“Linearity” in Arithmetic Differential

Equations Taylor Dupuy

UCSD Number Theory Seminar

Part 1: Origins/History

Part 2: Arithmetic Differential Equations

Part 3: “Linear” Diff ’l Eqns

PART 1

Origins of Arithmetic Differential Algebra

A/K

Sni=1(Ai + ai)

Ai = Ai(K)

Diophantine Geometryabelian variety

Ai A

ai 2 Ai(K)

configuration = finite union of translates of ab subvar

=

configurations form a basis of closed sets for a topology on called the configuration topologyA(K)

S ⇢ A(K)

S f.g. =) SConf = SZar

Mordell-Lang Problem

rf.r.

S f.g. if contained in f.g. subgroupDefn. f.r. f.r.

C/Q

g(C) � 2

Jac(C) = A

C ,! Jac(C)

C(Q) ⇢ Jac(C)(Q)

C(Q)Conf = C(Q)Zar ⇢ C(K)

#C(Q) < 1

S f.g. =) SConf = SZar

Mordell-Lang Problem

Example: (Mordell Problem)

rf.r.

C/K

K

g(C) � 2

A = Jac(C).

S = C(K) \A(K)tors

⇢ A(K)tors

SZar ⇢ C(K).

S f.g. =) SConf = SZar

Mordell-Lang Problem

example. (Manin-Mumford Problem)

rf.r.

History of Mordell-Lang Problem

• Manin - Mordell and Manin-Mumford when K = function field.

• Raynaud - Manin-Mumford for number fields

• Faltings, Coleman, Parshin - Lang Conjecture in General

• Buium - Lang Conjecture over function fields (using differential algebra), char 0

• Hrushovski - Relative Lang, char p

Finiteness hi(X) < 1

Vanishing hi(X) = 0 for i > 2n or i < 0

Trace Map Tr : H2n ⇠= K

Poincare Duality Hi ⇠= H2n�i⇤ via cup product

Kunneth H•(X ⇥ Y ) = H•(X)⌦H•(Y )

Lefschetz Axioms

Cycle Map Zi(X) ! H2i has some nice properties

Weak Lefschetz Re: pullback of hyperplane sections

Hard Lefschetz Hi ! Hi+2, ⇠ 7! ⇠ ^ !where ! = [H], H a hyperplane section.

H•: { Smooth Varieties/ k } ! { Graded K-algebras}

H•(X) =

2 dim(X)M

w=0

Hw(X)

Weil Cohomology Theory

�⌦F1 Z : SchF1 ! SchZ.

Spec(Z)F1 in the category SchF1 to get Spec(Z)F1

H•D : SchF1 ! Graded R-algebras

Setup for Desired Category

Deninger Cohomlogy Theory

Frechetness The spaces Hw(XD, j⇤C)

Vanishing Hw(X) = 0 for w > 2n or w < 0

Trace Map Tr : H2n ⇠= K

Poincare Duality Hi ⇠= H2n�i⇤ via cup product

Kunneth H•(X ⇥ Y ) = H•(X)⌦H•(Y )

Lefschetz Axioms ???, I’m not sure exactly how these should look but there should be something here.

Hodge ⇤ ⇤ : Hw ⇠= H2n�w

Real Action � : R⇥XD ! XD, which should be thought of as a replacement of the Frobenius.

H•D : SchF1 ! Graded R-algebras

Deninger Cohomlogy Theory

b⇣(s) =

Q̀b⇣(⇢)=0

s�⇢2⇡

s⇡ · s�1

⇡

.

=det1( s�✓

2⇡ |H1)

det1( s�✓2⇡ |H0)det1( s�✓

2⇡ |H2).

• ✓ acts as zero on H0

• ✓ acts as identity on H2

✓ = limt!0�t⇤�I

t

�⇤t

Deninger Says:

(induced action on cohomology)

thm

guess

hf, gi := Tr(f1 ^ ⇤f2)

f1, f2 2 H1

f1 ^ ⇤f2 = ✓(f1 ^ ⇤f2)= ✓(f1) ^ ⇤f2 + f1 ^ ✓(⇤f2)= ✓(f1) ^ ⇤f2 + f1 ^ ⇤✓(f2)

f1 [ ⇤f2 2 H2

✓ acts trivially on H2.

✓(f ^ g) = ✓(f) ^ g + f ^ ✓(f).

Pairing:

How derivatives act:

Get antisymmetry property:

Conditional Proof

bhf1, f2i = h✓(f1), f2i+ hf1, ✓(f2)i=) ✓ � 1/2 is antisymmetric on H1

=) �(✓ � 1/2|H1) ⇢ iR

=) �(✓|H1) ⇢ 1/2 + iR

Remarks

• Consistent with random matrix theory statistics

• Algebra with involution (H1(Spec(Z)F1), ⇤)

✓⇤ Noncommutative Geometric Object?

Prototype = BC system ?

Topological Hochschild Cohomology De Rham Witt

Cyclic Cohomology (NCG)

Borger-Buium Descent

X/K smooth projective

K = K char(K) = 0

� : K ! K

K� = {r 2 K : �(r) = 0}

Descent to the constants

1. KS(�) = 0

2. J1(X)

⇠=

TX as schemes over X

3. 9X 0/K�such that X 0 ⌦K� K ⇠

=

X

T.F.A.E.Theorem

CLASSICAL

Arithmetic Descent

1. DIn(X/R) = 0

2. J1(X)n

⇠=

FTXn as torsors

3. Xn/Rn admits a lift of the Frobenius.

R = W (Fp)

(FTXn)n�0

Theorem (D.)

“Canonically lifted” Frobenius Tangent Bundle

X/R

=)J1

(X)n torsor under FTXn

(*)

F1

�⌦F1 Z

Monad Geometries

Set Monoid Algebraic Geometries

Lambda Schemes

Blueprints

Generalized Rings

Sketch of Land

Connes-ConsaniMarcolliSouleBerkovichDeitmar Lorsch.

BorgerBuium

Schemes

subcat

Toen-VasquiesQuillen

Durov

�⌦F1 Z

(???)

PART 2

Wifferential Algebra

What is a p-derivation?

n� np = p · CRAP

CRAP = n�np

p

�p(n) =n�np

p

n ⌘ npmod p

8n 2 Z, 8p primeFermat’s Little Theorem

This is a p-derivation

�p(n) =n�np

p

Product Rule

Sum Rule

�p(ab) = �p(a)bp + ap�p(b) + p�p(a)�p(b)

�p(a+ b) = �p(a) + �p(b)�p�1X

j=1

1

p

✓p

j

◆ap�jbj

Zero mod p

Zero mod p

non-linear�(1) = 0

Kills Unit

p-derivations ring homomorphismsf : A ! W1(A)�p : A ! A

“dual numbers”

“Witt vectors”

“Wittferentiation”

derivations ring homomorphismsf : A ! A["]/h"2i� : A ! A

“infinitesimals”

“wittfinitesimals”

Witt Vectors.

(x0, x1)(y0, y1) = (x0y0, x1yp0 + y1x

p0 + px1y1)

(x0, x1) + (y0, y1) = (x0 + y0, x1 + y1 + Cp(x0, y0))

Cp(X,Y ) = Xp+Y p�(X+Y )p

p 2 Z[X,Y ]

�(a) ⌘ ap mod p

� : A ! B

lifts of the Frobenius ⇡ p-derivations

�(a) := ap + p�p(a)

�p : A ! B + rules

�p(a) :=�(a)�ap

p

p-torsion free

�p(ab) = �p(a)bp + ap�p(b) + p�p(a)�p(b)

�p(a+ b) = �p(a) + �p(b)�p�1X

j=1

1

p

✓p

j

◆ap�jbj

�p : A ! B

Always an -algebraA

8a, b 2 A

Defn. (Buium, Joyal 90’s)A p-derivation is a map of sets such that

R = Zp

�

p

(x) = x�x

p

p

R = Zp[⇣]

⇣

�

p

(x) = �(x)�x

p

p

�(x) =

=

(⇣ 7! ⇣p,

identity,

unique lift of the frobenius

= root of unity coprime to p

example.

example.

on roots of unity else

R = Zurp

= Zp[⇣ : ⇣n = 1, p - 1]example

�

p

(x) = �(x)�x

p

p

EXAMPLES

�p(pm) = pm�1 · ( unit mod p )

�p(p) =p� pp

p

= 1� pp�1

Decreases valuation:

�t =ddt

�t(tn) = n · tn�1

Decreases valuation:

Constants

(K, �)

K� = {c 2 K : �(c) = 0}

R = bZurp

R� = {r 2 R : �p(r) = 0

Constants of a p-derivation:

(ring with der)Constants of a derivation:

(subring)

(submonoid)(ring with p-der)

PART 2

Linear Arithmetic Differential Equations

The poor man’s model companion:

R = bZurp

unramified CDVR, residue field Fp

bZurp = Zp[⇣; ⇣n = 1, p - n]b

unique lift of the Frobenius: �(⇣) = ⇣p

9!� : R ! R

1)

2)

3)

R WILL ALWAYS MEANTHIS RING IN THIS

TALK

Properties:

Simplest Possible Equation:

�x = ↵x

(p)

x = (xij)↵ 2 GLn(R)

x

(p) = (xpij)

Remarks.x 7! �(x)(x(p))�1 almost a cocycle

proof.

↵ 2 gln(R) u0 2 GLn(R)(�u = ↵u(p)

u ⌘ u0 mod p

,

has a unique solution

|x� y|p 1 =) |f(x)� f(y)|p 1p |x� y|p

f : GLn(R) ! GLn(R)

f(x) = �

�1(✏x(p))

Contraction mapping:

Theorem (existence and uniqueness)

�(u) = ✏u(p)�u = ↵u(p) ()

u = limn!1

fn(u0)

✏ = 1 + p↵

matrix norms

(�u = ↵u(p)

u ⌘ u0 mod p

Theorem (coeffs in CDVR)

Ou0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O)

Complete Discrete Valuation Subring=

Proof. O = R�⌫

�⌫(✏) = ✏

�⌫+1(u) = �⌫(✏u(p))

�(�⌫(u)) = ✏(�⌫(u))(p)

�(u) = ✏u(p)

(characterization)

,�⌫(u0) = u0 �⌫(↵) = ↵,

✏ = 1 + p↵

(�u = ↵u(p)

u ⌘ u0 mod p

Theorem (coeffs in valuation delta subring)

O valuation delta-subring =

Proof.

u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)

O0/O finite extension of delta-subrings

Strategy: reformulate as dynamics problem!!

u0,↵, ✏ = 1 + p↵ 2 gln( bO)

9⌫ � 0,�⌫(u) = u=)CDVR regularity Lemma

complete

Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)

O0/O finite extension of delta-subrings

u = �⌫(u)

= �⌫�1(✏u(p))

= · · ·= �⌫�1(✏)(�⌫�2(✏)(· · · (✏u(p)) · · · )(p))(p)

w 2 GLn(R)

(w(p))(m) = (w(m))(p)

(vw)(m) = (vw(m))

w(m) = pick out mth column

1)2)

properties

Strategy: reformulate as dynamics problem!!

Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)

u = �⌫(u)

= �⌫�1(✏u(p))

= · · ·= �⌫�1(✏)(�⌫�2(✏)(· · · (✏u(p)) · · · )(p))(p)

Strategy: reformulate as dynamics problem!!

(w(p))(m) = (w(m))(p)

(vw)(m) = (vw(m))

properties

u(m) = ✏⌫�1(✏⌫�2(· · · (✏u(p)(m)) · · · )

(p))(p)

'(⌘) = ✏⌫�1(✏⌫�2(· · · (✏⌘(p)) · · · )(p))(p)' : An

F ! AnF

u(m) ' : An(Ka) ! An(Ka)fixed point of

Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)

u(m) = ✏⌫�1(✏⌫�2(· · · (✏u(p)(m)) · · · )

(p))(p)

'(⌘) = ✏⌫�1(✏⌫�2(· · · (✏⌘(p)) · · · )(p))(p)' : An

F ! AnFDynamics!

' : An(Ka) ! An(Ka)

# fixed points of < 1

Lemma.

F = Frac(O)

' : An(Ka) ! An(Ka)

# fixed points of < 1

Lemma.

general principle:

' : An(Ka) ! An(Ka)fixed points of ⇢ An(F a)

'/F =)

=) columns are in finite algebraic extensions

Theorem (coeffs in valuation delta subring)u0 2 GLn(O) and ↵ 2 gln(O) =) u 2 GLn(O0)

' : An(Ka) ! An(Ka)

# fixed points of < 1

Lemma. (Fornaes and Sibonay)

proof. '(x1, . . . , xn) = (x1, . . . , xn)

Y : 'j(x1, . . . , xn)� x

d�10 xj = 0

homogenize⌘ 7! ✏j⌘

⌘ 7! ⌘(p)built from'

zero dimensional in Y=) Pn

=)' : An(Ka) ! An(Ka)

fixed points of

Y \ {x0 = 0} = ;Claim

={ (1,u) }

Galois TheoryO ⇢ R

Gu/O

{c 2 GLn(O) : 9� 2 AutO(O[u]),� � � = � � �,�(u) = uc}

Galois Group=

, ↵ 2 gln(O)

�u = ↵u(p)

O[u] = O[uij ] = Picard-Vessiot ring= Ring obtained by adjoining entries of u

Gu/O

{c 2 GLn(O) : 9� 2 AutO(O[u]),� � � = � � �,�(u) = uc}

Galois Group of=

�u = ↵u(p)

Alternative Descriptions:

�u/O = {� 2 AutO(O[u]);� � � = � � � 2 GLn(O)}1)

2) 0 ! Iu/O ! O[x, 1/ det(x)] ! O[u] ! 0

StabGLn(R)(Iu/O) ⇠= Gu/O

�u/O ⇠= Gu/O

�u = ↵u(p)

�(uc) = ↵(uc)(p)

Question: For what do we get c 2 GLn(R)

?

�(uc) = u(p)�(c) + �(u)c(p) + p�(u)�(c) + {u, c}⇤

{u, c}⇤ = u(p)c(p)�(uc)(p)

p

Claim.{u, c}⇤ = 0

�c = 01)2)

=) �(uc) = ↵(uc)(p)

W

T

N = WT = TW

G�

= maximal torus of diagonals= permutation matrices

Main Example:

= matrices with roots of unity entries

= permutation matrices with roots ofunity entries

N � = T �W = WT �

GLn(Fa1)

subgroup of

�(uc) = u(p)�(c) + �(u)c(p) + p�(u)�(c) + {u, c}⇤ {u, c}⇤ = u(p)c(p)�(uc)(p)

p

O ⇢ R

O ⇢ R�⌫

Gu/O

delta subring

�u/O = {� 2 AutO(O[u]);� � � = � � � 2 GLn(O)}=

=) Gu/O finite

Theorem.

finite to begin with

u 2 O

Theorem A.9⌦ ⇢ Q2

8↵ 2 Z2 \ ⌦

Gu/O

�u = ↵u(p)

W= finite group containing

“thin set”9u 2 GLn(R),

⌦ ⇢ Zn2

X = {u 2 GLn(R);u ⌘ 1 mod p}

9⌦ ⇢ X

8u 2 X \ ⌦

�(u)(u(p))�1 2 gln(O)

O ⇢ R �-closed subring

Gu/O = N �

R� ⇢ O

Theorem B.

= ball around identity

of the second category

,1)2)

8

=)

9⌦ ⇢ GLn(Ka)

8u 2 GLn(R) \ ⌦u0 = ↵u(p)

dim((Z ·Gu/O)Zar) n

O 3 ↵ �-closed subring of R

Theorem C.Zariski closed

PART 4

Remarks on Picard-Fuchs and Painleve 6 and the Frobenius

E/K

K =

: E(K) ! Ga(K)

ker( ) = E(K)tors

Manin Map:

elliptic curvefunction field

(diff alg map)

E

A1

Et(C) =

Setup:

universal family

E : y2 = x(x� 1)(x� t)

px(x� 1)(x� t)

! = dx/y

R�1

! = ⌘1R�2

! = ⌘2

⌘1(t), ⌘2(t)

H1(Et(C),Z) = Span{�1, �2}

periods

(X,Y ) = ⇤R (X,Y )

dx/y

⇤ = 4t(1� t) d2

dt2 + 4(1� 2t) ddt � 1

Picard-Fuchs Equations:

kills periods

Manin-Map:

Buium’s Picard-Fuchs

⇤ = �2 + ↵1�+ p↵0

(P ) = ⇤

⇣1pr log!(P )

⌘Buium’s Manin Map