Algebraic curves of GL2-typertakloo/atkin/atkin2011/hoffman.pdfAtkin Memorial Conference Louisiana...

Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples

Algebraic curves of GL2-type

Jerome William HoffmanAtkin Memorial Conference

Louisiana State University

April 29-May 1, 2011

Jerome William Hoffman Atkin Memorial Conference Louisiana State University,

1 Galois representations of GL2-type

2 Explicit moduli for genus two curves

3 Humbert surfaces

4 Examples

For any number field K let GK = Gal(Q/K ). Let

ρ : GQ → GL(Vℓ) ∼= GL2d (Qℓ)

be a representation. Mostly we will be concerned with d = 2.Assume that

1 There is a number field E/Q of degree d and ahomomorphism E → End(Vℓ).

2 There is a finite extension K/Q such that ρ(GK ) commuteswith the image of E .

Then we have a factorization:

ρK := ρ | GK → GLE⊗Qℓ(Vℓ) ∼= GL2(E⊗Qℓ) ⊂ GL(Vℓ) ∼= GL2d (Qℓ).

We say that ρK has an E -linear structure. or ρ has anGL2-structure over K .

Example: Abelian varieties of GL2-type. For an abelian variety A/klet Vℓ(A) be its Tate module: a 2 dim A-dimenional representationof Gk .Theorem. (Ribet + Serre’s conjecture)

1 Let A/Q be a Q-simple abelian variety. Suppose thatE = Q⊗ EndQ(A) is a number field of degree = dim A. Thenthe Tate module Vℓ(A) defines a representation of Gal(Q/Q)with values in GL2(E ⊗Qℓ). Moreover A is isogenous to aQ-simple factor of J1(N) for some N ≥ 1.

2 Let C/Q be a elliptic curve. Then C is a quotient of J1(N)Qfor some N ≥ 1 if and only if C is a Q-curve, i.e., C isisogenous to each of its conjugates σC , σ ∈ Gal(Q/Q).

Problem: Explicitly construct Q-curves.

Theorem. (Shimura) Let f =∑

anqn be a normalized cuspidal

Hecke eigenform of weight 2 on Γ1(N) for some N ≥ 1. Then thereis an abelian variety Af defined over Q with an action of the fieldE = Q(..., an , ...). This is a quotient of J1(N). We havedim Af = [E : Q], E = EndQ(Af )⊗Q, and thus Vℓ(Af ) is ofGL2-type over Q.Example. N = 29. There exist Hecke eigenformsf , f ∈ S2(29, (∗/29)) with field of coefficients E = Q(

√−5).

Shimura’s Af is two-dimensional, and in fact it is isogenous toC × σC for an elliptic Q-curve C/Q(

√5). Also we have an isogeny

Af∼= Jac(X ) where X is the genus 2 curve

y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).

√−5).

y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).

√−5).

y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).

Note that for a given ρ, there can be more than one pair E ,K suchthat ρK has an E -linear structure.Examples are given by quaternion structures. Here d = 2. That is,assume:

1 there is a quaternion algebra B/Q and a homomorphismB → End(Vℓ) and

2 a finite extension K/Q such that the image of B commuteswith ρK .

Then ρK has an E -linear structure for every quadratic subfieldE ⊂ B .

From now on, we take d = 2Suppose ρ has an E -linear structure defined over K (so[E : Q] = 2).Let λ, λ be the places of E lying over the prime number ℓ (possiblyλ = λ ). Then we get a decomposition E ⊗Qℓ = ⊕

λ,λEλ hence

ρK ⊗ E = ρK ,λ ⊕ ρK ,λ, L(ρK , s) = L(ρK ,λ, s)L(ρK ,λ, s).

λ,λEλ hence

Special case: Suppose that:

1 K/Q is also a quadratic extension.

2 The generator s ∈ Gal(K/Q) induces an isomorphisms : ρK ,λ

∼= ρK ,λ.

Thenρ = Ind

QK (ρK ,λ)

thereforeL(ρ, s) = L(ρK ,λ, s).

Special case: Suppose that:

1 K/Q is also a quadratic extension.

2 The generator s ∈ Gal(K/Q) induces an isomorphisms : ρK ,λ

∼= ρK ,λ.

Thenρ = Ind

QK (ρK ,λ)

thereforeL(ρ, s) = L(ρK ,λ, s).

Let X be a projective nonsingular curve of genus 2 defined over Q.We assume an equation y2 = f (x) where f (x) ∈ Q[x] has degree 5or 6 with distinct roots.Then

Vℓ = H1(X ⊗Q,Qℓ) = H1(Jac(X )⊗Q,Qℓ)

gives a four dimensional representation of GQ. These are ofGL2-type if the Jacobian Jac(X ) has extra endomorphisms.For any abelian variety A defined over a field k we letEnd0

k(A) := Endk(A)⊗Q be the semisimple Q-algebra ofendomorphisms defined over k ; End0(A) = End0

Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:

2 a real quadratic field E/Q;

3 an indefinite quaternion algebra B/Q;

4 a quartic CM field K/Q.

Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?

The familiar invariants of an elliptic curve, e.g., j , g2, g3 arise as in-and covariants of the action of PGL(2) on binary quartic formsf (x , y) = a0x

4 + a1x3y + a2x

2y2 + a3xy3 + a4y

4. These wereworked out in the 19th century.Reason: every genus 1 curve can be expressed as a double cover ofP1 with four branch points. Then f (x , y) = 0 give the coordinatesof the branch points. This leads to coordinate systems on themoduli spaces of elliptic curves: these are the modular curves.Riemann already knew that the moduli space Mg of genus g ≥ 2curves had dimension 3g − 3, but explicit models of these asalgebraic varieties are not easy to find.

4 + a1x3y + a2x

2y2 + a3xy3 + a4y

4 + a1x3y + a2x

2y2 + a3xy3 + a4y

Let X be a projective nonsingular curve of genus 2 defined over afield k . Then X has a model in the form

y2 = f (x) deg f = 6, with distinct roots.

As for elliptic curves, the moduli for genus 2 curves then can beexpressed via the in- and covariants of the action of PGL(2) onbinary sextic forms. The expression of the moduli of genus 2 curvesvia projective invariants of binary sextics was done by Clebsch, andin more modern times by Igusa and Mestre.Note that dimM2 = 3, so there are three “J-invariants” j1, j2, j3.Recall also that the map X 7→ Jac(X ): M2 → A2 to the modulispace of principally polarized abelian varieties of dimension 2 is abirational correspondence.

Analytic moduli.Let

H2 = {τ =

τ1 τ2τ2 τ3

∈ M2(C) | Im(τ) > 0}

the Siegel space of genus 2.As an analytic space, Aan

2 = Γ\H2 where

Γ = {g =

∈ Sp4(Z)}

acting via τ 7→ (Aτ + B)(Cτ + D)−1.We get coverings by taking subgroups of finite index in Γ(congruence subgroups!)

Analytic moduli.Let

H2 = {τ =

τ1 τ2τ2 τ3

∈ M2(C) | Im(τ) > 0}

2 = Γ\H2 where

Γ = {g =

∈ Sp4(Z)}

Analytic moduli.Let

H2 = {τ =

τ1 τ2τ2 τ3

∈ M2(C) | Im(τ) > 0}

2 = Γ\H2 where

Γ = {g =

∈ Sp4(Z)}

Algebraic moduli. A natural set of coordinates on the covering ofM2 given by level 2 structure is gotten by taking the cross ratios ofthe roots ei , i = 1, ..., 6 of

y2 = f (x) = (x − e1)...(x − e6).

level 2 structure = an ordering of the 6 roots e1, ..., e6:Sp4(Z/2) = S6.So we can describe subvarieties of M2 by equationsH(e1, ..., e6) = 0, etc. Cross-ratios in the ei can be expressed bycertain Siegel modular functions (ratios of thetanullwerte.)

Algebraic moduli. A natural set of coordinates on the covering ofM2 given by level 2 structure is gotten by taking the cross ratios ofthe roots ei , i = 1, ..., 6 of

y2 = f (x) = (x − e1)...(x − e6).

level 2 structure = an ordering of the 6 roots e1, ..., e6:Sp4(Z/2) = S6.So we can describe subvarieties of M2 by equationsH(e1, ..., e6) = 0, etc. Cross-ratios in the ei can be expressed bycertain Siegel modular functions (ratios of thetanullwerte.)

We are interested in the subvariety H∆ ⊂ M2 of those genus 2curves X where Jac(X ) has endomorphisms by an order in a realquadratic field Q(

√∆). We have dim H∆ = 2 and these are called

Humbert surfaces.

1 When the integer ∆ is a square, H∆ is a product of modularcurves; Jac(X ) factors into 2 elliptic curves for X ∈ H∆.

2 When ∆ is a not a square, H∆ is a Hilbert modular surface forthe field Q(

√∆).

Humbert surfaces.

√∆).

Humbert surfaces.

√∆).

Explicit equations for Humbert surfaces were written down for∆ = 5, 8 by G. Humbert. Modern treatment given by P. Bending,Hashimoto, Hirzebruch, Murabayashii, R. M. Wilson, Sakai,Shephard-Barron, R. Taylor, van der Geer.Algorithms to compute equations of general Humbert surfaces havebeen developed by B. Runge, D, Gruenewald.

The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies

aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.

We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .

aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.

Proof (Cayley): Consider the dual projective plane:(P2)∗ = the variety of lines in P2.Let D∗ ⊂ (P2)∗ be the variety of lines tangent to D. This is also aconic.Define

E (C ,D) := {(P , ℓ) ∈ C × D∗ | P ∈ ℓ},the incidence correspondence.Then E (C ,D) is an elliptic curve, and a Poncelet n-goncorresponds to a point of order n on E (C ,D).

Humbert, Mestre: Given a Poncelet n-gon, one constructs ahyperelliptic curve X together with an action of Q(ζm + ζ−1

m ) onJac(X ). X is a double cover of C branched over the vertices of then-gon.When n = 5, 8 this curve X has genus 2, with endomorphisms byQ(

√5) and Q(

√2) respectively.

∆ = 8: −2τ1 + 2τ3 = 0. This corresponds to a Poncelet 4-gon.

√5) and Q(

√2) respectively.

√5) and Q(

√2) respectively.

Quadrilateral αβγδ is

Genus 2 curve X is double cover

of C branched over p, q, α, β, γ, δ.

Humbert 8 = Poncelet 4

inscribed on conic C, tangent to conic D

This configuration leads to the explicit equation for H8 = 0 interms of the roots ei of the sextic f (x) where the genus 2 curve isrepresented by y2 = f (x).

H8(e1, ..., e8) =

(e3 − e1) (e3 − e2) (e3 − e4) (e4 − e2)2(e3 − e5) (e6 − e1) (e6 − e2) (e6 − e4) (e6 − e5) (e1 − e5)

+ (e1 − e2) (e1 − e3) (e1 − e4) (e2 − e4)2(e3 − e5)

2(e6 − e2) (e6 − e3) (e6 − e4) (e6 − e5) (e1 − e5)

+ (e3 − e1)2(e4 − e1) (e4 − e2) (e4 − e3) (e2 − e5)

2(e4 − e5) (e6 − e1) (e6 − e2) (e6 − e3) (e6 − e5)

+ (e2 − e1) (e1 − e3)2(e2 − e3) (e2 − e4) (e2 − e5) (e4 − e5)

2(e6 − e1) (e6 − e3) (e6 − e4) (e6 − e5)

+ 16 (e1 − e2) (e2 − e3) (e1 − e4) (e3 − e4) (e2 − e5) (e3 − e5)

(e4 − e5) (e1 − e6) (e2 − e6) (e3 − e6) (e4 − e6) (e1 − e5)

∆ = 5: −τ1 + τ2 + τ3 = 0. This corresponds to a Poncelet 5-gon.

Humbert 5 = Poncelet 5

P" Pentagon αβγδεinscribes conic C

circumscribes conic D

Genus 2 curve X is the

double cover of C branched

above α, β, γ, δ, ε and

a point q in C intersect D.

The correspomdence

lifts to a correspondence

P −> P’+P"

2φ + φ −1=0of X with φ

in Jac(X).

This configuration leads to the explicit equation for H5 = 0 interms of the roots ei of the sextic f (x) where the genus 2 curve isrepresented by y2 = f (x).

H5(e1, ..., e6) =

(e2 − e3)2(e5 − e1) (e5 − e2) (e5 − e3) (e5 − e4) (e1 − e6) (e2 − e6) (e3 − e6) (e4 − e6) (e1 − e4)

+ (e1 − e2) (e1 − e3) (e3 − e4)2(e1 − e5) (e2 − e5)

2(e2 − e6) (e3 − e6) (e4 − e6) (e5 − e6) (e1 − e4)

+ (e1 − e2)2(e4 − e1) (e4 − e2) (e4 − e3) (e4 − e5) (e5 − e3)

2(e1 − e6) (e2 − e6) (e3 − e6) (e5 − e6)

+ (e3 − e1) (e3 − e2) (e3 − e4) (e4 − e2)2(e3 − e5) (e5 − e1)

2(e1 − e6) (e2 − e6) (e4 − e6) (e5 − e6)

+ (e2 − e1) (e2 − e3) (e3 − e1)2(e2 − e4) (e2 − e5) (e4 − e5)

2(e1 − e6) (e3 − e6) (e4 − e6) (e5 − e6)

Let B/Q be an indefinite quaternion algebra. Two-dimensionalprincipally polarized abelian varieties A with B ⊂ End(A) areparametrized by the points of the Shimura curve SB associated toB . There is a universal family X → SB of genus two curves whichhave Jacobians that have QM by B .Problem. Find explicit equations for these universal families, forsmall values of the discriminant of B .So far the only known examples are for discriminants 6, 10.

The idea of Hashimoto and Murabayashii is to consider theintersection of Humbert surfaces.If a simple abelian surface A has End0(A) ⊃ Q(∆1),Q(∆2) for twodifferent real quadratic fields, then End0(A) is a quaternionalgebra. Thus

H∆1∩ H∆2

= union of Shimura curves.

The idea of Hashimoto and Murabayashii is to consider theintersection of Humbert surfaces.If a simple abelian surface A has End0(A) ⊃ Q(∆1),Q(∆2) for twodifferent real quadratic fields, then End0(A) is a quaternionalgebra. Thus

H∆1∩ H∆2

= union of Shimura curves.

Example 1 (Brumer/Hashimoto) Let

f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4

− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3

+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)

Let C (a, b, c) : Y 2 = f (X ; a, b, c), assume f (X , a, b, c) has 6distict roots. Then C (a, b, c) is a genus two curve with RM byQ(

√5). These endomorphisms are defined over Q(a, b, c). Hence if

a, b, c ,∈ Q, the curve is modular.

f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4

− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3

+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)

f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4

− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3

+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)

Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define

B =Q(PA − Q) + 4P2 + 1

P2, C =

4(PA − Q)

Define αi by∏2

i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve

Y 2 = D

(X 2 − αiX + Pα2i + Qαi + 4P)

is defined over K and has RM by Q(√

2), also defined over K .Hence if K = Q, these curves are modular.

B =Q(PA − Q) + 4P2 + 1

P2, C =

4(PA − Q)

Define αi by∏2

Y 2 = D

(X 2 − αiX + Pα2i + Qαi + 4P)

B =Q(PA − Q) + 4P2 + 1

P2, C =

4(PA − Q)

Define αi by∏2

Y 2 = D

(X 2 − αiX + Pα2i + Qαi + 4P)

B =Q(PA − Q) + 4P2 + 1

P2, C =

4(PA − Q)

Define αi by∏2

Y 2 = D

(X 2 − αiX + Pα2i + Qαi + 4P)

B =Q(PA − Q) + 4P2 + 1

P2, C =

4(PA − Q)

Define αi by∏2

Y 2 = D

(X 2 − αiX + Pα2i + Qαi + 4P)

Example 3 (Bending/Hashimoto/Murabayashii) Let K ⊂ C be afield. Let N,D in K ∗ with N2 6= −1.Let C be the curve

Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]

Then End0K(Jac(C )) contains the division quaternion algebra B6

of discriminant 6, namely(

2,−3Q

. The action of Q(√

2) ⊂ B6 is

defined over K and the action of Q(√−3) ⊂ B6 is defined over

K (√−3). Hence if N,D ∈ Q these are modular.

Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]

2,−3Q

2) ⊂ B6 is

Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]

2,−3Q

2) ⊂ B6 is

Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]

2,−3Q

2) ⊂ B6 is

Example 4 (QM by B10) Let

C : y2 = x6 − 16x5 + 40x4 + 140x3 + 80x2 − 64x + 64.

Then Jac(C ) has multiplications by the quaternion division ring ofdiscriminant 10. We have

det(X − ρ(Frobp)) = gp(X )gp(X )

for gp(X ) ∈ Q(√

5)[X ].

p gp(X )

3 X 2 +√

5X + 3

11 X 2 + 3X + 11

13 X 2 + 3√

5X + 13

17 X 2 +√

5X + 17

19 X 2 + 19

23 X 2 + 23

29 X 2 + 9X + 29

31 X 2 + 31

37 X 2 + 37

41 X 2 + 41

43 X 2 + 43

47 X 2 + 5√

5X + 47

Example 5 (RM by Q(√

2)) Let

C : y2 = 21x5 + 50x4 + 5x3 − 20x2 + 4x .

Then Jac(C ) has multiplications by Q(√

2), but theendomorphisms are defined over Q(

√2). The ℓ-adic representation

here is not obviously automorphic. The conductor is 2?3272. This isspecial case of a 1-parameter family of Hashimoto/Sakai.

p det(X − ρ(Frobp))5 (X 2 − 5)2

11 (X 2 + 4√

2X + 11)(X 2 − 4√

2X + 11)13 X 4 + 6X 2 + 132

17 (X 2 + (4√

2 − 2)X + 17)(X 2 + (−4√

2 − 2)X + 17)19 X 4 − 10X 2 + 192

23 (X 2 + 4√

2X + 23)(X 2 − 4√

2X + 23)

29 (X 2 + 6√

2X + 29)(X 2 − 6√

2X + 29)31 (X 2 + 31)2

37 (X 2 + 10X + 37)(X 2 − 10X + 37)

41 (X 2 + (4√

2 − 2)X + 41)(X 2 + (−4√

2 − 2)X + 41)43 X 4 + 3X 2 + 432

47 (X 2 − 8X + 47)2

Example 6 (RM by Q(√

2)) Let

C : y2 = 9x5 − 210x4 + 165x3 + 16740x2 − 74844x

Then Jac(C ) has multiplications by Q(√

2), but the endomorphismsare defined over Q(

√−2). The ℓ-adic representation here is not

obviously automorphic. The conductor is 2?3?7?11?. This is specialcase of a 1-parameter family of Hashimoto/Sakai.

p det(X − ρ(Frobp))5 (X 2 − 5)2

13 X 4 − 6X 2 + 132

17 (X 2 + (4√

2 − 2)X + 17)(X 2 + (−4√

2 − 2)X + 17)

19 (X 2 + 4√

2X + 19)(X 2 − 4√

2X + 19)

23 (X 2 + 4√

2X + 23)(X 2 − 4√

2X + 23)29 SX 4 − 10X 2 + 29

31 (X 2 + 4√

2X + 31)(X 2 − 4√

2X + 31)37 (X 2 + 6X + 37)(X 2 − 6X + 37)

41 (X 2 + (4√

2 + 6)X + 41)(X 2 + (−4√

2 + 6)X + 41)43 (X 2 + 8X + 43)2

47 (X 2 + 8X + 47)(X 2 − 8X + 47)

Thanks to Ramin Takloo-Bighash andWinnie Li!

Algebraic curves of GL2-typertakloo/atkin/atkin2011/hoffman.pdfAtkin Memorial Conference Louisiana...

Documents

Transcript of Algebraic curves of GL2-typertakloo/atkin/atkin2011/hoffman.pdfAtkin Memorial Conference Louisiana...