0 . As this latter is not

the case, fk is a constant, which proves (I). QED

We now come to the main point of this section: we ask

whether we can coordinatize Jac C by using the function P(z)

and its derivatives along D,,:

p (k) (z) Dk p (z)

only. The fact that this is possible was discovered by

McKean-Val. Moerbeke in their beautiful paper*. Not only

is this possible, but this leads to an affine embedding of

Jac C-0 governed by a quite intricate algebra.

To be precise, we fix n and consider the morphism

0 : Jac C - 0 ---- Tn

z > (P (z) , p (1) (z) r ...,; (n-1) (z) )

*The spectrum of Hill's equation, Inv. Math., 30, 1975

3.162

Theorem 10.3. If n = 2g, 0n is an embedding, hence

(i) (z), 0 < i < 2g-1, generate the affine ring of Jac C - 0.

In fact, we may solve for Ui,Vi,Wi in terms of ir(k) and ak,

and for ir(k) in terms of Ui,Vi,Wi and ak by means of "universal

polynomials".

Proof: We shall not find the formulae relating the

{Ui,Vi,WiI and {p(k)I directly, but rather via a third set

of variables {U,V*,W I . Our first job is to introduce these.

We convert the identity

a) f = UW + V2

between polynomials in t to an identity between polynomials

in t-1.

b)f (t)

_ (U (t))(W

(t)) + 1(V (t) 12t2g+l tg tg+l t tg )

A polynomial in t-1, with constant term 1, has-a unique square

root in the ring of power series, with constant term 1, so we

write

2g+1f (t)t2g+1 a.=1 (1-ait-I) - (1 + alt-1+a2t-2....... ) 2

for suitable constants and write

0(t-l) = 1 + alt-1 + alt-2 +.......

Thus (b) can be written:

l(V(t)-t-

2

c) l = f9 l . t- g

O(t 1) O(t) ` 56 ct

3.163

Let U*(t-1) - U(t)-t-g

def (t1)= 1+Uit-1+U2t-2+......

V*(t-1) V(t) - t-gde

$6 (t-1)

= vt-1+v;t-2........

W*(t-1)do 0(t-1)

1 + w*t-1+W1t`2........

so that c) becomes

d) 1 =U*(t-1).W*(t-1)+t-1-V*(t-1)2

Note that the (U*,V!,W*) and the (Ui,Vi,Wi) determine each other

given the a i, by the universal polynomials obtained by equating

coefficients of t-n in:

(1+U1t-1+U2t2+---) = 1+a2t 2+-..)

e) (

V1t-]+V2t 2+,..)V1t+1+v2t-1+

(l+Wpt-

e.5.. U, Ul+alV1 = V1

WOw0+a1

3.164

On the other hand, (d) written out gives a recursive procedure

for finding the W from (U*,V*), viz.

U*+W* = 0

U2+U*W0*+w* = 0

f) U3+u*w*+u*w*+w*+Vi2 = 0

Note that

Un+Wn-1+ (univ. polyn. in 0

-W0).=(U1-WD) =1.

The flow D.0 can be easily written in terms of U*,V*,W*. It

comes out as

or

g)

W* =-(1-8p.t-l)V*

= it(-W*

Vi = f(-Wi+U* i+1-2U1UI*.)

W = -Vi+1+2U*.Vi

1

3.165

These give us, by induction, the formulae:

h)

*4p = U1

4p(1) _ U* = V*1 1

4}x 2) = V1 =2

(W1+UZ-2U' )

= U23

12(using W1 =

4x(3) = U* - 3U*U'*2 1 1

= V2 - 3UiVi

p(2k)= U*+1 + (Polyp. in

p(2k+1)=Vk+1 + in U1,-..,Ukfl,Vll...,

We may solve these backwards:

U1 = 4pV* = 4.p (1)

U2 = 4p(2)+24p2

i) V* = 4p (3)+48 (1)

* _ (2k) (1) (2k-1)

Uk+1- "'lyn' r ... ,z r call thisFk+l(p"P(1),.*.,p(2k)

Vk+l ,p (2k+1)+/polyn. in p,p (1) , - ..'p (2k)) , call this

i

(polyn.

).

3.166

It is easy to set up a recursive procedure which determines the

sequences {Fk},{Gk}. First of all, as

it follows

(10.4) Gk(p,...,p(2k-1)) = Fk(,p,...rV(2k-2)) -

The dot here means this; if F(p,p(1),---,P (n)) is any polynomial,

then:

* (1) ... J, (n+l))= aF (k+l)k =o ap (k)

= V*k

(-W*k + Uk+i - 8p.U ).

2((Vk+1_gVk)

+ Vk+1

V* - 4,p-Vk - 4 (VU*)*

hence

(10.5) Fk + 4,i-Fk + 4 (p-Fk)Gk+l Fk+l

Then (10.4) and (10.5) determine the polynomials {Fk},{Gk},

given the extra facts that Fk,Gk have no constant terms and that

the map

(1).p(2) ..I(E

IV,P(1) ,p(2) r...}

has no kernel except for constants. We also note for future use

3.167

that W* is given in terms of the p(k) by the 2n,d equation in (g):

Wi = Ui+1 - 2V - 8p-Uz1

= Fi+l - 2Gi -- 8p -Fi

Algebraically, we have shown that the 2 polynomial rings

RUrVrW = T [Ul,U2, .;VI , V2 r ... 1r .1 /identities (f)

Rp [p,p (1) ,V (2) , ...]

are isomorphic, by an isomorphism that carries the derivation

of RU*VrW defined by (g) to the derivation of R1P given by

(k) _ P(k+l), and carrying the subring

*,g * * *RU * *

g-1/(First g identities1CC[Ul , .. .rU g rV1 ' .. .rVgrW0' . .

in (f) )

to the subring

Rg = T[P,V (l) , ...,P (2g-1) I.

To finish the proof of the Theorem, note that by (e), the

functions ,...,vg and hence the whole affine ring

of Jac C-e

what we have

Thus 02g is

are polynomials in hence by

just said, polynomials p, p(1)r...rP(2g-1)

an embedding.QED

r

3.168

Still imitating the algebra of McKean and Van Moerbeke,'

we can go further and explicitly describe the equations in

p.P(l)I.--,V(2g) that define o2g+1(Jac C-0) C T2g+1. The

result is this:

Theorem 10.6: I) There are unique 12olynomials without constant

term

Hkk E[P,p (1) IV (2) r ... I

such that

Hkk = Gk.Fi.

In fact,

Hk,kE(n) 1, n = max(2k-2,2k-3), and Hk,O = Fk.

II) if ¢(t-1) = 1 + alt-1 + alt-2 then

02g+1(Jac C-0) is defned by

g+l

I ag+l-kHkk =k+l+g 0 k < g.

k=1

Proof: The method we follow is the most direct one, but it

unfortunately requires a rather nasty computation at the end.

If F(t-1) = is any power series, write

8F = ag+l +ag+2t-1 +......

for the "tail" of F starting at thet_g-l-terms,

and let

F =a0+a1t+...+agt-g

-1

3.169

be the "head" of F. Thus

Now

F = F + t-g-1.6F.

U(t) = O(t-1)U*(t-l)t

= ('+t-g-1. 6O) -

is a polynomial of degree g in tr1. For simplicity, we drop

the * and write U,6U for U*,5U*, and similarly for

V, 6V,fnl, 6W below. It follows:

6( -[T) + 60-10 mod t-9-1,

hence

a) 6U -Vl (6 (i-U) + 60 )mod t-g-l.

This formula enables one to solve for the terms in U* between

t-g-1

and t^2g-1 using the terms between 1 and t-g, given that

U* comes from a polynomial U in t of degree g. Similarly, we

get formulae

b) 6V + mod t-g-1

c) 6W E -r (6 ('W) + -W 9) mod t-g-l.

(In (c), the fact that W has degree g+l makes the formula

have an extra term.)

Now start with values of and ask whether

they give a point of C--G). From these values of

we define the numbers

3.170

Uir...FUg'V1'...'Vg,W0,....W9

by the universal polynomials of Theorem 10.3, hence the

polynomials 6,f and W as well as the one extra number W.

These in turn define unique polynomials U(t),V(t),W(t) such

that

CU (t) t gl ` U mod t.-g-lO(t1)

V t V modt-g-1(t)

O(t-1). g(W (t) t 1 + Wgt-g-l mod t-g-2.

56 (t-1)

The condition that we have a point of Im 562g+1

is that

d1)f = UW+V2

But we can rewrite this condition as

-gd2) 1 (Uwt-gl(W(t)t -g-11 + t-1 (V(t)t0(t-1 )J` 0(t ) ) 56 (t)

Now, mod t-2g-2, we have seen that

mod t-2g-2

U (t)t-g

t -g-lr'CS ( U)+aO-[)mod t-2g-2

56

(t-1) -

V(t)t-9 F - t-g-1 -1 S (gSV) +Sq . ) mod t-2g-2

S6(t-1)

W (t)t-g-1

=_ nl - t-g-1 -1 tS in1 +S . )modt-2g-2

90(t-1 )

3.171

Therefore equation (d2) is equivalent to

(a3) 1 = v-W+t`2-t_`1_1[ . S O-V) +2w"v.s9va CO.%

U- g+2t-'v-a ( i)+2t_' cOjmod t-2g-2

As the terms in t0,t-l,"--,t-g cancel automatically by definition

of the universal polynomials for the W*, this reduces to

(d4) a (-Wt-' ) , 2604.6 (97.-U) 4:U-(6 0-R) -W

+2t'v-6 ( v)mcd t9-1

First look at the constant terms in this equation.

To calculate this, note that 8(U*.W*+t-lV*2) = 0, hence

constant term in d (f 1.FS+t-l72)

-constant term in 6 (U*W*-f3'-V+t-l (V*2 _V 2)

(U9+1+Wg )

Altogether the constant terms give us:

-(U* +Wg*) = 2a g+l +(alU*g+---+agU)+(a1Wg-1+...+a W*)-W9+1 g g'

Since Wg = Wg+cx1W i_1+ . +agWO+rxg+l this reduces to

(e1) ag+1 Ug+1 + alu*,+..+ agUi

which is to be the equation in (II) for k = 0, i.e., set

Hk,0 ` Fk and then (e1) is:

g+1(e2) k1l ag+1-kHk,o g+l

3.172

To get the remaining equations, substitute into (d4)

Wg = W g * + a1Wg_k + - - - + agW0 - Ug+1 _ a1Ug .- _ agU*

and write (d4) as

Z as bus +2!U (S (- w _ (W*+- - .+ag 0 U-* --... -a Tj*) )

+ t-1-V6 ( V5 - rS (UW+t Y) mod t2g-2

Expand this into

1

g+lag+1--kg7ck(",p' .....V (2g))-t-Z(f 2) - I0 ak+1.+gt Q 10

k--

so that the coefficients of each t-P' give us the remaining

equations in the form required. It remains to check that

Hk Q = Gk-Fi . This should have a conceptual proof but it isnot too hard to check by directly differentiating.

We use D for - in this calculation.

Thus to start, note that

D (U) = V

2D(V) = -t(W + t-gr1W**) + t(U + tg-lU** )--BV U9 9+1

D(W) = -V + Gp-t-1(V - V*t-5)

from which one deduces

D(U-W)+t1V) = V-Wgt-g-l Wg+1t-9-1

Likewise, one has

3.173

(a+8pt-1(q Vg -9)) a (U) + W a

D (U S ( W)) = 'V-6W) + U- S[ ( V+8pt(V

D (2ta (sue)) = (- w+t-g-'W9*) + (u+t-'g`'v9*+1) - 8 t-rv) 5 (OV)

+t

Adding these up, we get a lot of cancellation, leading to

d(UW+t- )

g

= V- (Ug+lg) -8C7[ a -t-16 (S6-V)]

+ v La (QSti) -t xa -t-l(tu)I v[ 6 (W) 6

V'(U9+1+alUg+...+ag+l) - U

grlag+l-k(VQU Vk-U*Z) -t-

12.= k=00

Thus

which proves

D (-2NkRWk* R.) = QUk - VkUQ

D (Hk Q) _ U QVk = FR - Gk . QED

The cases g = 1,2 are given explicitly in the table below.

Note that the equation= 0 gives v(2g) as polynomial in the

lower derivatives, so that substituting this, we have exactly

g equations for 02g(Jac C-0) in C2g For g = 1, this is the

usual cubic equation in p,4'. Moreover, higher derivatives

3.174

F0=1

F1 = 4p

F2 = 4p"+24p2

F3 = 4 piv+40 (.,')

H10 = 4p

H20

= 4p"+24p2

H30 =

H12 2 = 8(p') 2+32;x3

H2 2 = 8 (p") 2+96x2-p"+288;4

Tables

G00G1 = 4p

G2 =

G3 = 4pv+l60,p' ,p"+80 +480p2.V '

2H1,1=8p

,1= 16p p"-8 (p') 2+64;3

73,1 =.prr?+8(p")

2 = 16p "piv-8 (p I I I) 2 +96;2 pxv-192pp I p "'+256; (p") 2+192 (p') 2p If

Curve of genus in 1 in C3embedded by

-a2 = al- (4p)+(4p"+24p2)

a3 = al- (8p2)+(16p.p"-8(p')2+64;3)

Abelian surface in C5 embedded by p,p',pof p fit ,piv

-a3 = a2-'(4,p)+a1- (4p"+24p2) + (4piv+40(;')2+80pp"+160;3)

-a4 = +8(p")2+320p2p"+480;4)

-a5 = a2(8(p')2+32;3)+a1(8p")2+96p2p"+288;4)

+ 16p"piv-8WIT)2"'+256;. (p") 2

2+ 192(p'),p"+1920p 3 5p"+2304;

3.175

V(n), n >2g, are expressed recursively in , ... , (2g-1) by

repeated differentiation of the equation k = 0, and substitution

of previous expressions for p(m), 2g < m <n. Likewise, the other

vector fields on Jac C can be given by elementary explicit

formulae. We sketch this.

We use the basis Dk' 1 < k < g, introduced in §3, i.e.,

gD = IP k=l k

for all P E C-(oo), a = t(P). Here D1 is the D

working with. We showed in §3 that

DkUQ = (viuVjui)i+j =k-t Q-1

max (k,2 )j<min(k,i)-1

we have been

Thus in the sum for DkUl we have only the one term j = 0,

and

DkU1= VkUO-VOUk

= Vk .

Therefore

(10.7) 4(DkJ) = DkU1

= Vk

= Vk+a1Vk_l+...+ak-1V1

kQI Gi(p,p(1),...).ak-i

3.176

Thus, in yet another basis Ek, 1 < k < g, the invariant vector

fields on Jac C are just given by

(10.7) ' 1 k(p,p(1),...).

(HereEkP(n)

is given by the rule

EkP(n)

= EkDnp = DnEk = G(n)(p,t(1),...).)

At this point, we have found the link to the famous Korteweg-

de Vries equation. Namely, we have

E2P = P (3) + 12p.p (1) .

This means that if V is restricted to a 2-plane in Jac C

tangent to the vector fields E2 and E1 = D = ', it gives a

solution of the KdV equation

3atf (x, t) = a 3f (x, t) + l2f (x, t) - axf (x, t)

ax

We want to explore this link further in the last section.

3.177

§11. The Korteweg-deVries dynamical system.

As with the Neumann dynamical system, our purpose now is

to introduce a dynamical system interesting in its own right,

and then to show that it can, in some cases, be integrated

explicitly by the theory of hyperelliptic Jacobians. More

precisely, we can, following the ideas in the previous section,

define an embedding of Jac C in an infinite dimensional space:

(Jac-O) --- kvector space R1 of analytic functions

( f (x) defaped in scene neigh. of 0 E cr00 n

z0 ----> p (z0+xe) p tn) (z0) . L,n 0

On Rj, we consider a simple class of vector fields X: those

which assign to f a tangent vector in Xf E TR f = R. given by1'

Xf = P(f,f,.,.,f(n)), P a polynomial.

Integrating this vector field means finding an analytic function

f(x,y) s.t.

ayP tf, ax, aff

axn

By the Cauchy-Kowalevski Theorem, for all f(x,O) analytic in

jxj < c, there exists f(x,y) analytic in jxj,jyj < n solving

this. What we want to do is to set up a sequence XI,X2'...

of such vector fields called the Kortweg-de Vries hierarchy

which1a) commute[Xi,X.] = 0 - we must define this carefully

and b) are Hamiltonian in a certain formal sense, such that

c) for all g, and for all hyperelliptic curves C of genus g:

3.178

Im(Jac-0) = orbit of all flows Xn,

i.e., all n are tangent to image

and a codimension g subspace ofIcn n are even 0 on Image

In fact, d) in some sense "fixing the value of these Hamiltonians"

gives the orbits of the Xn's: we will merely state some results

of this type without proof. Thus {Xn} may be considered an

infinite-dimensional completely integrable Hamiltonian system.

We first investigate what it means for 2 such vector fields

to commute. Let

Xf =

Yf = Q(f,f,---,f(m)).

Then, starting at a function f, the path through f obtained by

integrating Xf is

nf(x,t) = f+tP(f,

2 2I a(k) (f,...).(d)kP(f,...) +......

k=0 of

(because the t-derivative of the RHS is

tP (f, f, )

+ tof (k) -

(d) P (f, f, - ) - - - P (f+ tp, f+t P, ..... ) mod t) .

To go in 2 directions at once, one must be.able to define the

ts.t)-term unambiguously, i.e., the coefficient of s.t in

t P (f+sQ, f+sQ, - .... )

and

3.179

s Q (f+tP, f+tP, )

must be equal. This means

ap k aQ d )kP

of (k) dx Q £a f (k) dx

Theorem 11.2: (11.1) holds if and only if for all f E R1,

there is an analytic function f(x,s,t) (for IxI,IsI,ItI<s) such

that

of = p(f,f,...)atof = Q(f,f,...).as

Proof: The existence of f implies (11.1) by working out

the meaning of the equality of mixed derivatives

apas

a2fs=t=O

asst

_ aQ

s=t=0at

Given (11.1), we define f as follows:

s=t=0

a) let R3 = 0![XO,X1,X2,...] be a polynomial ring

b) let R3 be the map

X1 = x f

Thus R3 is a differential ring if we let Xi = Xi+l, and - isthe homomorphism of differential rings carrying X0 to f.

3.180

c) Let D: R3 -> R3 be the derivation such that

D(X0) = P(XO,XI,...,gn)

D (a) = D (a) -

d) Likewise, let E: R3 be the derivation such that

E(X0) = Q(XO1XI,...,Xm)

E(a) E(a)*.

e) Let

DIE3 (X )

t (s,t) _ I --_-, ; ti si ER3

([S, t1l.i,ja0

Note that (11.1) means precisely that (D,EI = 0 as derivations

of R3, and we check:

Moreover,

a

at(

ap D t(k)=k

aXk

= D(P(...,4) (k),---)).

Now both P(---,cD (k)---) and at satisfy the equations

T = DT, asT = Es

hence they are equal and

at as=Es.

(...,(k),...)) _axe at

and Tlc -0= P(...,Xk,...),

at

3.181

Likewise,

Therefore

satisfies

as

DIES (X )

f (x, s, t) 0 t1 s3iii!

k kat -

p(..-.(x)as

Q(...,(d--) p,...).

QED

Thus the differential ring R1 is very convenient for

integrating flows. However, the Hamiltonians that define these

flows do not exist on R1. instead, we need

R2 = f(differential) ring of C°° functions

f(x) with compact support

R2 has the advantage that there is a large class of functions

(usually called "functionals")

Op: R2 T ,

namely+co

O (f) =J

P(x,f,f,-.-,f(n))dx

where P is a polynomial in f1...,f(n), whose coefficients are Ca'

functions of x, and whose constant term has compact support.

These functionals may be called C"O-functionals because by the

calculus of variations they have excellent derivatives: i.e.,

3.182

5p (f+cg) -O (f) nlim ap (k)= i g dx

e-> 0 e_oo k-0 of (k)

+00k

( 1(ko(-1)k(dx)(afp(k)))'g(x) dx

_(integration by parts).

Define

Sf (-1)k (d d )k (of ap(k) )k=o

to be the variational derivative of p. We want to set up a

co-symplectic structure on R2, and define vector fields V56 forp

these Hamiltonians. These will turn out to be examples of the

same type of vector fields that we considered on R1: but on R1,

they can be integrated locally, on.R2 they come from Hamiltonians.

At the very end of this §, we will mention briefly yet another

approach: that of McKean, Van Moerbecke and Trubowitz, who used

R4 = ring of periodicC00

-functions f(x), and could do both at once.

However, the clearest and most elegant way to bring in the

co-symplectic structure is in a much larger vector space: a

space of differential operators. This approach goes back to

Lax and Gel'fand--Dikii and has been highly developed by M. Adler

and LebedeV-Manin*. Up to a point, we can develop the theory for

any of our differential rings R, but later we will restrict to R.

P4ark . ;,Adler, Ih a Trace Ptmctional for Formal Pseudo-Differential (aeratorsand the Symplectic Structure of the Korteweg-DeVries Tyne Equations, Inv. Math.,50, (1979), p. 219;

J.I. Manin,. Algebraic Aspects of non-linear differential equations, Modern problemsin Mathematics, (VINITI, 1978)

3.183

The central idea is to associate to f E R the differential

operator

(dd{) 2 + f (x)

and to consider R as part of an even bigger space, viz.

RCD ] = vector space of all differential operators

n= n ' dx

`an (x) E R

dI a (x)Dn D = d

In fact, we put this in a yet bigger space:

R{D} = vector space of "pseudo-differential

operators symbols"

dI an(x)D', an(x)E R

IL--00

In R{D}, we can introduce a ring structure as follows:

Note that

D(fg) = fDg + gDf.

Thus as operators on R,

(11.3) DOf = f*D + f .

Taking this as our golden rule, we get a ring structure

on R[D] such that:n

(k ) n+m-kk=O

n

= ky0 kT ak (fDn) *ak (gDm)

3.184

or more generally

Co

(11.4 ) XOY = I k11

aky* aky * = multiply byk=0 as though £,D

commute.

In fact, this extends to R{D} too, if we extend (11.1) via

fop-1 = D-1of + D°foD

hence D-1of = foD-1- D-lofoD-1

= feD-l --foD- 2

+DofoD2

feD-1 - f°D-2 + f6D-3 - D-1afOD-3

i.e.,

(11.3)' D-lOf =[foD-l -foD-2

+ faD-s+...+(-l)kf(k)aD-k-1+.....a

Note that again

D-1 f =k0 k! ak(D-1)*ax(f)

so the general rule for mult. is still (11.2):

XoY = I k' ;nX * axYCo

k=0

Associativity is very easy to check:

(X°Y)°Z =Q aD(I k!

akX * akY) * akZ

+p pk!p! (i-p) !

aD X * axaD- * x

kXa(YoZ) _ 1! aDX * ax

k (I 1-! aDY * ax Z)

1Z ! p ! (k-p) ! aDX * aD ax-pY * x} Z

1 ak+p*

Z-pak* axZ(Q-p)!p!k! D

3.185

Proposition 11.5: For all d E ZZ, every element

X = Dd + a1Dd-1 +... E R{D}

has an inverse

Corollary 11.6.

X-1 - D-d + b1D--d-l.......E R{D}.

The set of elements

1 + a1D-1 + a2D-1 +........

in R{D} is a group , celled the Volterra group by Lebedev-Manin.

Lie {a1D-1+a2D-2+ - - } is a Lie algebra under [ , 3

Proof of Prop.: Construct D-1 by induction. Suppose we

have b1,...,b such that

1+cD-n-1......

Then it follows that

(D-d+b1D-d-1+...+bnD-d-n-CD-d-n-1) m (Dd+a1Dd-l+ , - ) = 1 + (terms in D-n-2 or lower).

QED

For instance, one checks that(D2+q)-1

= D-2 -qD-4 + 24D-5 +.....

The following Proposition is due*, in fact, to I. Schur in 1904,

as P.M. Cohn pointed out to me:

I. Schur, uber vertauschbare lineare Differentialausdrficke,Berliner Math. Ges. Sitzber. 3 (Archiv der Math. Beilage (3) 8)(1904), pp. 2-8.

3.186

Proposition 11.7: For all d > 1 and all

X = Dd + a1Dd-1 +..... E R{(D)}

X has a unique d- root

x1/d = D + b1 + b2D-1 +.... E R{D}

and the commutator Z(X) of X in R{D} is the ring of Laurent series

ncix1/d, ciE T.

Proof: The main point is the calculation:

Lx, cDm I = d61) d+m-1 + lower terms, c E R.

From this it follows by easy induction that Z(X) has, mod scalars

and lower order terms, a unique element of each degree m E ZZ and

that it has the form (cDm+lower terms) c E (C. If Y E Z (X) has

degree 1, Y'E z(x) has degree -1, it follows that YOY' = c+W,

c E T, c 0 and deg W< 0. Therefore

CO

hence

Y-1 = !Y - I (-1) 1W1'/c1 E Z (X)c

,i=0

nZ(X) D {ring of Laurent series I ciYi}

i=-OD

hence "=" holds here because each side has one new element in

each degree. Thus Z(X) is commutative. Finally, X itself is

in z (X) so

3.187

dX= I ciY1, ci E o, cd 0i=-00

and, in a ring of Laurent series, such an element has a unique

dthroot (up to root of unity):

1/d 1/d cd-1 -l cd-2 -2 1/dX = cd YO (1 +

CY + Y

d cd

where the last term can be expanded by the binomial theorem. QED

Returning to the 2nd order operator X = D2+q, we can calculate

in terms of the universal polynomials introduced in §10. In

fact, expand:

17_ CO

D9n(q,4,...)l

n=02+q) -n

where fngn are universal polynomials without constant term,

except for f0 = 1. (Also go = 0). Now

00 9O (fnD"2n) O (D2+ q) n+1 = o(D2+q)

= (D2+q)oV5-1+-q

_ (D2+q).-g , o(D2+q)-n

(fnD_g2) 0(D2+q) n+1

0l

+r

(fnDI-2f x02-fnq 2 - g D)° (D2 ) n

hence

2gn

0 = ((f -g )D + 2f (D+ - 2f f 2 -nn n n q) nqT n4 - -7-) O(D +q)

3.188

hence

(11.8) gn = fn

__ l gnfn+i ng + 2fnq + 4

Thus, relating this to §10, if q = 2pr, then

4nfn(q.q,...) = Fn(2, S.,....)

Gn(2, ..... ).

One may while away an hour or more calculating this out a ways:

n = D + (q - ) e (D2+q)-1 + 13q - 96 ja (D2 t q) -2 +-

D +- 2- D-2 + (q$ ) D`3 + (6q-) D

4 16

mWe now choose our R to be the ring R2 of c -functions

on ]R with compact support. This enables us to integrate elements

of R as well as differentiate them. Many of our conclusions will

however be quite formal and for these we may afterwards go back

to the original R.

in calculations in R2{D}, we find that the coefficient of

D-1 has a very important special property, viz.:

Theorem 11.9 (Adler): For all X,Y E R2{D}, the coefficient a-1

of D-1 in X,Y is the derivative of a polynomial in the

coefficients of X and Y, hence

3.189

Proof: By linearity, it suffices to consider the case

X = aDk, Y = bDQ.

Clearly, if k+Q < -1 or if k > O,Q > 0 there is nothing to check.

We may as well suppose k > 1,Q < -1 and use:

XoY = (kab

(n) Dk+Q-nn

n=0with term

Likewise, is the coefficient of D-1 in

YGX. The difference is ab(k+Q+1)+(-l)k+Qa(k+Q+1)bwhich is

the derivative of lab (k+Q) - a b (k+ Q-1) + a b (k+Q-2)- ,. 1)k+Qa(k+Q) bl.

QED

We define

by

tr: R2{D} p- (C

+CO

tr X = J a1(x)dx, if X = akDk-ao

Now put the vector spaces

R2[ D 1,

in duality by

Lied

[ X,Y7 = tr(XoY) .

In particular, if

d CO

X I a Dn,Y =

I D-n-1b

n=0 n n=0 n

3.190

then

f d<X,Y> =

J

I (anbn)dxn=0

so that R2[D] is isomorphic to a subspace of (Lie )*, the linear

functions k on Lie g . Thus the lie algebra Lie 3 acts on Lie.

by the adjoint representation adX(Y) = [X,YI and on R2[Dl by

the co-adjoint representation. Explicitly, for all Y E Lie

define

ad*: R2[D ] > RJDI

by

<ad* (X),Y2> = -<X,adY (Y2)>1 1

-<X,[Y1,Y21 >.

Let

+:{R2

ID >R2

D

dn nbe the projection ( I a

nD)+ = E a

nD.

n=..-oo n= O

Corollas 11.10: ad*(X) = [Y,XI+

-<X,[Y1,Y2 ]>Proof: <ad* (X),Y2> _1

-tr(XOYImY2-XOY2DY1)

-tr((XoY1-Y1OX)oY2)

_ -tr ([X,Y1I +0Y2)

_ < [Y1,X I+, Y2>.

since deg

QED

2 < -1

3.191

We now recall a very general construction due to Kostant

and Kirillov which has many important applications. Let G be

an ordinary Lie group and its Lie algebra (which is finite

dimensional). Then j has a "co-symplectic structure" on it.

We explain this quite carefully to facilitate the infinite-

dimensional version to be used below:

a) Yx E *, identify Ti*,X =*

hence T*

Then for all a,S E T**,x define

92*(a,$) =<x,[a,9]a

Thus S2 is a skew-symmetric bilinear form on T**I ,X.

b) Now for all functions f,g on n*, we get

dfx,dgX E T**,x namely

< y,df > = lim f (x+ey) -f (x)X E

)y,dgx> = lim g(x-FE6)(W

Hence we define the Poisson bracket:

{f,g}X = S2* (dfX,dgX)

= < x, [dfx,dgx ]

3.192

c) For all functions f ont1*, this gives a vector field

Vf on U. Namely, if x ET, then (Vf)x E T%*,x =

is given by

or

<(Vf)x,s>

(Vf)x

= S2 (dfx,

= <x,[dfx,8]>

= <x, addf W>x

= -<ad* (x),S>x

-ad* f (x) .x

Note that for all functions g on *,

V* (g) def <Vf,dg>

= S2* (df,dg)

{f,g} ,

hence

{f,g}xd +C(Vf)x

0 E=0

d) Moreover, given any 2 vector fields V1,V2, we get their

bracket (V1,V2) = V31 which may be defined equivalently as

V3(f) = V1(V.2(f))-V2 (V1(f)), all functions f on

or directly by:

V = lim V2,xo+sxl V2,xo - lim Vl,xo+ex2Vl,xo3,x0 E £

where xl = Vl,xo, x2 = V2,xo.

3.193

The basic result -that this is a "good" co-symplectic

structure- is that

{f,{g,h}}+ {g,{h,f}}+ {h,{f,g}} = 0

or equivalently

[V'ffv9] = V{f,g} .

We prove this in 2 steps:

Step I: For all a E 41 , let Ra be the linear fcn. on

given by Qa (x) = <x,a>. Then (dia)x = a for all x E and

the definition tells us immediately

Thus, Jacobi's identity on tl gives us (11.9) when

are i 's.a

f,grh

Step II: We prove (11.9) at a point x0 under the assumption

that (df) x0

= 0. It merely states the equality of the mixed2nd

derivatives of f: i.e., let (d?f)x be the 2na derivative:

(d2f)x(Y,z) = aeart f(x+Ey+ r1z)I (0.0)

Then

{g,{h,f}} = ae({h,f}(x0+E(Xg)x

0))Ie=D

2

2a )

asarj,f (x0+e(Xg)x0 + n(Xh)x0+s(X9)x J

2a

= aEar,(xo+C(g)xx +"(Xh)x +0 0

s0, (pt. of -t)}.* dependingin C way on E

ignore this becausedfx = 0

(d2f)x ( (Xg)x r (Xh)x ).0 0 0

3.194

Thus

{g,{h,f}} = {h,{g,f}} and df = 0 = {f,{g,h}} = 0.x0

QED

Rather surprisingly, all of this works without essential change

for the infinite-dimensional case.

R2[D]

= Lie y r

provided we restrict ourselves to an appropriate class of functions

on R2[D]. We use the maps

R2 [D ] ---?P CC

+Go

OP(X) f P (x, - - -,k)...)dx,_CO d

if x = I akDk0

P a CCO functiondepending on afinite number of

the aa) ,k s

The main point is that, as above, ¢p is sufficiently differentiable:

OP(X+EY) = J P(x,...,(ak+Ebk) (Q),...)dx

is a coo function of e; and:

dg OP (X+EY) =J

a I b (PI) dxk,iBak

Ja aPW bkdxak

_ <Y, Dk

Q

((_l)L()i aPrr))>'

3.195

hence

(dop)X = I D-k ((l).L(d)Q 'PM) ELie 3.k aak

By (11.10), the corresponding vector field VP

is just:

(y6P )x IX, DI 2(&)!C

a(Q)

+

-k QaQ ap -x 1dkaQJ(x m {k, k

D' (-l) ( )

ar

x QD(-1) (E)

aq(Q> - dx.`plc 1

The Jacobi identity for { , } and the formula

(V0p,V5Q I = V{56prs6Q}

are proven exactly as before.

We now specialize all this to the submanifold M of R[DI:

M = {D2+q q ER} .

In general, one cannot restrict a co-symplectic structure from

a space. N to a submanifold M unless for all x E M, the 2-form

Sx factors through T*x

Sly TNT X - -- TNr x \PX

f

71

TM,xxTM,x

But if we compute

a,s ELie*we find the following. For all

2+q(a,S) _ < D2+q, lot,a l >

3.196

Let a = D _ ..

D-ls0+D-2$1+...

then [a,al =D-1a0D-1 S0

-D-1$0D-1ao +....

=

D-3 .

higher terms,

so

,*2+q (a, ) = J(&060-0a0)dx

= 2 f &00 dx.

This depends only on a0,$0, which give the restriction to a,to linear functions on TM,D2+q. Thus we have a co-symplectic

structure on M. In fact, it is non-degenerate now. This

non-degenerate 2-form was discovered by Gardner and Greene.

Now for all functionals on M:

OP(D2+q)= f P(x.q,q,...,q(n))dx

we see that using the variational derivative 6P/6q defined above:

(d96p)D +q = D-1.

dq{OP.0Q}D2+q 2 f

-1 -6P(V56 P)D+q[o2+q,

D 6ql+

= D6P 6 - 1 6 P D2

Sq Sq

3.197

We will also have occasion to compute the bracket of 2 vector

fields on M directly. Suppose Pi(x,q,q,-,q(n)) and

define 2 vector fields V1,V2 by the rule

(Vl) D2+q

(V2)D2+q

= P1(x,q,q,...,q(n.))

= P2(x,g,Q,--.,q(n))

then we can compute [V1,V2] directly as in (d) above:

P2 (x,q+eP (x,q)) -P2 (x,q)

[VlV2 )D2+q = limE

(11.12)'

P1(x,q+eP2(x,q))-P1(x,q))- lim

e

0o ap2 (k) aPl_ (k)

k0 aq Pl aq(k) P2

which is a vector field of the same form.

Formulae (11.12) have the following consequence:

Proposition 11.13. Given and Q(x,grq,---,g(m)

if there exists a polynomial H(x,q,4,-..,q(k)) such that

Coq)

then {OP,giQ} = 0.

Caq)

don't involve x, the converse is true. This can be proven

as follows. We use a purely formal result of the variational

calculus in the differential ring

3.198

R3 = (r[XO'X

Theorem 11.14. The se uence

R33

is exact, i.e., for all olynomials f(XO'X1'--.,Xn)'

Sfd X= 0 '"---> f= some g.

Sketch 9_f _proof: Working over Rz, we see that

f = g Of M 0 derivative Sq of 0f is zero.

Since this is purely formal, it holds in R3 too. To prove the

converse, use induction on the order n of the highest derivatives

in f, and argue like this:

8x = 0 f = Xn.f1+f3 3g s.t. f-g E T[XO,...,Xn-1]

f1, f2 E C [X0, .. , Xn-l ] QED

Corollary 11.15. If P(q,4,--.,q(n)), Q(q,q.---,q(n)) are

polynomials, then

{opIOQ} = 0 s polynomial Fi(q,q,-..,q(k)) such that

LpCB gI-(Sq)

Proof:{op,oQ} = 0 implies d{op,oQ} = 0, i.e.,

aq

((5q'\8q) ) - O

hence H exists by the Theorem. QED

3.199

This completes

structure on M. We

Hamiltonians:

1m+

tr ((D2+q) n+1/2)

Expanding / +q as above, we see that

n+3z ffn+l (q,4,---)dx

Note that n-k > 0, the kt term is a differential operator, hence

has'no trace and if n-k < 2, the kth term involves D-3 and lower,

so has no trace. Therefore

Hn

our general discussion of the co-symplectic

now introduce the Korteweg-de Vries

Hn = n+ tr ((D2+q) n (D2+q) n)

n+ I tr (fkD-2k)°(D2+q) n`k.k=0

+1)a(n2+q) -1>n

tr ( (fn+1D_ gT

= 56 (fn+1/n+h)

which is a function on M of the type we are considering. We

want to calculate the derivative of Hn

:

Lemma 11.17.

dq(D2+q)n+'J_

(n+/)[(D2+ql

1 1 '

or

6fn+l- (n+h)f

8q n

--1+q = D ofn(q,q,..).

3.200

Proof: This amounts to saying that for all a(x)E R,

den+35 ,.

_(n+/)tr((D2+q)n-koa)

Write

Then

(D2+q+sa)/ = E+e El (mod E2) .

m

de tr(En a (E+eE1)m)' _ tr(EnoEO...oEmE1OEo---OE)

= m tr (En+m-1p El)'

especially for m = 2, this says

so

dEtr(Eno(E+EE1)m) = z tr(En+m-2oa).

In particular, if n = 0, we get

m-dE

tr((D2+q+sa)m/2/ m tr(( D2+q) 2) 2 0 a ) . QED

Theorem 11.18. a) (VH)D +q = 2gn (q,q, - - - ) _ - f (D2+q) , [(D2+q)n-/

l]

n+

b) {Hn,Hm} = 0, all n,m.

Proof: In fact, the 1s- part of (a) is just the lemma,

and for the nd

[D2+q, (D2+q)n-/

-ID 2 +q,(D2+q) C/

-[D +q, (D2+q) nl ]+

[D2+q, D_1. fn

(q,q, )] +

-(Do.fn-D-1fnD2)+

_ -(fn+fn)

tr(Enoa) = ds tr(Eno(D2+q+Ea)) = 2 tr(En+1oE1)

= -2g n.

3.201

As for (b)

{Hn,Hm}D2+q = /IVH , dHm>D2+qn

- (n+2) [D2+q, (D2+q)n`" ], (m+2) (D2+q)dn1>

_ -(n+z) (m+Z)tr ( [D2+q, (D2+q)+-k ]o (D2+q)m-

but

tr ([A,B ]oC) = tr (ABC-BAC) = tr (B (CA-AC)) 0 if [A,C ] = 0.

These flows VH are the KdV dynamical system. Note thatn

they are defined by universal polynomials 2gn so in fact they

make sense for any differential ring R:

(VH1)D2+q =

q+g6q._ *(VH2' D2+q = ( 4 )

(which integrates to q(x+t)i.e., it is just transl.)

(VH u2+= +q (5)

3 q

etc.

We want to elaborate on the conclusions that we have drawn.

First of all, notice that combining the last Theorem with

Corollary 11.15, we have reproven the conclusion of §10:

for all k,Q, there is a polynomial such that

Hk2 = FkGi

Alternatively, we could have used this to prove {Hi,H.} = 0.

Secondly, notice that the conclusion

3.202

[ VH , VH ] = 0i a

makes sense over any differential ring R even when the Hi don't.

Namely the vector fields VH may be defined by part (a) of

Theorem 11.18, and their commutativity may be expressed by

(11.12)' by the polynomial identity

00

agi (k)CO ag3 , (k)

k10 aq (k)(gi ) =

k0 aq (k) (gi)

In particular, over the ring R1 of analytic functions, it follows

that starting with any analytic function f(x) defined by

Ixj < e, we can integrate any finite set of the flows VH

getting an analytic f(x,tIt...,tn) defined for 1xI,1tl1,." ,1tn1

such that

of =atl

of 6f f-f(3)at

2= 4

of 1 < i < n.

The seond form of these equations given in Theorem 11.18 is called

a Lax equation.' In general, if S is a vector space

and

is a

t - X (t) is, a 1-parameter of operators andway of transforming one operator into another,

equation for the family x(t) is an equation:

of operators X,

(D: S S

then a Lax

n

atx(t) = [x(t) ,(X(t) )).

3.203

The importance of such equations is that they say that the

operators X(t) are infinitesimally conjugate to each other, i.e.,

X(t0+St) = (X(t0)) ) mod 6t2

In good cases, this implies that any sort of spectrum of

X(t) is independent of t.

It is evident that this whole collection of flows on M

mirrors the flows on Jac C, as defined in §10. The precise

link is this:

Theorem 11.19: For a genus g, let C be a smooth hyperelliptic

curve of genus g, and let B,TJ,-,fix be defined as usual. Let

the vector field D on Jac C be written lei a/azi. Define

an embedding

Jac C -- ezo I-

M

> (the operator(ax) 2+2p (z0+xe) )

Then all the flows VEn on M are tangent to the image and VEn

restricts to the flow 4En on Jac C.

Proof: We simply combine the results of §10 and §11.

Note that

E (2p) =Ck (3p,p, - -) = 2. k-1 k-l

k 2 gk (2, 2#, ... ) = 4 (V)D2+2 .

3.204

Corollary 11.20. At the point E M, the vectors

VH span a finite-dimensional space of dimension g. In terms ofn

the moduli ai of C defined in §10, for all k > g

(VHk) D2+2px + 4 (VHk-1) D2+2p +.....+ 4k-l (VH1) D2+V = 0

Proof: Combine Theorem 11.19 and (10.7).

Corollary 11.21. For all C,z0, there is a differential

operator of degree 2g+l which commutes with D2+2p, namely

g+lI a

+l-Q.4Q-g-1(D2+2z)+-

£=l g

Proof: Combine Cor. 11.20 and Theorem 11.18a.

One case where the KdV dynamical system has been explored

much more deeply, first by McKean-Van Moerbeke, then by

McKean-Trubowitz, is over the ring

R4 = C°° periodic real functions on IR.

We sketch their theory very briefly. The operators

xq = (a) 2 + q (x)

with q periodic can be analyzed by the Floquet theory (cf. Magnus,

Hill's equation). In particular, for all h E C*, they have a

so-called h-spectrum:

3.205

h-spectrum = ( set of l s.t. there is an eigenfunction

f (x) withf(x)+q(x)f(x) = Xf(x)

f (x+1) = hf(x)

The fact that the K-dV flows can be written in the Lax form

q-Ed(dx)2+gt(x),((d

)2+gt(x))+_3

or

atXq [Kq ryt t gt

shows by standard results that the h-spectrum is constant as

a.function of t. We may now consider

I(q) = {(h,A) A h-spectrum} c (r 2

which is readily seen to be a 1-dimensional complex analytic

subset such that the projection J(q) D(A-plane) is 2-1.

In fact, for each a ,

f0 (x, a)let be the 2'solutions of f+gf = of with

f1(x,A)

f0(0'A) f0(0,A)

= 0f1(0,A) = 0., f1(0,A) = 1.

f (x+l,A)Then f0(x+l,A)} are again.2 solutions, so we can write them

3.206

f0(x+1,X) = a(X)f0(x,X)+b()L)f1(x,X)

f1(x+l,X) = c(X)f0(x,X)+d(X)f1(x,X)

a,b,c,d entire analytic functions of X such that

ad-bc _ 1.

Then J(q) is defined by

h2_ (X)+d (X))h + 1 = 0.

Let J*(q) be the normalization of I(q), i.e., the 2 sheets

separated at double zeroes of the discriminant 0(X) = (a+d)2-4

if any. Thus a "hyperelliptic curve" J*(q) usually of

infinite genus is associated to this situation. The basic result

of the theory of McKean and collaborators is that for all

J*(q) cc of finite or infinite genus the following sets

are equal:

1) {qlI the branch points of 1*(q1) _(:, I*(q)

are the same, hence 1*(ql) --c' J*(q)

2) {qll j(q1) = J(q) as subsets of T2}

3) the orbit of the KdV flows through q

4) the set of all q such that the IrdV Hamiltonians1

1 1

JF(q1i1i.)dxr

=J

Fn(q,q,...) are equal.

0 0

}

Moreover, this set is canonically isomorphic to a distinguished

component of the subgroup of real points on the Jacobian of C.

3.207

fl. The Prime Form E(x,y).

Given an arbitrary compact Riemann surface X, of genus g,

wouldn't it be handy if we had a holomorphic function

E: X x X --> T. such that E(x,y) = 0 if and only if x = y?

Although such a function doesn't exist, it turns out that it

"'almost" does! To'understand part of the problem and how to

fix it, let's look at the simplest case:

Example. Let X = IP1. The function x-y works on IP1--{-I but

not on all of IP1. So consider instead the "differential":

E (x,y) = x-y/dx /dy

where dx, dy are defined as follows:

Choose a line bundle square root L of and an

isomorphism L02 - 5211 1. Choose a section /dx C r (Ipl- {=},L)

such that ( dx)2 = dx (under this isomorphism). To check that this

is finite on all of IP1xIP1, let x' = 1/x be a coordinate on

IP1- {0 }. Then

dx' -'-dxx

so define v by

Then if x' = l/x, y' = 1/y,

3.208

1 1

R(x,y) _ -x y- =x Y = Xdx/d-y -dy'x' y'

For a general compact Riemann surface X, we will have to

modify this approach in several ways. First, choose an L and

an isomorphismL02 -- Std such that h0(L) = 1. In terms

of divisors, this means we want to find a divisor class

d E Picg-1(X) such that a) 28 and

b) 181 = a single divisor 8

We must show that such a 8 exists:

Lemma 1. 2 S E Picg-l(X) as above, i.e., there exists a

nonsingular, odd, theta characteristic.

Proof. We use Riemann's theorem, and Lefschetz' embedding

theorem. We want to translate the conditions on d into theta

functions:

S exists [S" J E 3 2Z2g/Z2g s.t.

61a) - [8 )(0,S2) = 0

and b) dz$'[a , J(o,O) 0

Now, Lefschetz' theorem states that:

g 2g-1,

z F----- ( .. 4 [ a I (Z' 2) .... 8',6" E ' fig/ Zg

3.209

ris an embedding. In particular, the differentials dz-9 [a ] (o,SZ)

must span the cotangent space. Note that 0 implies61 61] is even, hence S fa if ] (z,c ) is invariant under z 4---> --z,[a

,

hence ](o,S2) = 0. Thus if [VII satisfies (b), it alsosatisfies (a).

By Riemann's theorem, let

QED

a3[a;]=

a l a z (o) u. be ther

unique 1-form which is zero on 6, where corresponds as

above to 6. In fact, 2&, since (S)-6 = (an effective

divisor in K--S, i.e., 6). So t = (/)2, where / is a section

of L. We may think of x as a differential form of weight 2.

This will take the place of v'-d-x.

Next we modify the numerator x-y in E, using a theta functionr

-'9 [a:,, ]) (f Xw) for higher genus.

Definition. The prime form E(x,y) is given by

l r

[arr ](fx wE (x,Y) _

(x) /57YF

where: a) 6 is a fixed, nonsingular odd theta characteristic.6r

b) 6 corresponds to

c) TET, Y7 are as above.

This is a holomorphic differential form of weight on

X xX, where X is the universal cover of X.

3.210

A few remarks:

1. E is not defined on X xX since a choice of path of integration

from x to y must be made. To make this well-defined we simply pull

back to the universal cover.

2. Note, however, that whether this is zero or not only depends

on the image of x,y in X: E(1) = 0 <=- E ( c2,y2) = 0 whenN N

x1,x2)' resp. ylry2 have the same projection to X. Alternatively,

we can consider E(x,y) as a holomorphic section of a line bundle

on x x X.

The following properties make the prime form useful:

Pro2erties of E(x,y). Let x,y E X, x,y their images in X.

1. E(x,y) = 0 x = y. (This is its major property.)

2. E has a first order zero along the diagonal A = X x X.

3. E(x,y) _ - E(y,x).

4. Choose a local coordinate t about x E X such that = dt.

Then

E (x,y) = t (x) - t (y) (1 - O (t (x) -t (y)) 2) .dt x

5. If x or y is moved by an A-period, E(x,y) remains

invariant.

If x is moved by a B-period EmiBi to x',

yE (x,y) = +E (x,y) exp (-iri tmc m + 27ri tm f ri)x

If y is similarly moved to y':Y

E(x,y') = +E(x,y)exp(-iri tmcm - 2rri tm f -W).x

3.211

The main lemma that we need to prove this is:

g-lLemma 2. Given 6 as above, 181 Pi},

i=1yf w) = 0 > a) x = yx

then

or

h) x = some Pi or

c) y = some Pi.

Proof. By Riemann's theorem,

Sfa,,](Yfw)

= 0 sx

y-x+6 y # 0.

Now h°(EPi) = 1, so h°(y + EPi) = 1 or 2.

Case 1. h°(y + EPi) = 1.

So ly + EPii _ {y + EPi}

l Y- x+ EPi 34 0

Case 2. h°(y + EPi) = 2.

4==a either x = Y, or x = some Pi.

By Riemann-Roch, h°(K - y - EPi) = 1, but

K - EPi - EPi

So h° (EPi - y) = 1

sn y = some Pi.

The proofs of the properties above are now quite easy.

For instance, for (1): From the lemma we know:

QED

W yV (- { , ] ( f w) ) = V (x-y) U ( U P ix X) U ( .u x x Pi) .

x i i

3.212

The fact that it vanishes to order one is left to the reader.

But this is precisely why we divided by x), I:

()divisor = I Pi

so (E(x,y)) = A as divisors. For the others, (3), (5) are immediate,

and (4) is just a local calculation. QED

As one. application of the prime form, we will construct all

meromorphic functions on X, as well as the basic differentials:

(a) Given a1,...,an,b11...,b E X such that Eaj Dbi in

Pic X and suppose ai bj for all i,j. Then

n E(x,ai)f(x) = II E(x,b ) is a single-valued meromorphic function

iOl i

on X with zeros = Eai, poles = Ebi.

To prove this, note that all the etc., cancel out,

n w)[s 'so you are left with IT

and now just check

i=1'9 T I(J'w)

invariance under A,B periods.

(b) Construction of differentials of the 3rd kind.

We want Wa-b (x) = the unique differential 1-form on x with

a) zero A-periods

b) single pole at.a with residue 1

single pole at b with residue -1.

In fact, ma-b(x) = dx log E(x,b)' To check this, look

locally:

3.213

wa-b (x) = dx log (t (x) -t (a) ) - dx log (t (x) -t (b) ) + holcmorphicdifferential

dt (x) dt (x)= + holomorphic differential.t (xx))- -t

t(a) - t(xx))- -t

t(b)

(c) Construction of differentials of the 2nd kind.

We want na (x) = a 1-form on x with

a) zero A-periods

b) double pole at a E x.

Note that such an na is unique up to a multiplicative

constant.

Consider:

A

w (x, Y) = dX# y log E (X, y)

This is a well-defined 2-form on x x x, since

dX -dy log[ f (x,y) g (x) h (y) ] = dx dY log f (x,y) . For each fixedy = a, by choosing a basis for the tangent space to x at a, it

restricts to a 1-form on x equal to the above n a(x) up to a

multiplicative constant. In this manner, we can construct

differential 1-forms with any allowed divisor of zeros and poles.

3.214

92. Fay's Trisecant Identity

We now come to a very fundamental identity between theta

functions that holds for the period matrices of curves, but not

for general period matrices. Although the basic ideas behind this

identity go back to Riemann, it was not clearly isolated until

Fay made his beautiful and systematic analysis of the theory of

theta functions (J. Fay, Theta functions on Riemann surfaces,

Springer Lecture Notes 352, 1973).

Theorem [Fay, op. cit., p. 34, formula 45]. Let X be a compact

Riemann surface, X its universal covering space, ,9(z) its_,

associated theta function and E(x,y) its prime form. Then for

all a,b,c,d E 5, z E Tg.

c d

(z + f f w}E(c,b)E(a,d)a b

c d+(z + J w),E(c,a)E(d,b)

a

c+d

.(z +J

w). U (z),E (c , d) E (a, b)a+b

This type of identity is very special. The theta function

on general abelian varieties doesn't satisfy identities like

3cig (z+ai). 9 (z+bi) = 0.

J.--L

The proof of the theorem falls into several steps, each of

which is straightforward but sometimes tedious.

3.215

Step I. Check that all three terms satisfy the same functional

equations and are differentials of the same type. This way all

three terms will be sections of the same line line bundle L on

the space X xX xX xX X Jac(X).

Step TI. Next show that if both terms on the left are zero, then

the right hand side is also zero.

Step III. Let Dl,D2 be the codimension one subsets where the

two terms on the left, respectively, are zero. Then for all

components D3 of D1f, intersection is generically transversal.

Step IV. H1 (X X X X X X X X Jac (X) , L-1) = 0.

Step V. Assume: X smooth complete variety

L line bundle on X such that H1(X,L-1) = 0.

t,sl,s2 E H0(X,L) global sections s.t.

a) sl = 0, s2 = 0 are divisors D1,D2 without

multiplicity.

b) For all components D, of D1 D2, the

intersection is generically transversal.

c) s1 (x) = s2 (x) = 0 t (x) = 0.Then: 2X1,X2 such that

t = X1s1 + X2s2.

Stet VI. First, let a = b, secondly let b = c, to see that the

constants are one, finishing the proof.

3.216

We will not go through all the details but instead touch on

all the main points:

Ste : Everything is invariant under z 1j z+n, n E Mg.

z {-y z + Stm, M E Zg, the 3 terms are multiplied by

e-TrI tms2m- 2 Tr i tm (z+ ?w)a

e-7ri tmnm-2rri tm(z+bw)

and

e-Tri tmS2m- 2 Tr i tm (Z+bf W)

e rri txnm-27ri tm (z+lw)

c+de-7ri tmS2m-2Tri tm(z+a+bw) e-Tri tmQm--27ri

respectively, which are equal because

c d c d

fW + fW = fW + fW

a b b a

c+d

a+b

Under

on (X)4. Among the many substitutions in a,b,c,d, we consider only

c 1-> yc, y E Trl (x) . Let n+S2m be the period defined by YNote that the half-order differentials r;-FXT are sections of a

line bundle on X, hence are invariant by all such substitutions.

Thusc

E (Yc,b) = e-7r1 tmS2m r e-2Tri tm ( w +6") -2Tri tn 6' E (c,b) .

Collecting all the factors, you find that all 3 terms are multiplied

by 2ce-2Tri tmRm-27ri tm6"-2Tri tn. S *_2Tri tm(z+A+bw)

3.217

Std: One must look at all 16 combinations of one of the

4 factors of the 1st term with one of the 4 factors of the 2nd

term. Most combinations are obvious, e.g.,

E(a,d) = 0, E(c,a) _.0

dr

A(z + Jw) = 0, E(c,a) = 0b

E(c,d) = 0c+d

e(z + f w) = 0a+b

A slightly less obvious case is when

c c

e(z + Jw) = 0, B(z + fW) = 0.a b

If Dz is the divisor of degree g-l on X defined by z, this means

IDz+c - al 34 0

1Dz+c-bj 7-4 0

Then either a = b, or IDz+cj is a pencil or IDz+c-a-bl 0.

Therefore either a = b, or JDzj 0 or IDz+c+d-a-bl 0.c+d

Therefore either E(a,b.) = 0 or 0(z) = 0 or 0(z + f w) = 0.a+b

Step III: Let's look at the generic transversality ofc .c

8(z + fW) = 0 and 8(z + fw) = 0. We can ignore loci ofa b

codimension >2. Recall that the differential dz0 vanishes at

z = zo if and only ifI Dz I

is a pencil; and if JD ' is ao zo

single divisor Fz , this differential pulls back on X to the0

unique 1-form wz zero on Fz . Thus the loci where0 0

3.218

c cd1e (z + f w) = 0 or dze (z + (w) = 0

a b

are the loci where. IDz+c-al or IDz+c-bi are pencils: these

have higher codimension and can be ignored. we may also suppose

that of the 3 alternatives: (1) a = b, (2) IDz+cl pencil and

(3) IDz+c-a-b) 34 0, exactly one holds. Let wa, resp. wb, be

the 1-forms on X zero on the divisor in ADZ+c-al, resp. IDz+c-bI.

cIf Wa 34 Wb, this means that the differentials of e(z + fw)

acand 6(z + fw) in the z-direction are independent. If a = b,

b cbut wa(a) 0, this means that in the a--direction e(z + fw) has

ca

non-zero differential while 6(z + fw) has zero differential. Inb

both cases, the intersection is transversal.

But now there are 3 possibilities

Case 1: a = b, IDz+cl is one divisor, IDz+c-a-bj = 0.

Case 2: a 34 b, ADZ+cl pencil, IDz+c-a-bl = 0.

Case 3: a 76 b, IDz+c) one divisor, IDz+c-a-bi # 0.

It is not hard to show that in case 1, wa(a) 0 while in

cases 2 and 3, wa wb.

Step IV: Look at the projection

X XX XX XX xJ

p345

X xXxJ

where a,b vary in the fibres. L restricts to each fibre p315(z)

3.219

to a line bundle of the form pll(M1)®p21(M2) where M1,M2 have

positive degree. By the Kfnneth formula, Hi (L-1I ) _ (0) ,fib7rei = 0,1, hence by the Leray spectral sequence H1(L-1) (0).

Step V: Use the exact sequence

0 - LT1 (D 0X ---> 0(sl,s $z)

s1

where is the subscheme s1 = s2 = 0.

Step VI; Obvious.

Next, we want to give a geometric interpretation of the

identity.

In fact:

a) use 1201 to map Jac(X) to projective space.

The image is called the Kummer variety.

b) in this mapping, the trisecant identity will tell us

that the images of certain sets of three points

(C04 of them!) are collinear, i.e., the "Kummer

variety" has oo 4 trisecants.

First, we need to know what 1201 consists of:

Lemma.

1201 =the set of divisors of the form

c' )(zrz) = 0 ) r[n nI E g/'Zg

for all (c) E C 2g

Proof. First, it is easy to check that ([}z,-2

) = 0) E!201

3.220

We have 9 [n 01(z + n, z) [ ](z, 2)

and 9 [n ](z +nm, e-4Tri tnn. e-2Tri tmnm-4rri tmz-,[01{z,n z

Since the transition functions are the squares of those of l0I,

V(9[° ](z, a)) E 120

Next, we must check that these span 120l. One way is to find the

dimension of H0(20): it will be 29. Since the 9[o](z,2) are all

linearly independent, this shows they are a basis. To find thet

dimension, let f(z) = I a(n)e2Tri nz E H°(20). Thus:nEMg

.f (z+nm) = e-2Tri tmnm-4Tri tmz f(z)

This gives us a formula for a(k):

a(k+2m) = a (k) . e-27r tmom. k,m E Zg .

This gives us an upper bound of 2g for the dimension, which is what

we wanted.

Moreover,'recall from Chapter II, the fundamental:

Addition Theorem.

QED

2-g 1 ,&[07(x, ](y,2) .nE g/Z n

3.221

The geometric interpretation of the theorem is

Geometric Corollary. Let 1201 define

Then, Va,b,c,d E x, the three points

c+d a+c b+c

z f w), c(2 J W), w)

a+b b+d a+d

are collinear.

Jac (X) -->

Proof. The map 0, by the lemma above, is given explicitly

as

We want to use the trisecant identity to give us one relation.

Let V = vector space spanned by-9103(Z' 2) ,

Let Q = symmetric bilinear form on, V with these ,9 [n](z, 2) as an

orthonormal basis.

Now note,

1) For any a E Qg, i9(z+a,c),i9(z-a,c) E V.

(For instance, use the addition formula to get this.)

2) Let Ta .4(z,St) = -)g(z+a,S2). Then,

v f E V, Q (Ta$ - T* \9, f) = 2-g - f (a).

(Just check this on basis elements f (z) =.9 [n J (Z' 2) .

By the addition theorem, both sides are 3(a, 2)).

Now we can apply the trisect theorem. In the theorem, make the

substitution z f-->z - 2J

. We get:

a+b

3.222

b+c b+c

LHS = C 9(z + 2 J9(z_J)a+d a+d

a+c a+c

+ 2 J)(z_J)b+d b+d

c+d c+d

RHS _ c3,9(z + 1 Jw)

S(z -- 2 j w)a+b a+b

Note that these three products of 91s are of the form of the

function in notes 1,2. Let f (z) =,g Cn ] (z, 2) for any r1

2Mg/jg

Apply Q(_,f) to the equation to obtain

nb+c a+c

clg[o.](2 J Wr 2) +J

W.

a+d b+d

c+d

c3 [TO) ](2 J w 2a+b

where cl,c2,c3 are independent of ri. QED

3.223

§3. Corollaries of the identity

In this section we will study what happens to. Fay's identities

when the 4 points a,b,c,d come together in various stages. The

result will be identities involving derivatives of theta functions.

First, we need some notation. For the following formulas, let

a)

b)

z E Cg

a,b,c,d E X with distinct projections to X

c) (z) the theta function of X

d) for every a E X,"and local coordinates t on X near a, we

expand the differentials of the 1st kind:

3W = ( v t ) dt

j=0 j

and let

vj = (vi,,...,vgj).

(Note that the mapping

X >egx -)-x - J wa

is given near a by

CO+

tj+l

t 4-- V.( 7+l)j=0

3.224

We let

a

Da = constant vector field vo- az (i.e., voi azi

D = constant vector field v1 3ta

constant vector field v2

b bf

e) We abbreviate fw to J.

a a

The identities we will prove are:

(1) (Fay, Prop. 2.10, formula 38);c

(Z+ 1) , (Z+ 1)D log = c + c

b -9 (z) 12

(z+ C) - (Z)

where wa_c(b) = cldt (t a local coordinate near b)

E (a, C)2E(b,c)E(a,b) = c dt

(2)

DaDb log (z) = c1 + c2(z)2

where w(a,b) = _cldta dtb

(Fay, Cor. 2.12, formula 38):

19 (z+ f) &(z+

(ta,tb local coord. near a,b resp.)

1 2 = c2 dta dtbE (a,b)

3.225

(3) (Fay, Cor. 2.13, p. 27) :

D4 log tiA (z) + 6[ D2log,' (z) )2 - 2D D" log (z)a a a a

+ 3 D' 2 log'- (z) + cl D2 log- (z) + C2 = 0a

where c1,c2 are constants depending on the Taylor

expansions of E(a,b) and c)(a,b).

As explained in Ch. II, there are 3 ways to get meromorphic

functions on XQ from :

--a as productsTr9(z+ai)

7T-O(z+biif I ai -- 1 bi

as differences of logaritignic derivatives D log (z+a). (z)

-!P. as 2nd logarit snic derivatives D D' log ,9 (z) .

The above identities give basic identities between meromorphic

functions formed in these 3 ways. Identities (1) will appear as

the limiting case of the trisecant identity when d >b, while

a,b,c are still distinct. Identity (2) will appear when in (1)

we let c --- * a while a,b are still distinct. Identity (3) will

appear when finally we let b >a.Before proving the formulas, we need the following lemma:

LemmaE (x,b) I 1

: a) d x E x,a)x=b

= E(b,a

3.226

b) W (x) E (x, a) d E (x,b)a-b E(x,b) x E(x,a)

c) dxa_x(b)

x=a= -w (a, b)

(To prove (a), use the local expansion of E near x = b; (b) and

(c) are restatements of the definitions.)

Proof of formula one:

We want:

(lb)

C C

Dbo (z + f) -9 (z) - IDb, (z) IS (z + fa a

c c bf) (z) +c219(z+ f)e(z+ f )a b a

Take the trisecant identity, and divide by E(c,b)E(a,d):

c d(z + f )s(z + f

a b

C dE (c,a) E (d b) j fz + J )+ f jE (c, b) E (a, d) l

b a

c+dE(c,d)E(a,b) fE (c, b) E (a, d)

a+b

Now differentiate w.r.t d (as scalar functions on our

R.S..), and let d -->b:

3.227

9 (z +CJ c b).D,(z) + E(c,a) d E(x,b) I -(z + J)9(z + f)b E(c,b X E a,x Ix=b

a b a

c\E (a, b) d E (C, x) 9 (z)9(z + f)E (c,b) x E(a,x)

x=ba

c

+-7(z) f)a

Now use the lemma (a),(b) to get:

c

J)Db9(z) E (c, a)z + + E(c,b)E(a,b)a

19

cz + J)t9(z +

b

a-c (b) J(z ),- (z +

This gives us our formula.

Proof of formula two:

We want:

(2b)

c cc

1) + 9'(z)Dbi9(z + f)a a

[ (z)] $(z) - Da& (z) .Db.9 (z)b a

zcl-9(z) + c2'5 (z + f) (z + f)a b

Now, take formula (lb), differentiate w.r.t. c, let c -.> a,

while noticing that cl,c2 in (lb) are not constants w.r.t. c,

and in fact both vanish when c = a:

3.228

[DaDb.9 (z)'1,9(z) - Da-9 (z)

do W a-c (b) I c=a S (z)

E (a, c)+ do E (b,c)

1E (a,b)

c=a

a b

+ f P(z + f) -b a

But now use the lemma (a), (c), to get:

[DaD O(z)),&(z) -- Db&(z) - Da,} (z)

= -w(a,b) ti9(z)2 + 12

,4 `zE (a, b)

which is exactly what we wanted.

Proof of formula three:

From (2) we have:

a

D z

(3b) E (a,b) 2 [DaDb log 9(z)]9(z)2 = -W(a,b) E (a,b) 2 (z) 2b b

+-az 19

z

a a

The idea now is:

Let a,b be in the same coordinate neighborhood,'t(a) = 0t(b) = t

and expand(3b) in terms of t, and pick off the first non-trivial

term, which will be the t4 term!

3.229

(1) Locally we have:

wi = (j=o vi t 1 dt3

7

bt2 t3

jwi = vi0t + 11 2 + v.2 6 +...a

t3 11 ab=i t(j-0 Vij Jaz

xDa + t D'a + 2 Da +---

E(a,b) = (t + clt3 + c2t5 +...) -1

dt FOT

E (a,b)2 = (t2 + 2c1t4 + (2c2+c2) t6 +- .. ) dt(o1

) dt

Calculating from the definition, one easily checks:

w (a, b) _ ( - 2c1 + (6c1 -12c2) t2 +- ... ) dt (o) dt.

Hence

w(a,b)E(a,b)2 = 1 + (3c1 2 -- lOc2)t4 +......

(2) Locally the term E(a,b)2 DaDb logi9(z) is

(t2 + 2ct4+- ..) (D2 log 14 (z) + t DaDalog 14 (z) +

+.. )2 DaDa log &(Z)

t - D2 DaD. log i9 (z )2 log9(z) + t3

+ t4 [2c, Da log-9 (z) + 2 DaDa log 9 (z) l +-.

3.230

(3)

1 ]

-& (z+6) --(z-S) -$(z) + 1 16ii9i (z) + 3 6 6 (z)LLLL i 7 ij

1 g6i6 6i9 (z)+...I+ 6

i,jfk=1 3 k ilk

g g9(z) - 6i 9 i (z) + 3 ij ij (z)3=1

ga i6.6kki.k(z) +---

.1r] rl

(z) 2 +4 6.6. (-8(z) -Si. (z) - (z)cg. (z))

i,3=l 1 3 1 D

Now expand in t via 6i =

+ Sk6Qf12(z)'7ijki(z) -irj rkrlC=a

+.. .+ JJw.(t)dt:0

S(z)2+t J. J(v v..+v. V 9ij Y3 2 io31 30 il 13

+ t v.o o kovko([2kr =1 ijkQ- 3 i)3 jki + 4 aij kQ)

2Let '9i az1'0 r -tlij azaz. , etc. Then

(iov 2 ilv'1 vizv.o)(,g .

irj=1 6 4 3

3.231

+t2(Vl D'log+ t4(1 D4& - D D3$ + !(D2, )2a a s 12 a 3 a a 4 a

+

3

(DaDalog t9) ' 2 +4

(Da21og9 )_323 .

Now substitute what we have in (3b). Remarkably, the ti-terms for

i < 4 cancel and the t4 terms give:

2.2c1 Dalog19 + 2DaDa log - (z) 29

= (10c2 -

Da!' 3'DagDad+

4

(Da2 log 5) _j2.

Use the following lemma:

44 2 D2log f + 3[ D log f lLemma.

2

D -2

2 Df D3f + 3 (D2f) 2f2 2 f2f

Proof: Completely straightforward.

We have:

2c1 ' D2 log 'j +6

DaD" log,4

(10c2-3c12) +

4

Da2 log-'- + 12 Da

4log-

+2Da iog 9

QED

3.232

As in §2, these analytic identities have a geometric

interpretation in terms of the Kummer variety c(Jac(x)) c IP(29`1)

Geometric Corollary ]P(2g'"1)y of (1) : Let 1201 define Jac (X) a

Then for all a,b,c E X, then the ima es under c ofc

1) the point2 Jw

ac

2) the infinitely near point (2 f w) + s-Db

a+c

3) the point2

f w

2b

a

are collinear, i.e., there is a line in IP (2g-1)tangent toc _ _ .. ..,.

O(Jac(x)) at

c(1

2 fw) along the direction Db and meetinga

0(Jac(X)) a1 a+c-,(2

f w) also.2b

Proof: This is clearly the limiting form of the Geometric

Corollary in §2. Alternatively, we can write (1) as

Dby) [3(+) - (z-y)

a

where

c C a+c a+c

(zr+ 2 J).9(- z f) + c2 (z+22b) -2 f

a a 2b

(y)Db = voi a/ayi , and wi(b) = voidt

3.233

Applying the addition theorem and Q as in §2, this gives us

c a+c

](y, 2)c

= cl9[°](zJ,

) + c2.9 [° ](2 L, 2 ) .

Jw aa

where cl,c2 are independent of 1. QED

cWhen c approaches a, J) approaches (O) which is a

a

singular point of the Kummer Variety. In fact, the local coordinates

in x(29-1) at (0) all pull back to even functions on Jac(X). In

this situation, elements of Symm2(TJac(X),0) define tangent vectors

to 4(Jac(X)) by the formula:

if (Aij) is a symmetric gxg matrix, let tAbe the

vector at cp(0) given by

2

to (f )

=1 az9 .

(0) iji,7 1 7

for all coordinate functions fa on IP (2g-1) near 4 (0) .

In particular, if a,b E X, then we get a tangent vector t

by

t(a,b) (f a) = DaDb

(faoW0) .

(This is the case .A.. = 1 (v.a) v.b) + v.a) v(b)) , where1] 2 1,O ],O ],0 10

Wi(a) = v, 1,0a)dta, wi(b) = v(1,0b) dtb).

a,b)

Geometric Corollary of (2): For all a,b E X, the point (0),a

the vector the point 0( fw) are collinear.b

3.234

Proof: To see that this is the correct limiting form of the

previous assertion, note that (2) can be rewritten:

Da(Y) DA(Y) , (z+X) . 9 (z-y)Y=O

b b

=c15(z)2+c29(z+ J)i(z - f).a a

Applying the addition formula and Q, we get

](z,2)z=0

b

c2-&[on )(f ui. 201a

where cl,c2 are independent of n. QED

A different limiting case of (1) is when c approaches b rather

than a. Analytically, the constants c1,c2 will approach -, but

geometrically the meaning is that (Jac(X)) will have a point ofb

inflection at 02 f w). This has been used very effectively bya

Welters and Arbarello-de Conchini in their work on the Schottky

problem: cf. Introduction.

Another interpretation of formula (1) shows how the Riemann

surface x is intertwined with the function theory of Jac(X). For

a,c E X, let Va,c be the vector space of second order theta functions

on Jac(X) spanned by the functionsc c

19 (z + 2 J.i9z - 2 f )a a

c c

+ z f - 2 f (za a a

c

az. tg(z + 2 f ).a

1 < i < g.

3.235

Lemma: dim Va,c = g+l.

Proof: If not, then for some constant c and vector

D = I ai a/3zi there would be an identity:

tc c' c c

2 J)9(z - 2 f(

(z +2

J).D(z - 2 f)a a a a

c c19(z-2f).D(z+zJY.

a ac

Let w = z Then

a

c

(w) = 0 (w + J).D(w) = 0.a

cSince (w + f) 54 0 for almost all w such that -9 (w) = 0, this

a

means that -&(w) = 0 D 19(w) = 0 which we have seen never

holds unless D = 0. QED

Using Va,c and formula (1), we can recover X as follows:

locus of decomposableCorollary: V f (functions

a,c (z+e)-9(z-e),eE1C

set of fungttions l c+a+ 2 f ) 1 z - 2 f

2b 2b

b EX

cone over X.

3.236

Proof: n: This follows from identity (1).

Suppose -9(z+e).-g(z-e) E Va,c. Note that if

c C

+ 2 J) _ S (z - z J) = 0 then all functions in Va,c vanish.

a a

Therefore

c c

(z + 2 J) _ (z - 2 J) = 0 --- (z+e) = 0 or -9 (z-e) = 0.a a

c c

Substituting z + i J for z and e - f for e, this says:

a a

c c(z + J) _.(z) = 0 (z + e) 0 or 19 (z + J -- e) = 0.

a a

We will show that if this holds, then e

substituting back, is what we want.

Our hypothesis can be written

(*) 0ftO(c1 = 0e U 0c

\(f w} ifw-e)a a

c

.1.b

or e =

b

wJwa

which,

where Of is 0 translated by f. Next, use Riemann's theorem to

express this in terms of divisors. To fix notation, let z E Mg

define the divisor class D(z) by

Wz

= zJ

D (z)

and let 6 be the divisor class of degree g--l such that

z EO > ID(z)+61 0.

3.237

Let Dz = D (z) +6. So, z E S n do Dz 0 and I Dz + c - a l 0.

a

Let W = set of divisors Dz such that z E Ol ®c . Clearly W

Jag-2

contains the subset Wa = {divisors Do + a: Do = Qi}. ouri=l

hypothesis (*) tells us that:

Dz E W' - either I Dz+D (e) l 0 or IDz + c -- a - D(e) I 0.

Since Wa is an irreducible set, it must lie entirely in one of

these two sets:

g-2QiD = DO+a E W either ID0+a + D ( e ) l 34 0 V D0 =

i=l2

or ID0+c - D()I 0 V D0 = 1

The following lemma then finishes this proof.

Lemma: If D(e) is a divisor of degree zero such that for all

g-2Do

ID0+a+D(e)I 3 0

then D(e) . b-a for some b E X.

Proof. Left to the reader.

So, we have used formula (1) to construct the cone over X,

and hence X. We can ask whether we can also use formula (2) to

construct X. As a possible approach, start out as above, and let

3.238

V0 be the vector space spanned by the functions

-3(z) 2,9 (z) a-a a\9-

az.az. - az1 J 1 71 < i < 7 < g.

As above: a) V0 c vector space of second order $.functions

b) dim VO < l + 9(9+1).

decomposition fConsider v0 (1 (z+a).

unctions

)

Formula 5.2 tells us that this contains the set:

br b

{$(z + 1 ) ,3(z - f) some a,b E X}a a

which is isomorphic to a cone over Symm2 X.

Question 1. Are these two spaces equal?

This would follow, as above, from the following question:

Question 2. If D is a divisor class of degree 0 on X such that

for all divisors E of degree g-l for which JEl is a pencil, then

either JD+EI # 0 or ID-El 0, then does it follow that

D n a-b for some a,b E X?

3.239

§ 4. A lications to solutions of differential equations

The corollaries of Fay's trisecant identity can be used to

construct special solutions to many equations occurring in

Mathematical Physics. In this section we will consider the

following equations:

1) Sine-Gordan: utt - uxx = sin u.

2) Korteweg-de Vries (K- dV) : u, + uxxx + u ux = 0.

3) Kadomtsev-Petviashvili (K-P) :uYy

+ (ut+u +u-ux)x = 0.

Many other equations also have solutions constructed via

theta functions, such as

4) Non-linear Schr6dinger: iut = uxx +

5) Massive Thirring model, i ux = v(l + uv)

i vy = u(1 + vu),

but we will not consider these here (for the non-linear Schrtdinger

equation, see the PhD thesis of E. Previato, Harvard, 1983).

We will give some solutions in terms of 9 -functions to the

first three equations. In the last section, we will indicate

how one uses the generalized Jacobian to relate our solutions to

the famous "soliton" solutions to the K-dV equation.

The easiest solutions to obtain are some for the K-P equation.

Corollary. For all a E X, 12 D2 log9 (z0 + xv0+ti/3yv1 -2tv2)+2c1

satisfies k-P, where:

c1 is the constant appearing in formula (3), §3.

vi = (vij,...,vg7)

wi = Ivi7 t3 dt, (t a local coordinate near a)

3.240

Proof. Take Da of formula (3) and set u(z) = D2 logS(z)to get:

D4 u(z)+ 12D2u(z)-u(z) + 12(Dau(W))2 + 2c1D2a u(z)

- 2DaD" u(z) + 3 (Da) 2 u(z) = 0.a

Let v = 12u+2c1 = 12 D2log.9 (z) + 2c1; then:

3 D'2v(z) + Da(D3v(z) + v(z)-Dav(z) - 2 Da v(z)) = 0.

Finally, note that by definition,

Dau(z) = ax u(z +

D'au(z) = ay u(z + y -vl) ,

I-)

Thus

Dau(z) _ u(z + t-v2).

v(z0 + xv0 + y v1 - 2tv2)

solves K-P, as wanted. QED

In order to find solutions to KdV and Sine-Gordan, we need

to consider hyperelliptic curves. Let X be hyperelliptic,

T: X >IP1 the double cover, and let is X ;,X be the

involution.

Let a E X be a branch point of "?T and let t be a local

coordinate about a such that the hyperelliptic involution i is

just" t +--- e -t. But i*wj = -wj (see Ch. (IIIa, I2.) so ifwj = Vi (t) dt, vi (t) dt + vj (-t) d (-t) = 0. Thus vi is an evenfunction of t, hence vjl = 0 and D' = 0.a

3.241

Corollary. 12 Da loge (0 + xv0 + tv2)) + 2c1 satisfies KdV

where:cl, v0,v2 are as in the previous corollary,

and a E X, X hyperelliptic, and the local coordinate t

a satisfies i*t = -t-

Proof. Take Da of formula (3), and use the above fact that

DI = 0 to get the result.

Next we would like to tackle Sine-Gordan. Recall from Cb. lila

that if a,b E X are branch points, then fw E 2 L i.e., if

b a

a,b are branch points, fw = (n+cm) for some n,m E Zg,

a

To solve Sine-Gordan: Let X be hyperelliptic a,b E X branch points.b

Start with Formula (2). Substitute z ---.;- z + f and subtract thea

original formula:

b'9 (Z+P

DaDblog -= c2 b 2

g(z+ f)a

b , b b(z+ f) (z+ f -- 2 f )

a a a--(z)2

b

It f W =z

(n+.SZm) and get, using the functional equation fora

b(z+ $) -Tri* n -2Tri mz i9 (z) 2

DaDb log = c2 e e b -(z) -0 (z+ f

2'-(Z)

a

19(Z+I 2

bg(z+2f }S(z)

a

Ti 2Tri mz a-e e .'9 (z)

3.242

b9 (z+ f ) b

Let u(z) 2i log (z) - 21r tm(z + 2 J); then

iu(z)-e_iu(z)DaDb u(z) _ -

c 2 = -4c2 e- Tri - tm Sam+ r i tmn

a

So u(z) satisfies DaDbu(z) = C;-sin u(z). Thus for any zo, the

function

v(x,t) = u(z0 + x(a2 + t(a2b))satisfies

a2 32v (x, t) --

av (x, t) = cz . sin v (x, t) ,

where a,b are proportional to (w1(a),...,wg(a)) and (w1(b),...,wg(b))

respectively. We pass over the interesting question of when v and

c; are real and what these solutions "look like".

3.243

§5. The Generalized Jacobian of a Singular Curve and itsTheta Function

In this section we will define and describe the generalized

Jacobian of the simplest singular curves: the curves obtained

by identifying 2g points of IP l in pairs. We will then determine

their theta functions and theta divisors. Finally, we will apply

this theory to understand analytically and geometrically the limits

of the solutions to the T(dV equation that were discussed in the

previous section, when the hyperelliptic curve becomes singular of

the above form.

Let C be a singular curve of genus g, and let S = Sing(C).

Suppose the singularities of C are only nodes pl,...,p9 and that C

has normalization 7: IPl >C. If 7-1 (pi) = {bi,ci}, i = 1...g,

this means that c is just IP1 with the g pairs of points {birci}

identified. We assume - bi "i Vi. Now, in general we define

Pic C =

group of divisors D = E nixi, xi E C-S

mod: D "S 0 if D = (f) forscam f E T (C) , f continuous and

finite, nonzero at each p.

In our case we can pull back to fl 1 and we get

group of divisors b = E nixi, xi E IPl - 7r-1(S)

Pic C mod: D 0 if D = (f), f E C(IP

and f(bi) = f(ci) for all i = 1...g

We define Jac(C) to be the piece Pic°(C) of Pic(C) corresponding

to divisors nixiof degree 0, i.e., Eni = 0. The structure of

3.244

Jac(C) is easy to work out n start with D of degree 0. As a

divisor on IP1, it equals the divisor of zeroes and poles of some

rational function f. The ratios f(bi)/f(ci) represent the

obstruction to,D being zero in Pic(C). It is easy to verify that

they set up an isomorphism of groups:

Jac(C) > ( *)g

f(b1) f(b )D I > (f(c1) ,..., f(cg /

As in chapter IIla, we can add to any divisor D the divisor

x-x0 and get a family of divisors D+x-x0 depending on a point x near

x0. Letting x approach x0 this gives a tangent vector to Jac C

near D, and as D varies, an invariant vector field Dx on Jac C.0

For later use we can work out this vector field in terms of

coordinates X1, ... on (T*) q:

If D = (f (t)) , then D+x-x0 = (f (t) t_x ) ; hence the0

coordinates of D+x-x0 in Jac C are

?x0

f (ci) -

Then

b.-xf(bi).

b i-x0

c.-x

ci-x0

f (bi) ci-x0 bi_i- T . brx0 - (c x)

= x.Y

b.-c.

= 0 (b-)x0 (c1- 0

3.245

Thus the vector field Dxo is given by

Dxo it (bi_x0)ci_x0) `i aai -c

Now, Jac C is not compact: we want to construct a natural

compactification of it. N.B. This will no longer be a group

however! It is clear what we need to do to compactify: we need

to allow the support of our divisors to approach the singular

points. But considering divisors Enixi, arbitrary xi E C does

not work very well. We need to encode the "multiplicity" of the

singular point in a more subtle way. This is done as follows.

In general let

Pic C =set of coherent oC-module

up to isomorphism

Translating this to more down-to-earth language, this becomes

kset of all divisors D = n.x. along with finitely generated

i=1I idC,xi modules Mx. c C(C), (xi E C) are arbitrary, where

if xi is not singular, MX is simply t ni-x

local coordinate near x

and if xi is singular, ni is determined via:

r(,C,x

ni = dimM

ti---- - dim 1inoC'x. Mx MC,x.

to

By convention, b& = dC,x if x t {xl;...,xk}.

mod: D N D" if 3 f E (C) such that x = f 1, t1x E C

3.246

The module, Mx can be thought of as a refined way of measuring theY

multiplicity ni at the singular points: we will call them the

multiplicity modules. Pic C always has a natural structure of

projective variety but let's just think of it as a set.

In our case of g nodes, we know exactly what the Mx 's must

look like:

Lemma. If p E C is an ordinary double point obtained byY

glueing two points b,c in a smooth curve C, then for all M c T(C)

which are finitely generated OP ,C-modules, either:

a) M= f -@p,C for some f E VC)

or

b) M = f- r(j'p C for some f E VC), where dP C = Ob,C.rnOc,(5,

normalization of (,P,C-

Proof. Let Mk'k = iciodule of functions f such that

ordaf a k, ordbf > £,. Take k,i the largest integers so that

M c Mk,Z Then almost all functions f E M satisfy ordaf = k

and ordbf = Q. So choose such an f E M. We have

f-(DP,C c M c Mk,R

But now Mk,i = f-M0,0 and M0,0 is just OP,C. Moreover, 0P ,C

is the subspace of @rP,C defined as {gJg(b)=g(a)} so it has

codimension 1 in 0P,C. Therefore

dim Mkrk/f (0P,C = dim f -UP,C/f qPC

= dim P,C/OP,C = l.

So either M = f-dP,C or M = Mk ,Z = f-P,C, as wanted. QED

3.247

From this lemma, we get immediately:

Corollary. For any subset T c {P1,...rP9}, let

CT = [C with Pi separated into bi and i1 for i E T3 = CIP1 with b.,ci

identified for i T3. Then, as a set:

Pic C = fl Pic (CT)T

(f = disjoint union)

Proof: In fact, divide up all divisors D = {Znixi,Mi}

according to whether their multiplicity modules are isomorphic to

(OPTIC or &Pi01C = (Dbi,tP1 fl ddiIIpI at each singular point.

For each subset T c {P1,. . Pg,}, let Pic(C)(T) be the set of

D whose multplicity module is bP C exactly for Pi E T. We

claim:

Pic (C) (T) Pic (CT) .

In fact, if D E Pic(C)(T), then when Pi T, Pi singular, there

exists an f. such that PIP = fide C It's not hard to see thata a.'

one can choose a single rational function f such that this holds for all such P

Let D' be defined by the multiplicity modules f-1. MP. It defines

a divisor on CT with "trivial" multiplicity (np,C

at all thei

singularities of CT. Two such are equivalent in Pic((:) if and

only if they are equivalent in Pic(C) because the condition

f (bi) = f(c) in the definition of equality in Pic (CT) is the sameas the condition

C = (0P C included in the definition ofQ

equality in Pic(C).

3.248

Actually, we can be much more explicit, and make the degree 0

component Jac C into a compact analytic space as follows:

Theorem Jac(C) (IPl) g/.....)

with equivalence relationn

(Wk.Xl...., ...,C g g) ,., ( "0'... 1, g), for all k.

kthspot kth spot

where i'i

(bib .) (ci-c . )ij = (b-c -b.) .) (c

Sketch of proof: Fix some n > g and let

S = nordered sets (xl, ... , n) : xi E IPl ; for each i, a at most one j s.t. xjE {bi,c.

Define two maps

(IPA') g Jac (C)

n bl^xiby 7'1(xl,,..'xn)`

nlcl-xi

1T2(xlI...,xn) _ (the divisor

the multiplicity module is

at pi).

n b -x.1 \ , andcg ji=1

y where

i(the maximal ideal of

The following things are not hard to prove:

a) if n is sufficiently large, e.g., 2g, then

surjective

if xi = bj or c.,

functions zero

7, . 72 are

b)Tr2

is constant on the fibres of 7l so that there is a

unique map gyp: (IPl) g >Jac (C) satisfying cp°'Tl = 7T2'

3.249

c) (A is independent of n and defines an isomorphism of

(IP l) g/- with Jac (C) .

d) (A restricted to (C*)g is the isomorphism

(C*) g -->Jac (C)

defined above.

e) More generally, if T{1,---,g} is any subset,

h = g-#T and E: T. --a{O,w} any function; then p

restricted to

H {s(i)} x H C*iET iET

c (mil) g

is the same isomorphism of (C*)h _..Jac(CT) Jac(C)

up to multiplication by a constant in (C*)h

The idea of the crucial step b is this:

Say Tr1(xl,---,xn) = Tr 1(yl,...,yn), and

xi,yi E ]Pl - U {bk,ck}. Letk

fi(t) 11 (t-x.)/ JH (t-y)l<i<n 1 l<i<n 1

xi yip

then the hypothesis says that

f (bk) = f (ck) ,

hence

all k

in Pic (C) .(ixi - n-°°) ^' (Eyi - n-)

3.250

In Step C, the w1 s come in because for .any x2 , - - - ,xn S IlP k{},we have v2(bk,x2,..-,xn) _ IT 2(ck,x2' 2'..." x and

b . n xibl . bk-b n xi-bbk n \ bk-cl i=2 xi g

ratioWkl

The details of the proof are not central to the exposition and

are omitted.

Several points in this proof are useful below. Firstly, note

that Jac C has one "most singular" point at infinity, namely the

point corresponding to (X1,...,X ) where all ki are either 0 or =.

We will call this P... Secondly, the map Trl enables us to construct

an analog of 0 for Jac C. To do this, let's calculate

dim Trl Xg)

Let Tr1(xl,...,xg) = (al,...,ag). Up to an undetermined

constant, let cp(t) = c TI(t-xi), where if xi = = that term

is omitted. So deg ((p) < g. Write (P (t) _ IcDit he (pii=0

depend on xi,...,xg and determine uniquely up to

permutation. Now,

3.251

Ak for k = l...g .

So

g i b') = 0 for k = 1...g.i =0

7rll(l1,-.-,1g) is given by the set of solutions in of these

equations so

dim Trll (al, - - . , ag) = g - rank (Akci - b )

i=0...gk=1...g

In particular, iT1

is generically 1-1.

Next, let us determine the analog of the theta divisor 0

using the above. We want equations for the locus where the

divisor I xi - W is effective. From the discussion above,i lthis is exactly when deg 0 < g-l, i.e., g = 0. Over a given

point there is such a if and only if:

1-A 1 .. r . 1-A

detb1-A1cl . bg- gcg

bg-1 - cg-1. cg-1

1 1 1 . g g g

= 0

This determinant is the analog of -9 and its zeroes, as a subset

of (]P1) g/- or via (P as a subset of Pic C) , are the analog of 0.

3.252

We shall call this function TC has a useful expansion.

First recall the Vandermonde determinant

1 ... 1

det

a1 a2 ag

a2 a2 ... a21 2 g II (ai-a.)

i>j

g-l g-l g-1)a1

a2a3 J

In'the above determinant, this enables us to work out the

coefficient of the H A. term:iES 1

IT (c.-c.) H (b-b.) II (b.-c.) (-1)#S - a(S)i.>j 1 3 i>j 1 3 icy . 7

i,jES i,j S jES

where a(S) = the sign of the permutation changing 1..g to

(S,{l..g)-S) and preserving the order of each set (e.g., a({l,31)=-1).

TC, therefore, can be expanded:

c.-b(*) TC = IT (b.-b.) (-l)#S If (A. fl -Z ]I w 0.

i<j 1 jES jaibi -j

i<j

iji,jES

Note that the worst boundary point, p.0 = is not on 0, and

correspondingly, det(0,...,0) 0.

I claim that this determinant is also a limit of theta

functions of our non-singular curves C. Formally, we can see a

link as follows:

3.253

Let 0ij(t) be a family of period matrices in which

Tm Hii (t) -- > oG

and

as t ----> 0, 1 < i < g,

Stij(t) are continuous for It l< E, if i j.

Then consider .9(z, Q (t)) . The limit of this function as t -- >0

will be just 1. A better thing to do is to translate the functions

by a vector depending on t first:

Let 80(t) = diagonal of H(t); then

(z - 6, (t) , St (t) ) =eTri tmSkn+2iri m (z - 62 (t)

)

Y II eTFl (mini) 03- ai(t) ITe27Ti m.mj"ij (t) .e2lra t

mEXg i=l i<j

As t ->0, this function approaches:

m =(ml " ... ,mg)

m.=0 or 11

II e2Tri.mizijt2ij (0) e2Tri mz

i<j

I ITe2h1 ij (0) -

II- e27rizi

S={l..g} i<j iESi,jES

Now if

2Tr i S2J

(0) = (bi-b .) (ci-c . )' - =(bi--c (ci-b Wijj

3.254

and

e2Trizi = _ 1 - ITi

clJ#i bi-b:

it equals TC up to a constant. In fact, if Ct is a family of smoothcurves of genus g "degenerating" to C, it can be shown that its

period matrix behaves exactly like this. Correspondingly, in the

lattice Lq (t) =229 + c2 (t)2Lg, the B-periods S2(t) zg go toinfinity, but the A-periods 2Zg remain finite. Thus

(Cg/7Lg+S2 (t)Zg tends to Cg/Zg, which is just (C*) g withcoordinates e27'1zi, We do not want to describe this in detail,

referring the reader to Fay, op. cit., Ch. 3.

In the limit, is there anything left of the quasi-periodicity

of -& with respect to its B-periods? At first it would seem not

but there is, in fact, something. In fact, the three methods by

which we formed from -9 meromorphic functions on Xn now give us

rational functions on the compactification Jac CC which are

continuous maps

Jac C - (codim 2 set of indeterminacy) ---> ]Pl

The point is that the induced rational maps

(3P l)g - (codim. 2 set) ----> IP l

are compatible with the equivalence relation - of the above theorem.

Let's check this for the second logarithmic derivative with

respect to the invariant vector fields Xi a/aai, Xj a/aai, i.e.,

xi aa. - j aa. (log TC(Xl,...,Xg)).3

3.255

Note that this is the analog ofap a

r of-b] 1 JII If k irjr then

jai 1 7

lim aiaa. . aj

Ak-roo 1 J

l -}W 1k

be abk =

= lien a, -3-.9X

Let

a.a

[log a' + log (-1) -#S 1 (1 if kEsax ] S wi. - IIa-j l

i<j iES ` 1-l1rjES ilk ak if ks

= ai aa. A. a (1og (-1)#S

IT wi. w it a'.]1 j S with i<j iES-k iES-k

kES i,jES-k

-a a [log (-l) #S+1fl w 11

j S with i<j ij iEskE S i,jES

lim Xi aA- aj aa.[log wgkAg)]

ak+0 a.

xi)

Now let's apply this to give solutions of KdV. We want C to

singular limit of hyperelliptic curves. This occurs if

-ck for all k. In fact, when this. is satisfied, if t is the

coordinate on P1, let

x = t 2

y = t ]T (t2-b. ) .i=l

3.256

Then x(bk) = x(ck), y(bk) = y(ck), and the 2 functions x,y embed

the singular curve c-{-} into a2. The image is defined by

y2 = x - II (x--b.) 2, which is a limit of equations y2 = f2g+l (x)

for smooth hyperelliptic curves of genus g.Recall from the beginning of this section that the invariant

vector field on Jac C associated to a point x0E C is

bi - cia

bi--x0 ci-x0

If bk = -ck, then

In . -.r-2

D - 2- i 0 -I-x

0 i 1--b? x0-2 i

= 2- (x-2 bia + x4 biaiaai -a

+...).

The vector field associated to the point at infinity is therefore:

D.biai aa. r

a. i

and the singular p-function is:

2 2 #Sbi-b 2 b +bi

TC(al....,Ag) = ( biai ,.) 1og (-1) II b b) - II (xi - II b bSCl..g it j 3 iES jai

i,7ES

To obtain a solution to KdV, we need merely substitute

(ei+bix-2b1 t)Xi e

3.257

b.+b.for any el,...,eg; or absorbing the factor - n in the ei,

j74i ] 1

f(x,t) _ (ax)2log

IT

i<7i,jES

bi b. 2 (ei+bix-2b3 t)(-L-1)

) iESe

These are precisely the g-soliton solutions of KdV.

The famous asymptotic properties of g-solitons (that for

t << 0, it splits up into g widely separated blobs, which interact

for moderate values of t, and which for t >>O split up again into

the same g blobs, with the same shape but with a phase shift) can

all be deduced very simply from the above formula and the fact that

D2 log TC extends to a continuous function on the compactification

Jac C described above of the generalized jacobian. To get a real-

valued function f(x,t), assume that all bi are real, and define

c: 2((C*) g Jac C c Jac C

by

Then

b.-b. e +b x-2b 3t0(x,t) (... _]T

ijai b3-+ e

f (x,t) = (D2.log TC) (a (x,t)) .

As shown above, D. log TC extends to a continuous function on

Jac C. In fact, it is zero at the "most singular" point P

given by letting all coordinates ai on .((E*)g tend to 0 or

To see this, write

3.258

aSXSS-{i,",g}

Then if

bS I b.S iES

D2 logaS-(EaSbS?S)2

g TC -(EaSaS)2

Note that all termsX2S in the numerator cancel out while for

every S, the denominator has a a2S term since aS 0. Thus

D2 log TC(P0) = 0.

Therefore, for all e > 0, there is a neighborhood UE of P.

in Jac C such that:

P E UE: ID22 log TC(P)I < E.

Therefore, there is a constant c such that if

x-2bit I> C, all 1< i < g c (x, t) E UE

If(x,t)I <E.

Thus the effective support of f(x,t) is a set of g bands

x-2bi t I < C

representing "blobs" moving with distinct positive velocities 2bi.

3.259

Moreover, if t ---a +oo and we stay in the loth band, then

x--2bi t I ----a Co , i iand lim a(x,t) will lie on the is h 1-dimensional strata

Ji0

= {(A1,...rX

g)Iai

0

E T*, but ai C- {0,-} if i 34 i0}.

In fact, fix the value z.= x-2b? t and let t - >+- . Then for1 0

some choice of e iE {0,00}, (i # i0) ,

3.260

lim a(x,t) = (elf-.rei-l' Ai , Ei +l1 ...rE9)

t-r-OO 0 0 0

lim Cr (x, t) = (e 11 r ... rEil'1' Ai r

Ei1+1' ... fegt->+co 0 0 0

where Ai depends only on 2Z. By the theorem describing how0

(]P'-)g is "glued" together to produce Jac C, we see that for some

constant fl.10

(E1r...,Ei 0_1, A, Ei0

+11...re9)

n .

(Ellr...,Eil-l,e i01,

ei1+1r..,re;1), all A

0 0

(- meaning equality in Jac C).

Therefore,

lim f (x, t) = lim f (x, t)t+-00 t

x-2b . t= x. x-2bt-,7-+° i ° (r1 i °/i ° )

i.e., for t ->- or t -- -, f(x,t) has the same shape in each band except

for a phase shift. The fact that this shape is a single "wave"

moreover is more or less a consequence of the simple fact that on

each 1-dimensional stratum Ji , the rational function TC(A) tends0

asymptotically to (l+A)AS (in a suitable coordinate A E (C*),

i.e., up to the scale factor AS, has a single negative zero. When

you set A = ebx and take logarithmic derivatives, f will have a

single pair of complex conjugate poles closest to the real axis and

these give its wave'shape. More generally, the zeroes of TC on

Jac(C) give poles of f(x,t) but only for complex values of x,t

and f will have large values along the real points near these poles.

3.261

Resolution of algebraic equations

by theta constants

Hiroshi UMEMURA

The history of algebraic equations is very long. The

necessity and the trial of solving algebraic equations existed

already in the ancient civilizations. The Babylonians solved

equations of degree 2 around 2000 B.C. as well as the Indians

and the Chinese. In the 16th century, the Italians discovered

the resolutions of the equations of degree 3 and 4 by radicals

known as Cardano's formula and Ferrari's formula. However in

1826, Abel [1] (independently about the same epoch Galois [71)

proved the impossibility of solving general equations of degree

5 by radicals. This is one of the most remarkable event in

the history of algebraic equations. Was there' nothing to do in

this branch of mathematics after the work of Abel and Galois?

Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can

solve the algebraic equation of degree 5'by using an elliptic

modular function. Since = exp((l/n)log a) which is alsoa

written as exp((l/n) (1/x)dx), to allow only the extractions1

of radicals is to use only the exponential. Hence under this

restriction, as we learn in the Galois theory, we can construct

only compositions of cyclic extensions, namely solvable exten-

tions. The idea of Hermite and Kronecker is as follows; if we

use another transcendental function than the exponential, we can

solve the algebraic equation of degree 5. In fact their resulta

is,analogous to the formula P-a = exp(l/n) (1/x)dx). In the1

3.262

quintic equation they replace the exponential by an elliptic

modular function and the integral r (1/x)dx by elliptic

integrals. Kronecker [15] thought the resolution of the equa-

tion of degree 5 by an elliptic modular function would be a

special case of a more general theorem which might exist.

Kronecker's idea was realized in few cases by Klein [11], [13].

Jordan [10] showed that we can solve any algebraic equation of

higher degree by modular functions. Jordan's idea is clarified

by Thomae's formula, §8 Chap. III (cf. Lindemann [16]). In this

appendix, we show how we can deduce from Thomae's formula the

resolution of algebraic equations by a Siegel modular function

which is explicitely expressed by theta constants (Theorem 2).

Therefore Kronecker's idea is completely realized. Our resolu-

tion of higher algebraic equations is also similar to the

aformula exp((1/n) 5 (1/x)dx). In our resolution the expo-

1

nential is replaced by the Siegel modular function and the

integral ` (1/x)dx is replaced by hyperelliptic integrals.

The existance of such resolution shows that the theta function

is useful not only for non-linear differential equations but

also for algebraic equations.

Let us fix some notations. We follow in principle the

covention of Chap. II. Let F(X) be a polynomial of odd degree

2g+l with coefficients in the complex number field C. We

assume that the equation F(X) = 0 has only simple roots so

that Y2 = F(X) defines a hyperelliptic curve C of genus g.

Then C is a two sheeted covering of JP1 ramified at the roots

of F(X) = 0 and at -. Let be the roots

3.263

of F(X) = 0. Let us set B' = For two subsets

S, T of B', we put. SeT = S 0 T S n T. n2i__1 is defined

ithplace2

as the 2xg matrix

ith place(0...0 0...0

and Ti 2i is the 2xg matrix

For all T c B, the sum E nk is denoted

by q T. Classically the period matrix S2 of C is calculated

with respect to the normalized basis of H1(C, 7Z) in §5, Chap.

=. Thus Q is determined when we fix not only F(x) but also

the order of its roots. Finally we put U =

the subset of B' consisting of all the odd numbers of B'.

For row vectors m1,m2 E ]Rg, z c Cg and a symmetric gxg matrix

T with positive definite imaginary part, we define the theta

function 0[ml] (Z' -U) = E (+mi)t(Z+m2))m2 e2J g

mwhere e(x) = exp(27ix). The theta constant 0[m1](0,T) will

2mbe denoted by 0[m](T) for short.

2

Theorem 1. The following equality holds;

x1-x3x1-x2

(1 ::i0-p

_ (e2(Q)4e lp 2 .... 0J (0)4 + e

[ 0, (a2)4

0 ... 0 p 10 (Q) 40[ 2 ( Q ) 4) / (20(2 1 (Q) 4e 2 2

0...0J ' 0 ... 01 l0 ... 00 . 0](0)4).

7 2

3.264

The theorem is deduced from Theorem 8.1, §8, Chap. III, by

carrying out precisely the calculation indicated in the proof

of Corollary 8.13 and form the formula 0[m1 1](z,T) = e(m1E2)t

m 2

+2

e[m1](z,T) for c Z39 (see for example Igusa [9], Chap. I2

§10, (0,2) p-49). In fact for a division B = V1 U V2 u {k}

with #V1 = #V2 = g, it follows from Theorem 8.1, §8, Chap. III,

4 #(U-(V 2+k) )(1.1) 6 [n (V +k) oU] (St) = c (-1) IT (xi-x,)

2 itN2+k, j ev1

because (V2 + koU)oU = V2+k here the union V2L{k} is denoted

by V2+k. Theorem 8.1, 98, Chap. III for S = (V1+k)oU gives

#(U-(V1+k))-l(1.2) e [n (V +k) oU] (S2)

4 = C (-l) E (xi - xj )1 icv1+k,jcV2

Dividing (1.1) by (1.2), we get

40 In (v2+k) oU] (n)

e In (V1+k) oU] (0) 4

II (xi-xm)#(U-(V2+k))+#(U-V1+k)) iEVI+k,mcv2

- (-1)JI (xi-xj)

ieV2+k,jcV1

IT (xk-xi)iEV2

TI (xk- x i)icV1

Let us consider a division B' ={1,2,3}u{2nl2snsg}i.{2n+112sn5g}.

Putting V3 = {2nj25nsg}, V4 = {2n+112sn5g}, we apply (1.3) for

k=l, V1=V3+2, V2= V4+3;

4 IT (xx.)0 Ln

(V4+3+1)oU] (S2) iEV4+3 1- 1

II (x -x.e Ln (V3+2+1) oU] (S2) icV3+2 1 1

3.265

Next (1. 3) for k =1, V1 =V4+2, V2 = V3+3 is

)(x111-xi8 Cn (V3+3+1) oUl (SZ)

4ieV3+3

(1.5)4

= (x -x.)'e [n (V4+2+1) oUl (S2) iEV4+2 1 a.

Multiplying (1.4) with (1.5), we get

e [ n (V4+3+1) oUl (52) 46 [ n+ 3 + 1 )

1 ( 2 )(2) 4 (xl-x3) 2

( V] ( Q ) 4 (x1-x2) 2'

For the above division B' = {1,2,3},u. {2nj2!5n5g}u {2n+1j2snsg}

if we interchange 1 and 2, then (1.6) becomes

(1.7)

e [n (V4+3+2) oUl (SZ) 4e [n (V3+3+2) QUl (S2) 4(x2-x3) 2

4 4e n (V3+1+2) oU] S ) e n (V4+1+2) oUl S) (x2--xl) 2

We notice the following identity,

(1.8)x-x3 1{1 + ( x 1 -x 3 ) 2 - x 2 -x1 3)2}.x1-x2 - 2 x1-x2 x2-x1

It follows from (1.6),(1.7) and (1.8)

(1.9) xl-x2 = (6 In (V3+2+1) oUl (52)46

[n (V4+2+1) oU] (S2)4

+ e [n (V4+3+1) oU] (SZ)4e

[n (V3+3+1) oU] (S2) 4

- e [n (V4+3+2) oU] (SZ) 4e [n (V3+3+2) oU] (S2) 4)/

( V

The theta characteristics in (1.9) are half integral. Theorem

now follows from the following formula: for in Mg,,

3.266

®[m21 +11 (z,,r) = e(m1 2)e [m11 (z,T)2 2 2

We notice that by the transformation formula, the right

hand side of the equality in Theorem 1 is a Siegel modular func-

tion of level 2 (see Igusa [9), Chap. 5 §1, Corollary).

A marvellous application of Theorem 1 is the resolution of

the algebraic equation by a Siegel modular function.

Theorem 2. Let

(2.1) a0 0, ai E T (Osisn)

be an algebraic equation irreducible over a certain subfield of

T, then a root of the algebraic equation (2.1) is given by

0:::i.((1 1 O...p 0.011. 0 0...0(2.2) (e [2

0)4010 ..... 01(S'Z)4 +0 OJ(S2)46 2.... 0]()4

0 0

(0461 (Q)4)/(20{: (SZ)4eiT 21

(2)42

0 0 0

where is the period matrix of a hyperelliptic curve C : X2

= F (X) with F (X) = X (X - 1) (a0Xn +a1

Xn-l +...+ an) for n odd

and F (X) = X (X - 1) (X - 2) (a0Xn + alxn-"l +. -+ ate) for n even.

More precisely let be the roots of equation (2.1).

Then S2 is the period matrix of the hyperelliptic curve C

with respect to the classical normalized basis of x1(C, 7Z)

when the roots of F(X) = 0 are ordered as"f"ollows : for n

odd xl = 0, x2 = 1, xi+2 = ai (lsisn) and for n even xl ="01,

x2 = 1, xi+2 = ai (lsisn), xn+3 = 2. The root a1 of equation

(2.1) is given by (2..2).

3.267

Proof. It follows from the assumption that the equation

is irreducible over a subfield of C, F(X) = 0 has only simple

roots. Since (x1 -x3)/(x1 - x2) = x3 = al , Theorem 2 followsfrom Theorem 1.

To determine the period S2 we have to number the roots of

the algebraic equation. Even if we don't know the precise roots

of the equation, the numbering can be done once we can separate

the roots of the algebraic equation. The complex Sturm theorem

says that there exists an algorithm of separating the roots of

the algebraic equation (Weber [19], 1 §103, §104). Therefore

Theorem 2 is a resolution of an algebraic. equation by a Siegel

modular function. Compared with the formula = exp((l/n)log a)

a= exp((1/n): (l/x)dx), in our theorem the exponential is re-

1.

placed by the Siegel modular function (2.2) and the integral

(1/x)dx is replaced by hyperelliptic integralsSa105isg-1 which determine the period Q.

Let u:s compare our Theorem with the result due Hermite [8],

Kronecker [15]. and Klein [12] on the resolution of the quintic

algebraic equation by an elliptic modular function. Their

theory s,ti.cks to the modular variety of elliptic curves with

level five structure (cf. Chap. 1). Let H be the upper half

plane and Fn be the principal congruence subgroup of level n

{( a a).ESL2(2)Ib c 0, a d = 1 mod n}. rn operates on

H' in,usual way and the quotient variety H/rn is the modular

variety of elliptic curves with level n structure. The func-

tion-field (G (H/rn) has, a model (Q (H/rn) over W and the

3.268

morphism Tr: H/ Fn -r H/ I'1 descends giving an inclusion Q (H/ T'1)

'-, Q(H/rn) (see Deligne et Rapaport (41). The natural projec-

tion H/rn + H/F1 is a Galois covering with group I'1/±In'

Therefore H/I'5 -r H/I'1 is a Galois covering with group r1/±r5

which is isomorphic to.the alternating group ot5 of degree 5.

Since H/I'1 is a rational curve and its coordinate ring Q[H/r11

is a polynomial ring Q[j(w)], Q(H/I'5)/Q(H/I'1) is a one para-

meter family of Galois.extensions with group 615. The key point

is this family contains any Galois extension with group z5 in

T. To be more precise,c since°t5

has a subgroup of index 5,

there exists an extension (resolvent) Q(H/I'5) F Q(H/I'1)

with [F, Q(H/r1)1 = 5. Moreover one can show among such re

solvents there is a particular one described explicitely by

using the Dedekind n function : There exists a resolvent of

degree 5 of Q(H/I'5)/Q(H/F1) given by an equation

( 2 . 3 ) w5 +b1

w4 +b

2w

3 +b3

w3 +b

4w + b5 = j (.w) , bi c Q (lsis5)

and the solutions wi(w) (15is5) of equation (2.3) are expli-

citely written by the Dedekind r1 function. Now given a general

quintic equation over a subfield k of T

(2.4) X5 + a1

X4 + a

2X

3 + a3

X2 + a

4X + a5 = 0, ai E k, (lsis5) .

Then it is easy to see that by a Tschirnhausen transformation

involving only the extractions of square and cube roots, the

resolution of the given equation (2.4) is reduced to that of

(2.5) X5 + b1

X4 +b

2X

3 + b3

X2 +b

4X+ a5 = 0

where a is in a solvable extension of k(ai)1<i,,5 obtained

by adjunction of square and cube roots (Weber [191, 1 §60, §80,

e

3.269

981) . Next we look for a point w0 c I - i such that a; = b5 - j (w0) .

This procedure depends on elliptic integrals. Recall for an

elliptic curve C : y2 = 4x3 - g2x - g3 the modular invariant j

of C is equal to 26.32.g3/(g3 - 27g3We solve in T b5 - a;2 2

= -27b 2) for unknowns a,b. This is done by ex-

tractions of a square or cube root. Then the period w0 of

the elliptic curve C : y2 = 4x3 - ax -b is calculated by ellip-

tic integrals 1/ 4x 3ax-b dx for suitable paths y and,7 Y

j (w0) = b5 - a5. Therefore wi (w0) (1s i s 5) are the solutionsof the equation (2.5) hence the given equation (2.4) is solved.

If we try to solve a quintic equation by Theorem 2, it is simpler

than the above mentioned classical method because in our theory

the Tschirnhausen transformation is not involved. But we need

a modular function of genus 3.

Remark 3. Let

(3.1) f(X) = a0Xn+ a1Xn-l an = 0 a0 34 0, ai c CC (0si5n)

be a general algebraic equation of even degree n= 2g+2 4

over a subfield k of C. We do not want to clarify the word

"general". Then considering f(X) itself as F(X) instead of

multiplying X, (X - 1 ) or (X - 2) , we can show that for f(X) =

F(X), the values of the modular function in Theorem 1 for all

the orders of the roots of F(X) = 0, generate the Galois ex-

tension of (3.1) over k. in this form, the back ground of our

theorem is clear. Let Xg(2) be the moduli space of (C,(xl,x2,

..,x2g+2)), C a hyperelliptic curve of genus g and (xl,x2,

,x2g+2), the (ordered) set of the Weierstrass points as in §8

3.270

Chap. in. The symmetric group G2g+2 operates on Age) as

permutations of the Weierstrass points. By Chap. TTC §2, Lemma

2.4 and §6, Proposition 6.1, k(2) is a subvariety of the

modular variety M2 of the principally polarized abelian varie-

ties of dimension g with level 2 structure. Let M1 be the

modular variety of the principally polarized abelian varieties

of dimension g. Then there is a canonical morphism DI2 -} M1

of forgetting the level 2 structure. This morphism is a Galois

covering with group Sp29 (2Z/22Z). The Galois group of (3.1)

which is a subgroup of Cy2g+2, interchanges the Weierstrass

points of the hyperelliptic curve C : X2 = F(X). This opera-

tion of r2g+2 on the Weierstrass points induces a faithful

representation 8;2g+2 -> Sp (J (C) 2) = Sp2g (2Z/22Z) by Chap. M §6,

Proposition 6.3. Therefore the equation (3.1) is solved in a

specialization of the Galois covering M2 - M1. The specializa-

tion involves the modular function in Theorem 1 and the hyper-

elliptic integrals.

Remark 4. Finally we notice that Theorem 2 is similar to

Jacobi's formula : Setting K = Sdx//l_x2) (l-k2x2), iK' =

1/k\J dx/ (/,_x2)

(1-k2x2

) and w = iK'/K, we have k= 610(0,w)/1

e00(0,w). Jacobi's formula solves a quadratic equation 1-k2x2

= 0 by theta constants and elliptic integrals.

3.271

Bibliography

[1] Abel, N., H., Beweis der Unmoglichkeit algebraische

Gleichugen von hoheren Graden als dem vierten allgemein

aufzulosen, J. fur die reine and angew. Math., Bd. 1 (1826)

65-84.

[2] Belardinelli, G., Fonctions hypergeometriques de plusieurs

variables et resolution analytique des equations algebriques

generales, Memorial des sc. math., Gauthier-Villars, Paris

(1960).

[3) Coble, A., B., The equation of the eighth degree, Bull.

Amer. Math. Soc., 30 (1924), 301-313.

[4] Deligne, P. at Rapaport, M., Les schemas de modules de

courbes elliptiques, modular functions of one variable 11,

Lecture Notes in Math., 349, Springer-Verlag, Berlin,

Heidelberg, New York (1973).

[5] Enriques, F., Sur les probl?ames gui se rapportent a la

resolution des equations algebriques renfermant plusieurs

inconnues, Math. Ann., Bd. 51 (1899), 134-153.

[6] Fricke, R., Lehrbuch der Algebra, Friedr. Vieweg & Sohn,

Braunschweig (1924).

[7] Galois, E., Ecrits et memoires mathematiques, Gauthier-

Villars, Paris (1962).

[8] Hermite, Ch., Sur la resolution de l'equation du cinquieme

degre, C. R. Acad. Sc., t. 46 (1858) 508-515.

[9] Igusa, J., Theta functions, Springer-Verlag, Berlin,

Heidelberg, New York (1972).

[10] Jordan, C., Traite des substitutions et des equations

algebriques, Gauthier-Villars, Paris (1870).

3.272

[11] Klein, F., Gleichungen vom siebenten and achten Grade, Math.

Annalen, Bd. 15 (1879) 251-282.

[12] , Vorlesungen uber das Icosaeder and die Auflosung

der Gleichungen vom funften Grade, Teubner, Leipzig (1884).

[13] , Sur la resolution, par les fonctions hyper-

elliptiques, de l'equation du vingt-septi me degrd de

laquelle ddpend la determination des vingt-sept droites

d'une surface cubique, Atti Rend. R. Acad. dei Lincei, Ser.

5a, vol. 8 (1899).

[14] , Lectures on Mathematics, Evanston Colloquium,

Macmillan, New York (1894).

[15] Kronecker, L., Sur la resolution de 1'equation du cinquieme

degrd, C. R. Acad. Sc., t. 46 (1858) 1150-1152.

[16] Lindemann, F., Ueber die Auflbsung algebraischer Gleichungen

durch transcendente Functionen I, II, Gottingen Nach. (1884)

245-248, (1892) 292-298.

[17] Mellin, Hj., Rdsolution'de 1'dquation algdbrique gdndrale

l'aide de la fonction F, C. R. Acad. Sc., t. 172 (1921),

658-661.

[18] Thomae, J., Beitrag zur Bestimmung von durch

die Klassenmoduln algebraischer Functionen, J. fur die

reine and angew. Math., Bd. 71 (1870) 201-222.

[19] Weber, H., Lehrbuch der Algebra, Reprint, Chelsea New York.

Coavriahted Material

David MumfordTata Lectures on Theta 11

The second in a series of three volumes surveying the theoryof theta functions, this volume gives emphasis to the specialproperties of the theta functions associated with compactRiemann surfaces and how they lead to solutions of theKorteweg-de-Vries equations as well as other non-lineardifferential equations of mathematical physics.

This book presents an explicit elementary construction ofhyperelliptic Jacobian varieties and is a self-contained intro-duction to the theory of the Jacobians. It also ties togethernineteenth-century discoveries due to Jacobi, Neumann. andFrobenius with recent discoveries of Gelfand. McKean, Moser,John Fay, and others.

A definitive body of information and research on the subjectof theta functions, this %olume will be a useful addition toindi,,idual and mathematics research libraries.

ISBN 0-6176-3110-0

BirkhauserBoston Basel - Berlin