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 1 / 54

Outline Geometric versus arithmetic Torelli theorem 1 Siegel modular forms 2 A special modular form : χ h 3 Main result and ingredients of the proof 4 Explicit computations and application to optimal curves 5 Primes dividing χ 18 6 New strategies ? 7 2 / 54

Geometric versus arithmetic Torelli theorem Geometric Torelli theorem k algebraically closed field. A g : space of (isom. classes of) g -dimensional p.p.a.v. M g : 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 ) M g − − − − → A g is injective. 3 / 54

Geometric versus arithmetic Torelli theorem Arithmetic Torelli theorem k arbitrary field. ( A , a ) / k ∈ A g such that ( A , a ) ≃ k ( Jac X 0 , j 0 ) . Arithmetic Torelli theorem (Serre) There is a model X / k of X 0 such that : 1 If X 0 is hyperelliptic, there is a k-isomorphism ∼ ( Jac X , j ) − − − − → ( A , a ) . 2 If X 0 is not hyperelliptic, there is a quadratic character ε of Gal ( k sep / 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). 4 / 54

Geometric versus arithmetic Torelli theorem Jacobians in dimension 2 and 3 g = 2 (resp. g = 3) dim M g = 3 g − 3 = dim A g = g ( g + 1 ) / 2 = 3 (resp. 6 ) . ( A , a ) / k indecomposable : ( A , a ) is not isomorphic to a product of p.p.a.v. abelian variety of dimension g, with an Curve of genus g ≤ 3 Jac ⇐ ⇒ indecomposable over k principal polarization 5 / 54

Geometric versus arithmetic Torelli theorem Applications to maximal curves of genus g ≤ 3 To construct a genus g curve with many points N over F q : find an abelian variety A over F q with trace of Frobenius q + 1 − N ; 6 / 54

Geometric versus arithmetic Torelli theorem Applications to maximal curves of genus g ≤ 3 To construct a genus g curve with many points N over F q : find an abelian variety A over F q with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ; 7 / 54

Geometric versus arithmetic Torelli theorem Applications to maximal curves of genus g ≤ 3 To construct a genus g curve with many points N over F q : find an abelian variety A over F q with trace of Frobenius q + 1 − N ; prove that A has an indecomposable principal polarization ; use arithmetic Torelli to conclude. 8 / 54

Geometric versus arithmetic Torelli theorem Applications to maximal curves of genus g ≤ 3 To construct a genus g curve with many points N over F q : find an abelian variety A over F q 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 F q such that | # C ( F q ) − q − 1 | ≥ 3 m − 3 . 9 / 54

Geometric versus arithmetic Torelli theorem Applications to maximal curves of genus g ≤ 3 To construct a genus g curve with many points N over F q : find an abelian variety A over F q 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 F q such that | # C ( F q ) − q − 1 | ≥ 3 m − 3 . Serre’s Question (letter to Top, February 2003) : how to compute the twist ε ? 10 / 54

Siegel modular forms Analytic Siegel modular forms Siegel upper half space of genus g � � τ ∈ M g ( C ) | t τ = τ, Im τ > 0 H g = . R g , h ( C ) : space of analytic Siegel modular forms of weight h on H g , i.e. holomorphic functions φ ( τ ) on H g satisfying φ ( M .τ ) = det ( c τ + d ) h φ ( τ ) � a � b for any M = ∈ Sp 2 g ( Z ) . c d 11 / 54

Siegel modular forms Geometric Siegel modular forms A g : moduli stack of p.p.a.s. of relative dimension g π : V g − − − − → A g (universal p.p.a.s) Ω 1 : rank g bundle induced by relative regular differential forms on V g / A g . Relative canonical line bundle ω V g / A g = � g π ∗ Ω 1 over A g : ω V g / A g  g �  Ω 1 k [ A ] = H 0 ( A , Ω 1 Ω 1 � A ⊗ k ) , ω k [ A ] = k [ A ] . A g Space of geometric Siegel modular forms of weight h over a ring R : S g , h ( R ) = Γ( A g ⊗ R , ω ⊗ h ) 12 / 54

Siegel modular forms Analytic and geometric modular forms A = ( A , a ) p.p.a.v. of dimension g defined over k ⊂ C . 13 / 54

Siegel modular forms Analytic and geometric modular forms A = ( A , a ) p.p.a.v. of dimension g defined over k ⊂ C . ω 1 , . . . , ω g basis of Ω 1 k [ A ] and γ 1 , . . . γ 2 g symplectic basis (for a ) such that   � � γ 1 ω 1 · · · γ 2 g ω 1   ( A , a ) ≃ ( C g / Ω a Z 2 g , J 2 g ) Ω a = [Ω 1 Ω 2 ] = . . . . . .   � � γ 1 ω g · · · γ 2 g ω g satisfies τ a = Ω − 1 2 Ω 1 ∈ H g . 14 / 54

Siegel modular forms Analytic and geometric modular forms A = ( A , a ) p.p.a.v. of dimension g defined over k ⊂ C . ω 1 , . . . , ω g basis of Ω 1 k [ A ] and γ 1 , . . . γ 2 g symplectic basis (for a ) such that   � � γ 1 ω 1 · · · γ 2 g ω 1   ( A , a ) ≃ ( C g / Ω a Z 2 g , J 2 g ) Ω a = [Ω 1 Ω 2 ] = . . . . . .   � � γ 1 ω g · · · γ 2 g ω g satisfies τ a = Ω − 1 2 Ω 1 ∈ H g . Proposition Let ω = ω 1 ∧ · · · ∧ ω g ∈ ω k [ A ] and � f ∈ R g , h ( C ) . � f ( τ a ) f (( A , a )) = ( 2 i π ) gh ( det Ω 2 ) h · ω ⊗ h . Then the map � f �→ f is an isomorphism R g , h ( C ) − → S g , h ( C ) . 15 / 54

Siegel modular forms Analytic and geometric modular forms A = ( A , a ) p.p.a.v. of dimension g defined over k ⊂ C . ω 1 , . . . , ω g basis of Ω 1 k [ A ] and γ 1 , . . . γ 2 g symplectic basis (for a ) such that   � � γ 1 ω 1 · · · γ 2 g ω 1   ( A , a ) ≃ ( C g / Ω a Z 2 g , J 2 g ) Ω a = [Ω 1 Ω 2 ] = . . . . . .   � � γ 1 ω g · · · γ 2 g ω g satisfies τ a = Ω − 1 2 Ω 1 ∈ H g . Proposition Let ω = ω 1 ∧ · · · ∧ ω g ∈ ω k [ A ] and � f ∈ R g , h ( C ) . � f ( τ a ) f (( A , a )) = ( 2 i π ) gh ( det Ω 2 ) h · ω ⊗ h . Then the map � f �→ f is an isomorphism R g , h ( C ) − → S g , h ( C ) . Notation : f (( A , a ) , ω ) = f (( A , a )) /ω ⊗ h ∈ C . 16 / 54

A special modular form : χ h Thetanullwerte and the form � χ h Thetanullwerte of characteristic ( ε, η ) ∈ { 0 , 1 } g × { 0 , 1 } g : � ε � � � � � � i π ( n + ε 2 ) .τ. ( n + ε i πη. ( n + ε θ ( τ ) = exp 2 ) exp 2 ) . η n ∈ Z g Even characteristics : ε.η ≡ 0 ( mod 2 ) . Theorem (Igusa-Ichikawa) If g ≥ 3 , then � ε � � ( − 1 ) gh / 2 h = 2 g − 2 ( 2 g + 1 ) . χ h ( τ ) = � 2 2 g − 1 ( 2 g − 1 ) · θ ( τ ) ∈ R g , h ( C ) , η even χ ( τ a ) � ( det Ω 2 ) h · ω ⊗ h ∈ S g , h ( Z ) . χ h (( A , a )) = ( 2 i π ) gh 17 / 54

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 ∈ ω k [ A ] . 1 χ 18 (( A , a )) = 0 if and only if ( A , a ) is hyperelliptic. 2 With the previous notation � ε � � even θ ( τ a ) χ 18 (( A , a ) , ω ) = ( 2 π ) 54 η · 2 28 det (Ω 2 ) 18 is a square in k ∗ if and only if ( A , a ) is a non hyperelliptic Jacobian. Quadratic character given on σ ∈ Gal ( k sep / k ) by ε ( σ ) = d σ � d , d = χ 18 (( A , a ) , ω ) 18 / 54

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. 19 / 54

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 ω 1 , . . . , ω g ∈ Ω 1 k [ A ] , ω = ω 1 ∧ · · · ∧ ω g ∈ ω k [ A ] . 20 / 54

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 ω 1 , . . . , ω g ∈ Ω 1 k [ A ] , ω = ω 1 ∧ · · · ∧ ω g ∈ ω k [ A ] . Let γ i = φ ∗ ( ω i ) and γ = γ 1 ∧ · · · ∧ γ g ∈ ω k [ A ′ ] . 21 / 54