Serres obstruction for genus 3 curves Christophe Ritzenthaler - - PowerPoint PPT Presentation

Serre’s obstruction for genus 3 curves

Christophe Ritzenthaler

Institut de Mathématiques de Luminy, CNRS

Geocrypt, Guadeloupe, April 27 - May 1, 2009

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Outline

1

Geometric versus arithmetic Torelli theorem

2

Siegel modular forms

3

A special modular form : χh

4

Main result and ingredients of the proof

5

Explicit computations and application to optimal curves

6

Primes dividing χ18

7

New strategies ?

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Geometric versus arithmetic Torelli theorem

Geometric Torelli theorem

k algebraically closed field. Ag : space of (isom. classes of) g-dimensional p.p.a.v. Mg : space of (isom. classes of) curves of genus g. j canonical principal polarization on Jac X. Geometric Torelli Theorem (Weil) Torelli’s morphism θ : X → (Jac X, j) Mg − − − − → Ag is injective.

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Geometric versus arithmetic Torelli theorem

Arithmetic Torelli theorem

k arbitrary field. (A, a)/k ∈ Ag such that (A, a) ≃k (Jac X0, j0). Arithmetic Torelli theorem (Serre) There is a model X/k of X0 such that :

1 If X0 is hyperelliptic, there is a k-isomorphism

(Jac X, j)

∼

− − − − → (A, a).

2 If X0 is not hyperelliptic, there is a quadratic character ε of

Gal(ksep/k), and a k-isomorphism (Jac X, j)

∼

− − − − → (A, a)ε where (A, a)ε is the twist of A by ε (if ε is not trivial, then (A, a) is not a Jacobian).

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Geometric versus arithmetic Torelli theorem

Jacobians in dimension 2 and 3

g = 2 (resp. g = 3) dim Mg = 3g − 3 = dim Ag = g(g + 1)/2 = 3 (resp. 6). (A, a)/k indecomposable : (A, a) is not isomorphic to a product of p.p.a.v. Curve of genus g ≤ 3

• ver k

Jac

⇐ ⇒ abelian variety of dimension g, with an indecomposable principal polarization

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Geometric versus arithmetic Torelli theorem

Applications to maximal curves of genus g ≤ 3

To construct a genus g curve with many points N over Fq : find an abelian variety A over Fq with trace of Frobenius q + 1 − N ;

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Geometric versus arithmetic Torelli theorem

Applications to maximal curves of genus g ≤ 3

To construct a genus g curve with many points N over Fq : find an abelian variety A over Fq with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ;

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Geometric versus arithmetic Torelli theorem

Applications to maximal curves of genus g ≤ 3

To construct a genus g curve with many points N over Fq : find an abelian variety A over Fq with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ; use arithmetic Torelli to conclude.

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Geometric versus arithmetic Torelli theorem

Applications to maximal curves of genus g ≤ 3

To construct a genus g curve with many points N over Fq : find an abelian variety A over Fq with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ; use arithmetic Torelli to conclude. g = 2 : all curves are hyperelliptic : ok. g = 3 : curves can be non hyperelliptic the quadratic twist is a Jacobian and its number of points is minimum. Theorem (Lauter 2002) Let m = ⌊2√q⌋. For all q there exists a genus 3 curve C over Fq such that |#C(Fq) − q − 1| ≥ 3m − 3.

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Geometric versus arithmetic Torelli theorem

Applications to maximal curves of genus g ≤ 3

To construct a genus g curve with many points N over Fq : find an abelian variety A over Fq with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ; use arithmetic Torelli to conclude. g = 2 : all curves are hyperelliptic : ok. g = 3 : curves can be non hyperelliptic the quadratic twist is a Jacobian and its number of points is minimum. Theorem (Lauter 2002) Let m = ⌊2√q⌋. For all q there exists a genus 3 curve C over Fq such that |#C(Fq) − q − 1| ≥ 3m − 3. Serre’s Question (letter to Top, February 2003) : how to compute the twist ε ?

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Siegel modular forms

Analytic Siegel modular forms

Siegel upper half space of genus g Hg =

• τ ∈ Mg(C) | tτ = τ, Im τ > 0
• .

Rg,h(C) : space of analytic Siegel modular forms of weight h on Hg, i.e. holomorphic functions φ(τ) on Hg satisfying φ(M.τ) = det(cτ + d)hφ(τ) for any M = a b c d

• ∈ Sp2g(Z).

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Siegel modular forms

Geometric Siegel modular forms

Ag : moduli stack of p.p.a.s. of relative dimension g π : Vg − − − − → Ag (universal p.p.a.s) Ω1 : rank g bundle induced by relative regular differential forms on Vg/Ag. Relative canonical line bundle ωVg/Ag = g π∗Ω1 over Ag : ωVg/Ag  

• Ag

Ω1

k[A] = H0(A, Ω1 A ⊗ k),

ωk[A] =

g

• Ω1

k[A].

Space of geometric Siegel modular forms of weight h over a ring R : Sg,h(R) = Γ(Ag ⊗ R, ω⊗h)

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Siegel modular forms

Analytic and geometric modular forms

A = (A, a) p.p.a.v. of dimension g defined over k ⊂ C.

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Siegel modular forms

Analytic and geometric modular forms

A = (A, a) p.p.a.v. of dimension g defined over k ⊂ C. ω1, . . . , ωg basis of Ω1

k[A] and γ1, . . . γ2g symplectic basis (for a) such that

Ωa = [Ω1 Ω2] =   

• γ1 ω1

· · ·

• γ2g ω1

. . . . . .

• γ1 ωg

· · ·

• γ2g ωg

   (A, a) ≃ (Cg/ΩaZ2g, J2g) satisfies τa = Ω−1

2 Ω1 ∈ Hg.

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Siegel modular forms

Analytic and geometric modular forms

A = (A, a) p.p.a.v. of dimension g defined over k ⊂ C. ω1, . . . , ωg basis of Ω1

k[A] and γ1, . . . γ2g symplectic basis (for a) such that

Ωa = [Ω1 Ω2] =   

• γ1 ω1

· · ·

• γ2g ω1

. . . . . .

• γ1 ωg

· · ·

• γ2g ωg

   (A, a) ≃ (Cg/ΩaZ2g, J2g) satisfies τa = Ω−1

2 Ω1 ∈ Hg.

Proposition Let ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A] and f ∈ Rg,h(C). f ((A, a)) = (2iπ)gh

• f (τa)

(det Ω2)h · ω⊗h. Then the map f → f is an isomorphism Rg,h(C) − → Sg,h(C).

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Siegel modular forms

Analytic and geometric modular forms

A = (A, a) p.p.a.v. of dimension g defined over k ⊂ C. ω1, . . . , ωg basis of Ω1

k[A] and γ1, . . . γ2g symplectic basis (for a) such that

Ωa = [Ω1 Ω2] =   

• γ1 ω1

· · ·

• γ2g ω1

. . . . . .

• γ1 ωg

· · ·

• γ2g ωg

   (A, a) ≃ (Cg/ΩaZ2g, J2g) satisfies τa = Ω−1

2 Ω1 ∈ Hg.

Proposition Let ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A] and f ∈ Rg,h(C). f ((A, a)) = (2iπ)gh

• f (τa)

(det Ω2)h · ω⊗h. Then the map f → f is an isomorphism Rg,h(C) − → Sg,h(C). Notation : f ((A, a), ω) = f ((A, a))/ω⊗h ∈ C.

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A special modular form : χh

Thetanullwerte and the form χh

Thetanullwerte of characteristic (ε, η) ∈ {0, 1}g × {0, 1}g : θ ε η

• (τ) =
• n∈Zg

exp

• iπ(n + ε

2).τ.(n + ε 2)

• exp
• iπη.(n + ε

2)

• .

Even characteristics : ε.η ≡ 0 (mod 2). Theorem (Igusa-Ichikawa) If g ≥ 3, then

• χh(τ) =

(−1)gh/2 22g−1(2g−1) ·

• even

θ ε η

• (τ) ∈ Rg,h(C),

h = 2g−2(2g + 1). χh((A, a)) = (2iπ)gh

• χ(τa)

(det Ω2)h · ω⊗h ∈ Sg,h(Z).

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Main result and ingredients of the proof

Main result

Theorem (Lachaud-R.-Zykin) Let A = (A, a)/k be a p.p.a.t. defined over a field k ⊂ C. Assume that a is

• indecomposable. Let ω ∈ ωk[A].

1 χ18((A, a)) = 0 if and only if (A, a) is hyperelliptic. 2 With the previous notation

χ18((A, a), ω) = (2π)54 228 ·

• even θ

ε η

• (τa)

det(Ω2)18 is a square in k∗ if and only if (A, a) is a non hyperelliptic Jacobian. Quadratic character given on σ ∈ Gal(ksep/k) by ε(σ) = dσ d , d =

• χ18((A, a), ω)

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Main result and ingredients of the proof

First ingredient of the proof : action of twists

Let φ : (A′, a′)/k − → (A, a)/k be a k-isomorphism of p.p.a.v.

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Main result and ingredients of the proof

First ingredient of the proof : action of twists

Let φ : (A′, a′)/k − → (A, a)/k be a k-isomorphism of p.p.a.v. Let ω1, . . . , ωg ∈ Ω1

k[A], ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A].

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Main result and ingredients of the proof

First ingredient of the proof : action of twists

Let φ : (A′, a′)/k − → (A, a)/k be a k-isomorphism of p.p.a.v. Let ω1, . . . , ωg ∈ Ω1

k[A], ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A].

Let γi = φ∗(ωi) and γ = γ1 ∧ · · · ∧ γg ∈ ωk[A′].

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Main result and ingredients of the proof

First ingredient of the proof : action of twists

Let φ : (A′, a′)/k − → (A, a)/k be a k-isomorphism of p.p.a.v. Let ω1, . . . , ωg ∈ Ω1

k[A], ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A].

Let γi = φ∗(ωi) and γ = γ1 ∧ · · · ∧ γg ∈ ωk[A′]. Let ω′

1, . . . , ω′ g ∈ Ω1 k[A′] and ω′ = ω′ 1 ∧ · · · ∧ ω′ g.

Let Mφ ∈ GLg(k) the matrix of the basis (γi) in the basis (ω′

i).

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Main result and ingredients of the proof

First ingredient of the proof : action of twists

Let φ : (A′, a′)/k − → (A, a)/k be a k-isomorphism of p.p.a.v. Let ω1, . . . , ωg ∈ Ω1

k[A], ω = ω1 ∧ · · · ∧ ωg ∈ ωk[A].

Let γi = φ∗(ωi) and γ = γ1 ∧ · · · ∧ γg ∈ ωk[A′]. Let ω′

1, . . . , ω′ g ∈ Ω1 k[A′] and ω′ = ω′ 1 ∧ · · · ∧ ω′ g.

Let Mφ ∈ GLg(k) the matrix of the basis (γi) in the basis (ω′

i).

Proposition f ∈ Sg,h(k), f ((A, a), ω) = det(Mφ)h · f ((A′, a′), ω′). Assume g odd, h even and char k = 2. Let f ∈ Sg,h(k) and φ : A′ − → A a non trivial quadratic twist. There exists c ∈ k \ k2 such that f ((A, a), ω) ≡ ch/2 · f ((A, a), ω′) (mod k∗h).

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Main result and ingredients of the proof

Second ingredient : Teichmüller modular forms

Mg : moduli stack of smooth and proper curves of genus g π : Cg − − − − → Mg (universal curve) Ω1 : rank g bundle induced by relative regular differential forms on Cg/Mg. Relative canonical line bundle λCg/Mg = g π∗Ω1 over Mg : λCg/Mg  

• Mg

Ω1

k[X] = H0(X, Ω1 X ⊗ k),

λk[X] =

g

• Ω1

k[X].

Space of Teichmüller modular forms over R, genus g, weight h : Tg,h(R) = Γ(Mg ⊗ R, λ⊗h).

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Main result and ingredients of the proof

As for Siegel modular forms, we define f (X, λ) = f (X)/λ⊗h ∈ k if f ∈ Tg,h(k) and λ is a basis of λk[X]. The Torelli map θ : Mg − → Ag satisfies θ∗ω = λ, and induces a linear map θ∗ : Sg,h(k) − − − − → Tg,h(k) such that f ((Jac X, j), ω) = [θ∗f ](X, θ∗ω). Theorem (Ichikawa) There exists µh/2 ∈ Tg,h/2(Z) such that θ∗(χh) = µ2

h/2.

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Main result and ingredients of the proof

The proof

Argument 2 implies that for any X/k in Mg : χ18((Jac X, j), ω) = µ9(X, θ∗ω)2 ∈ k.

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Main result and ingredients of the proof

The proof

Argument 2 implies that for any X/k in Mg : χ18((Jac X, j), ω) = µ9(X, θ∗ω)2 ∈ k. Argument 1 implies that there is a c ∈ k \ k2 and χ18((Jac X, j)ε, ω′) ≡ c9 ·χ18(Jac X, j), ω) ≡ c9 ·µ9(X, θ∗ω)2 (mod k∗18) is not a square.

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Main result and ingredients of the proof

The proof

Argument 2 implies that for any X/k in Mg : χ18((Jac X, j), ω) = µ9(X, θ∗ω)2 ∈ k. Argument 1 implies that there is a c ∈ k \ k2 and χ18((Jac X, j)ε, ω′) ≡ c9 ·χ18(Jac X, j), ω) ≡ c9 ·µ9(X, θ∗ω)2 (mod k∗18) is not a square. By-products :

1 Klein’s formula : µ9 = ± Disc ; 28 / 54

Main result and ingredients of the proof

The proof

Argument 2 implies that for any X/k in Mg : χ18((Jac X, j), ω) = µ9(X, θ∗ω)2 ∈ k. Argument 1 implies that there is a c ∈ k \ k2 and χ18((Jac X, j)ε, ω′) ≡ c9 ·χ18(Jac X, j), ω) ≡ c9 ·µ9(X, θ∗ω)2 (mod k∗18) is not a square. By-products :

1 Klein’s formula : µ9 = ± Disc ; 2 Beyond genus 3

cannot work for g even ; need to find forms of weight h such that h/2 is odd ; cannot take χh (because it is only a square and not more) ; but another Klein’s formula for a genus 4 curve X

• χ68(τa)

det(Ω2)68 = c · ∆(X)2 · T(X)8.

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Main result and ingredients of the proof

The proof

Argument 2 implies that for any X/k in Mg : χ18((Jac X, j), ω) = µ9(X, θ∗ω)2 ∈ k. Argument 1 implies that there is a c ∈ k \ k2 and χ18((Jac X, j)ε, ω′) ≡ c9 ·χ18(Jac X, j), ω) ≡ c9 ·µ9(X, θ∗ω)2 (mod k∗18) is not a square. By-products :

1 Klein’s formula : µ9 = ± Disc ; 2 Beyond genus 3

cannot work for g even ; need to find forms of weight h such that h/2 is odd ; cannot take χh (because it is only a square and not more) ; but another Klein’s formula for a genus 4 curve X

• χ68(τa)

det(Ω2)68 = c · ∆(X)2 · T(X)8.

3 Still works over finite fields of char = 2 . . . 30 / 54

Explicit computations and application to optimal curves

. . . but how to compute it ? An example

• R. 09 : worked out the procedure in the case A = E 3 where E is an elliptic

curve with CM. ∃ ? optimal curve C/F47 : A = Jac C ∼ E 3 with E CM by O = Z[τ] where τ = (1 + √−19)/2.

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Explicit computations and application to optimal curves

. . . but how to compute it ? An example

• R. 09 : worked out the procedure in the case A = E 3 where E is an elliptic

curve with CM. ∃ ? optimal curve C/F47 : A = Jac C ∼ E 3 with E CM by O = Z[τ] where τ = (1 + √−19)/2. Cl(O) = 1 ⇒ A = E 3.

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Explicit computations and application to optimal curves

. . . but how to compute it ? An example

• R. 09 : worked out the procedure in the case A = E 3 where E is an elliptic

curve with CM. ∃ ? optimal curve C/F47 : A = Jac C ∼ E 3 with E CM by O = Z[τ] where τ = (1 + √−19)/2. Cl(O) = 1 ⇒ A = E 3. a0 the product polarization on A : {a p.p. on A} ← → {M = a−1

0 a ∈ M3(O) positive definite

hermitian of determinant 1}.

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Explicit computations and application to optimal curves

. . . but how to compute it ? An example

• R. 09 : worked out the procedure in the case A = E 3 where E is an elliptic

curve with CM. ∃ ? optimal curve C/F47 : A = Jac C ∼ E 3 with E CM by O = Z[τ] where τ = (1 + √−19)/2. Cl(O) = 1 ⇒ A = E 3. a0 the product polarization on A : {a p.p. on A} ← → {M = a−1

0 a ∈ M3(O) positive definite

hermitian of determinant 1}. classification by Schiemann of such matrices for some orders. There is

• nly one in this case :

M =   2 1 −1 1 3 −2 + τ −1 −2 + τ 3   .

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Explicit computations and application to optimal curves

lift E as a CM curve over Q : ˜ E : y2 = x3 − 152x − 722 ;

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Explicit computations and application to optimal curves

lift E as a CM curve over Q : ˜ E : y2 = x3 − 152x − 722 ; find a period matrix associated to (˜ E 3, a0M) w.r.t. wedge product of the pull back ω0 of the differential dx/(2y) on ˜ E : c1(a0M) = 1 Im(ω1ω2)

tM

where [ω1, ω2] is a period matrix of ˜ E w.r.t. dx/(2y) ;

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Explicit computations and application to optimal curves

lift E as a CM curve over Q : ˜ E : y2 = x3 − 152x − 722 ; find a period matrix associated to (˜ E 3, a0M) w.r.t. wedge product of the pull back ω0 of the differential dx/(2y) on ˜ E : c1(a0M) = 1 Im(ω1ω2)

tM

where [ω1, ω2] is a period matrix of ˜ E w.r.t. dx/(2y) ; compute an analytic approximation of χ18((˜ E 3, a0M), ω0) = (219 · 197)2;

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Explicit computations and application to optimal curves

lift E as a CM curve over Q : ˜ E : y2 = x3 − 152x − 722 ; find a period matrix associated to (˜ E 3, a0M) w.r.t. wedge product of the pull back ω0 of the differential dx/(2y) on ˜ E : c1(a0M) = 1 Im(ω1ω2)

tM

where [ω1, ω2] is a period matrix of ˜ E w.r.t. dx/(2y) ; compute an analytic approximation of χ18((˜ E 3, a0M), ω0) = (219 · 197)2; since it is a square (over F47), such an optimal curve C exists. Guàrdia (09) : ˜ C : x4+(1/9)y4+(2/3)x2y2−190y2−570x2+(152/9)y3−152x2y = 1083.

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Explicit computations and application to optimal curves

Values of χ = χ18((˜ E 3, a0M), ω0)

˜ E : Gross’ models with discriminant d3 (when class number is 1).

d M, τ = (1 + √ d)/2 χ # Aut( ˜ E3, a) −7 @ 2 1 1 1 2 τ 1 τ 2 1 A (77)2 2 · 168 −19 @ 2 1 −1 1 3 −2 + τ −1 −2 + τ 3 1 A (25 · 197)2 · (−2) 2 · 6 −43 @ 3 1 1 − τ 1 4 2 1 − τ 2 5 1 A (26 · 437)2 · (−47 · 79 · 107 · 173) 2 · 1 −163 @ 2 1 −τ 1 2 1 − τ −τ 1 − τ 28 1 A (25 · 74 · 114 · 1637)2 · (−2 · 7 · 11 · 19 · 127) 2 · 6 −15 @ 2 −1 −1 + τ −1 2 1 − τ −1 + τ 1 − τ 3 1 A 22769095299822142340569171645771726299/4+ 10182522603020834484863085151244322675 · √ 5/4+ 4462640909353821881995695647429476869 · √−15/4 +9978330617922886443823982755114202445 · √−3 2 · 24 39 / 54

Explicit computations and application to optimal curves

Values of χ = χ18((˜ E 3, a0M), ω0)

˜ E : Gross’ models with discriminant d3 (when class number is 1).

d M, τ = (1 + √ d)/2 χ # Aut( ˜ E3, a) −7 @ 2 1 1 1 2 τ 1 τ 2 1 A (77)2 2 · 168 −19 @ 2 1 −1 1 3 −2 + τ −1 −2 + τ 3 1 A (25 · 197)2 · (−2) 2 · 6 −43 @ 3 1 1 − τ 1 4 2 1 − τ 2 5 1 A (26 · 437)2 · (−47 · 79 · 107 · 173) 2 · 1 −163 @ 2 1 −τ 1 2 1 − τ −τ 1 − τ 28 1 A (25 · 74 · 114 · 1637)2 · (−2 · 7 · 11 · 19 · 127) 2 · 6 −15 @ 2 −1 −1 + τ −1 2 1 − τ −1 + τ 1 − τ 3 1 A 22769095299822142340569171645771726299/4+ 10182522603020834484863085151244322675 · √ 5/4+ 4462640909353821881995695647429476869 · √−15/4 +9978330617922886443823982755114202445 · √−3 2 · 24

Consequences : d = −19 : defect 0 curve for q = 47, 61, 137, 277 (see Top, Alekseenko et al.). d = −67 : defect 0 curve for q = 233.

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Primes dividing χ18

Geometric interpretation of χ18

Two remarkable geometric forms defined from even Thetanullwerte : — the product χ18 (0 iff at least one Thetanullwerte is 0) ; — the 35th symmetric function Σ140 of the 8th powers (0 if two Thetanullwerte are 0).

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Primes dividing χ18

Geometric interpretation of χ18

Two remarkable geometric forms defined from even Thetanullwerte : — the product χ18 (0 iff at least one Thetanullwerte is 0) ; — the 35th symmetric function Σ140 of the 8th powers (0 if two Thetanullwerte are 0). Theorem (Igusa) Let (A, a) ≃ C3/τaZ3 + Z3 ∈ A3(C).

1 (A, a) is decomposable if and only if

χ18(τa) = Σ140(τa) = 0.

2 (A, a) is a hyperelliptic Jacobian if and only if

• χ18(τa) = 0

and

• Σ140(τa) = 0.

3 (A, a) is a non hyperelliptic Jacobian if and only if

χ18(τa) = 0.

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Primes dividing χ18

An arithmetic version

Proposition (Ichikawa) χ18 is primitive (i.e. it is not zero modulo any prime). We need a similar result for Σ140.

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Primes dividing χ18

An arithmetic version

Proposition (Ichikawa) χ18 is primitive (i.e. it is not zero modulo any prime). We need a similar result for Σ140. Proposition Σ140 = (2iπ)140 2218 ·

• Σ140(τa)

det(Ω2)140 (ω1 ∧ ω2 ∧ ω3)⊗140 is a primitive Siegel modular form of weight 140 over Z.

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Primes dividing χ18

An arithmetic version

Proposition (Ichikawa) χ18 is primitive (i.e. it is not zero modulo any prime). We need a similar result for Σ140. Proposition Σ140 = (2iπ)140 2218 ·

• Σ140(τa)

det(Ω2)140 (ω1 ∧ ω2 ∧ ω3)⊗140 is a primitive Siegel modular form of weight 140 over Z. Conjecture : if χ18((A, a)) = Σ140((A, a)) = 0 then (A, a) is decomposable. Corollary : if the conjecture holds then Igusa’s theorem is valid over finite fields as well.

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Primes dividing χ18

Basic results

Mestre’s observation : if 12|#Aut(˜ E 3, a) then the primes which divide χ are the prime factors of j(˜ E) − 1728 apart from 3.

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Primes dividing χ18

Basic results

Mestre’s observation : if 12|#Aut(˜ E 3, a) then the primes which divide χ are the prime factors of j(˜ E) − 1728 apart from 3. Let (A, a) = (˜ E 3, a) with a indecomposable defined over a local ring S of a number field with residue field F at a prime p. Proposition ? : Assume that ˜ E has CM by O maximal and ˜ E ⊗ F is smooth. If (A, a) ⊗ F is decomposable then ˜ E ⊗ F is supersingular. The converse is true if char F = 2. In particular p|χ.

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Primes dividing χ18

Basic results

Mestre’s observation : if 12|#Aut(˜ E 3, a) then the primes which divide χ are the prime factors of j(˜ E) − 1728 apart from 3. Let (A, a) = (˜ E 3, a) with a indecomposable defined over a local ring S of a number field with residue field F at a prime p. Proposition ? : Assume that ˜ E has CM by O maximal and ˜ E ⊗ F is smooth. If (A, a) ⊗ F is decomposable then ˜ E ⊗ F is supersingular. The converse is true if char F = 2. In particular p|χ. Question : how to detect hyperelliptic reduction ? d = −15 : √−3 √ 5 results over F193 −4 9 a defect 3 non hyperelliptic curve −4 −9 a defect 3 non hyperelliptic curve 4 −9 a minimal defect 3 non hyperelliptic curve 4 9 a hyperelliptic curve

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New strategies ?

To move out from the border by isogeny

Let k be an arbitrary field of characteristic not 2. E/k : y2 = x(x − α)(x − β) an elliptic curve and A = E 3/W where W ⊂ E 3[2] is a certain isotropic subgroup.

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New strategies ?

To move out from the border by isogeny

Let k be an arbitrary field of characteristic not 2. E/k : y2 = x(x − α)(x − β) an elliptic curve and A = E 3/W where W ⊂ E 3[2] is a certain isotropic subgroup. Let a0 be the product polarization on E 3 and a be the descent indecomposable p.p. on A from 2a0. Theorem (simplest case of Howe-Leprevost-Poonen (2002)) (A, a) is a Jacobian if and only if 3α + β ∈ k2.

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New strategies ?

To move out from the border by isogeny

Let k be an arbitrary field of characteristic not 2. E/k : y2 = x(x − α)(x − β) an elliptic curve and A = E 3/W where W ⊂ E 3[2] is a certain isotropic subgroup. Let a0 be the product polarization on E 3 and a be the descent indecomposable p.p. on A from 2a0. Theorem (simplest case of Howe-Leprevost-Poonen (2002)) (A, a) is a Jacobian if and only if 3α + β ∈ k2. Question : can we say something about this value a priori ?

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New strategies ?

To move out from the border by isogeny

Let k be an arbitrary field of characteristic not 2. E/k : y2 = x(x − α)(x − β) an elliptic curve and A = E 3/W where W ⊂ E 3[2] is a certain isotropic subgroup. Let a0 be the product polarization on E 3 and a be the descent indecomposable p.p. on A from 2a0. Theorem (simplest case of Howe-Leprevost-Poonen (2002)) (A, a) is a Jacobian if and only if 3α + β ∈ k2. Question : can we say something about this value a priori ? In the same spirit, when k = F2n. Theorem (Nart-R. (2008 and 2009)) If n > 5 is even then there is always a maximal curve ; If n is odd and ⌊2 √ 2n⌋ ≡ 1, 5, 7 (mod 8) there is always a maximal curve.

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New strategies ?

To move out from the border by change of polarization

Can we algebraically link the Thetanullwerte on A corresponding to two non isomorphic principal polarizations ?

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New strategies ?

Thank you for your attention !

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