From the curve to its Jacobian and back

Transcription

1 From the curve to its Jacobian and back Christophe Ritzenthaler Institut de Mathématiques de Luminy, CNRS Montréal web: ritzenth/ Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

2 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

3 Link with the conference Why do we care? CM method: CM-type + fundamental { unit lattice + polarization the curve over C period matrix ThetaNullwerte curve /F q. invariants AGM for point counting: curve /F q lift quotients of ThetaNullwerte canonical lift + info on Weil polynomial Weil polynomial. Other applications: fast computation of modular polynomials, class polynomials, isogenies... Caution: work over C but try to show why it works in general. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

4 Period matrices and ThetaNullwerte Period matrices 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

5 Definitions Period matrices and ThetaNullwerte Period matrices Let C be a curve over k C of genus g > 0. The Jacobian of C is a torus Jac(C) C g /Λ where the lattice Λ = ΩZ 2g, the matrix Ω = [Ω 1, Ω 2 ] M g,2g (C) is a period matrix and τ = Ω 1 2 Ω 1 H g = {M GL g (C), t M = M, Im M > 0} is a Riemann matrix. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

6 Construction Period matrices and ThetaNullwerte Period matrices v 1,..., v g be a k-basis of H 0 (C, Ω 1 ), δ 1,..., δ 2g be generators of H 1 (C, Z) such that (δ i ) 1...2g form a symplectic basis for the intersection pairing on C. Ω := [Ω 1, Ω 2 ] = [ δ j v i ] i = 1,..., g j = 1,..., 2g. Magma (Vermeulen): can compute Ω for a hyperelliptic curve. Maple (Deconinck, van Hoeij) can compute Ω for any plane model. Rem: there is a polarization j involved in the definition of Ω with Chern class ( ( ) ) i Ω t Ω. 1 0 Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

7 Example Period matrices and ThetaNullwerte Period matrices Ex: E : y 2 = x 3 35x 98 = (x 7)(x a)(x ā) which has complex multiplication by Z[α] with α = and a = [ ā dx 7 ] Ω = 2 a 2y, 2 dx = c [α, 1]. a 2y (Chowla, Selberg 67) formula gives c = 1 8π 7 Γ(1 7 ) Γ(2 7 ) Γ(4 7 ) with Γ(x) = 0 t z 1 exp( t) dt. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

8 Period matrices and ThetaNullwerte ThetaNullwerte 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

9 Period matrices and ThetaNullwerte Projective embedding ThetaNullwerte The polarization j comes from an ample divisor D on Jac(C) (defined up to translation). Theorem (Lefschetz, Mumford, Kempf) For n 3, nd is very ample, i.e. one can embed Jac(C) in a P ng 1 with a basis of sections of L(nD). For n = 4, the embedding is given by intersection of quadrics, whose equations are completely determined by the image of 0. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

10 ThetaNullwert Period matrices and ThetaNullwerte ThetaNullwerte A basis of sections of L(4D) is given by theta functions θ[ε](2z, τ) with integer characteristics [ε] = (ɛ, ɛ ) {0, 1} 2g where [ ] ɛ θ ɛ (z, τ) = exp (iπ (n + ɛ2 )τ t (n + ɛ2 ) + 2iπ (n + ɛ2 ) )t (z + ɛ 2 ). n Z g When ɛ t ɛ 0 (mod 2), [ε] is said even and one calls ThetaNullwert [ ] [ ] ɛ ɛ θ (0, τ) = θ (τ) = θ[ε](τ) = θ ab ɛ where the binary representations of a and b are ɛ, ɛ. ɛ Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

11 Example Period matrices and ThetaNullwerte ThetaNullwerte Let q = exp(πiτ). There are 3 genus 1 ThetaNullwerte: [ ] 0 θ 00 = θ (0, τ) = q n2, 0 n Z [ ] 1 θ 10 = θ (0, τ) = q 0 (n+ 1 2) 2, n Z [ ] 0 θ 01 = θ (0, τ) = 1 n Z( 1) n q n2. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

12 Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

13 Period matrices and ThetaNullwerte Case g = 1 Gauss, Cox 84, Dupont 07 From the ThetaNullwerte to the Riemann matrix Let z = θ 01 (τ) 2 /θ 00 (τ) 2. Duplication formulae vs AGM formulae : θ 00 (2τ) 2 = θ 00(τ) 2 +θ 01 (τ) 2 2 a n = a n 1+b n 1 2, θ 01 (2τ) 2 = θ 00 (τ) θ 01 (τ) b n = a n 1 b n 1, θ 10 (2τ) 2 = θ 00(τ) 2 θ 01 (τ) 2 2 AGM(θ 00 (τ) 2, θ 01 (τ) 2 ) = lim θ 00 (2 n τ) 2 = 1 AGM(1, z) = 1 θ 00 (τ) 2. θ 10 (τ) 2 = θ 00 (τ) 4 θ 01 (τ) 4. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

14 Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Transformation formula : θ 00 (τ) 2 = i ( ) 1 2 τ θ 00, θ 10(τ) 2 = i ( ) 1 2 τ τ θ 01. τ AGM(θ 00 (τ) 2, θ 10 (τ) 2 ) = i τ lim θ 00(2 n 1 τ )2 = i τ 1 AGM(1, 1 z 2 ) = i τ 1 θ 00 (τ) 2. Proposition i AGM(1, z) AGM(1, 1 z 2 ) = τ. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

15 Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Difficulty: define the correct square root when the values are complex. Rem: one cannot get Ω from the ThetaNullwerte. But from the curve: Theorem (Gauss, Cox 84) If E : y 2 = x(x a 2 )(x b 2 ) then [ω 1, ω 2 ] = [ a period matrix relative to dx/y. π AGM(a,b), iπ AGM(a+b,a b) ] is Use the same ingredients as above and, as first step, the Thomae s formulae ω 2 a = π θ 00 (τ) 2, ω 2 b = π θ 01 (τ) 2. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

16 Case g 2 Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Particular case: real Weierstrass points and g = 2 (Bost-Mestre 88). General case (Dupont 07): under some (experimentally verified) conjectures. Proposition One can compute τ in terms of θ[ε](τ) 2 /θ[0](τ) 2 in time O(g 2 2 g M(n) log n) for n digits of precision (M(n) is the complexity of the binary multiplication). Question: what about the period matrix? Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

17 Period matrices and ThetaNullwerte From the Riemann matrix to the (quotients of) ThetaNullwerte 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

18 Period matrices and ThetaNullwerte The work of (Dupont 07) From the Riemann matrix to the (quotients of) ThetaNullwerte Naive method: O(M(n) n) for g = 1 and O(n 2+ɛ ) for g = 2. New method: invert the AGM. Complexity for n bits of precision on the quotients O(M(n) log n) for g = 1, O(n 1+ɛ ) for g = 2 (conjectural algorithm). Main idea for g = 1: let f (z) = i AGM(1, z) τ AGM(1, 1 z 2 ). Then f (θ 01 (τ) 2 /θ 00 (τ) 2 ) = 0. Do a Newton algorithm on f. can we get rid of the conjectures? can we generalize to all genera? can we compute the ThetaNullwerte alone? Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

19 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

20 Thomae s formulae From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Let C be a hyperelliptic curve C : y 2 = 2g+1 i=1 (x λ i). Theorem (Thomae s formulae) ( ) det 2 θ[ε](τ) 4 Ω2 = ± (λ i λ j ) π g (i,j) I with the choice of the basis of differentials x i dx/y (the set I depends on [ε] and on the basis of H 1 (C, Z)). Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

21 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Proof: see (Fay 73) using a variational method. Proof for the quotients: study the zeroes of the section s ε (P) = θ[ε](φ P0 (P)) where P 0 C and φ P0 (P) = P P 0 Jac(C). c f (P) = sε(p)2 s ε (P) 2 for an explicit f C(C). c = sε(p 1) 2 s ε (P 1 ) 2 f (P 1 ) = sε(p 2) 2 s ε (P 2 ) 2 f (P 2 ) for P 1, P 2 such that sε(p 2) 2 s ε (P 2 ) 2 = s ε (P 1) 2 s ε(p 1 ) 2. Rem: work in progress by Cosset for non-integral [ε]. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

22 From the curve to its Jacobian Main result on s ε (P) Hyperelliptic case and the first tool: s ε Let C be a curve of genus g > 0, P 0 C. Theorem If s ε (P) is not identically zero, then s ε (P) has g zeroes P 1,..., P g such that the divisor D = P P g is characterized by D gp 0 ε + κ where κ is a constant depending only on the homology basis and on P 0. Geometrically, let Θ = {z, θ[0](z, τ) = 0} Jac(C), L be the corresponding ample line bundle. Poincaré s formula (φ P0 (C) Θ) = g. s ε is a section of the line bundle L ε = φ P 0 t εl = O C (P P g ). Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

23 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Riemann s theorem: κ 0 a theta characteristic such that Sym g 1 C κ 0 = Θ. L 0 = φ P 0 L = O C (P P0 g ) with P P0 g κ 0 + P 0. Indeed P 0 i P 0 ( j i P0 j κ 0 ) Θ = Θ. Canonical isomorphism: φ P 0 : Pic 0 (Jac(C)) = Jac(C) Pic 0 (C) = Jac(C) is an isomorphism with inverse φ L. φ P 0 (t εl L 1 ) = φ 1 L where κ = κ 0 (g 1)P 0. φ L(ε) = O C (ε) = O C ( P i P 0 κ 0 ) = O C ( P i κ) Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

24 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Lemma s ε 0 ε + κ D gp 0 with D Sym g (C) and i(d) > 0. s ε 0 P, P P 0 ε Θ P P 0 D + gp 0 + κ 0 (g 1)P 0 Sym g 1 (C) κ 0 P D + 2κ 0 Sym g 1 (C) D P Sym g 1 (C) i(d) > 0 Corollary The zero divisor D of s ε is completely determined by the equivalence D gp 0 ε + κ. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

25 From the curve to its Jacobian Non hyperelliptic case and the second tool: Jacobian Nullwerte 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

26 From the curve to its Jacobian Non hyperelliptic case genus 3 Non hyperelliptic case and the second tool: Jacobian Nullwerte Let C be a smooth plane quartic. Theorem (Weber 1876) ( ) θ[ε](τ) 4 θ[ε = [b i, b j, b ij ][b ik, b jk, b ij ][b j, b jk, b k ][b i, b ik, b k ] ](τ) [b j, b jk, b ij ][b i, b ik, b ij ][b i, b j, b k ][b ik, b jk, b k ] where the b i, b ij are linear equations of certain bitangents of C and [b i, b j, b k ] is the determinant of the matrix of the coefficients of (once for all fixed) equations of the bitangents. Weber s proof uses s ε (P). Question: can we find a formula for a Thetanullwert alone like in the hyperelliptic case? Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

27 From the curve to its Jacobian Derivative of theta functions Non hyperelliptic case and the second tool: Jacobian Nullwerte When ɛ t ɛ 1 (mod 2), [ε] is said odd and we write [µ] instead. Definition The theta gradient (with odd characteristic [µ]) is the vector ( ) θ[µ](z, τ) θ[µ](z, τ) θ[µ] := (0, τ),..., (0, τ). z 1 z g The theta hyperplane is the projective hyperplane of P g 1 defined by a theta gradient. We denote the matrix θ[µ] (X 1,..., X g ) = 0 J[µ 1,..., µ g ] := ( θ[µ 1 ],..., θ[µ g ] ) and [µ 1,..., µ g ] its determinant (called Jacobian Nullwerte). Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

28 From the curve to its Jacobian Non hyperelliptic case and the second tool: Jacobian Nullwerte Case of Riemann-Mumford-Kempf singularity theorem Let C be any curve of genus g > 0. Theorem Let φ be the canonical map φ : C P g 1, P (ω 1 (P),..., ω g (P)). Let D be an effective divisor of degree g 1 on C such that h 0 (D) = 1. Then ( ) θ(z, τ) θ(z, τ) (D κ 0, τ), (D κ 0, τ) Ω 1 2 t (X 1,..., X g ) = 0 z 1 z g is an hyperplan of P g 1 which cuts out the divisor φ(d) on the curve φ(c). Rem: this can also be re-interpreted geometrically in terms of Gauss maps. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

29 where S is the set of all g + 2-tuples {[ε g+1 ],..., [ε 2g+2 ]} even theta characteristics such that {[µ 1 ],..., [µ g ], [ε g+1 ],..., [ε 2g+2 ]} forms a fundamental system. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40 From the curve to its Jacobian Non hyperelliptic case and the second tool: Jacobian Nullwerte Generalization of Jacobi s derivative formula [ ] 1 dθ (z, τ) [ ] [ ] [ ] (0, τ) = π θ (0, τ) θ (0, τ) θ (0, τ). dz Theorem (Igusa 80) Let [µ 1 ],..., [µ g ] be distinct odd theta characteristics such that the function [µ 1,..., µ g ](τ) is contained in the C-algebra C[θ] generated by the functions θ[ε](τ) for all even characteristics [ε]. Then [µ 1,..., µ g ](τ) = π g [ε g+1 ],...,[ε 2g+2 ] S ± 2g+2 i=g+1 θ[ε i ](τ),

30 From the curve to its Jacobian Non hyperelliptic case and the second tool: Jacobian Nullwerte Sketch of the proof of Weber s formula (Nart, R. unpublished) Let [ε], [ε ] be two even characteristics in genus 3. create two fundamental systems of the form {[µ 1 ], [µ 2 ], [µ 3 ], [ε], [ε 4], [ε 5], [ε 6], [ε 7]}, {[µ 1], [µ 2], [µ 3], [ε ], [ε 4], [ε 5], [ε 6], [ε 7]}. #S = 1 and [µ 1, µ 2, µ 3 ] [µ 1, µ 2, µ 3 ] = θ[ε] θ[ε ]. An odd 2-torsion point µ is given by D κ 0 where D is a degree 2 divisor, support of a bitangent of equation b µ = 0. [b µ1, b µ2, b µ3 ] = det(ω 2 ) 1 (λ µ1 λ µ2 λ µ3 ) [µ 1, µ 2, µ 3 ] where λ i are constants depending on the choice of scalar multiplier for b i and of τ. use several quotients to get rid of the λ i. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

31 Remarks From the curve to its Jacobian Non hyperelliptic case and the second tool: Jacobian Nullwerte For g = 4, #S = 2 and for g = 7, #S = 960. For g 5 it is known that [µ 1,..., µ g ] is in C[θ].In general, it is not true but [µ 1,..., µ g ] can be expresses as a quotient of two polynomials in the ThetaNullwerte. There is also a precise conjectural formula (Igusa 80). Could we directly invert the formula, i.e. express a ThetaNullwert is terms of Jacobian Nullwerte (at least for g 5)? (Nakayashili 97, Enolski, Grava 06): Thomae s formula for y n = m i=1 (x λ i) n 1 2m i=m+1 (x λ i). a general theory exists (Klein vol.3 p.429, Matone-Volpato 07 over C, Shepherd-Barron preprint 08 over any field). Their expressions involve determinants of bases of H 0 (C, L(2K C + µ)). But no formula or implementation has been done. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

32 From the Jacobian to its curve Even characteristics 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

33 From the Jacobian to its curve Torelli theorem: classical versions Even characteristics Let C/k be a curve of genus g > 0. Theorem C is uniquely determined up to k-isomorphism by (Jac(C), j). Corollary C is uniquely determined up to C-isomorphism by Ω or by the ThetaNullwerte. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

34 From the Jacobian to its curve Even characteristics From the Jacobian to its curve: hyperelliptic case 2g 1 C : y 2 = x(x 1) (x λ i ). Idea: invert quotient Thomae s formulae (Mumford Tata II p.136, Takase 96, Koizumi 97) i=1 λ k λ l = i c θ[ε 1] 2 θ[ε 2 ] 2 λ k λ m θ[ε 3 ] 2, c {0, 1, 2, 3}. θ[ε 4 ] 2 For genus 1: λ 1 = θ 4 1 /θ4 0. For genus 2 (Rosenhain formula): λ 1 = θ2 01 θ2 21 θ 2 30 θ2 10 For genus 3 (Weng 01):, λ 2 = θ2 03 θ2 21 θ 2 30 θ2 12, λ 3 = θ2 03 θ2 01 θ10 2. θ2 12 λ 1 = (θ 15θ 3 ) 4 + (θ 12 θ 1 ) 4 (θ 14 θ 2 ) 4, λ 2 = (θ 4θ 9 ) 4 + (θ6 θ 11 ) 4 (θ 13 θ 8 ) 4,... 2(θ 15 θ 3 ) 4 2(θ 4 θ 9 ) 4 Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

35 From the Jacobian to its curve Even characteristics From the Jacobian to its curve : non hyperelliptic genus 3 (Weber 1876) shows how to find the Riemann model: q q q C : x(a 1x + a 1 y + a 1 z) + y(a 2x + a 2 y + a 2 z) + z(a 3x + a 3 y + a 3 z) = 0 with a 1 = i θ 41θ 05 θ 50 θ 14, a 1 = i θ 05θ 66 θ 33 θ 50, a 1 = θ 66θ 41 θ 14 θ 33, a 2 = i θ 25θ 61 θ 36 θ 70, a 2 = i θ 61θ 02 θ 57 θ 34, a 2 = θ 02θ 25 θ 70 θ 57, a 3 = i θ 07θ 43 θ 16 θ 52, a 3 = i θ 60θ 20 θ 75 θ 16, a 3 = θ 20θ 07 θ 52 θ 75. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

36 From the Jacobian to its curve Odd characteristics 1 Link with the conference 2 Period matrices and ThetaNullwerte Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte 3 From the curve to its Jacobian Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte 4 From the Jacobian to its curve Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

37 From the Jacobian to its curve Torelli theorems: odd versions Odd characteristics Theorem (Grushevsky, Salvati Manni 04) A generic abelian variety of dimension g 3 is uniquely determined by its theta gradients. Theorem (Caporaso, Sernesi 03) A general curve C of genus g 3 is uniquely determined by its theta hyperplanes. Rem: the second result is not a corollary of the first. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

38 From the Jacobian to its curve Odd characteristics Hyperelliptic case: genus 2 example (Guàrdia 01,07) Let [µ 1 ],..., [µ 6 ] be the odd characteristics. Then C admits a symmetric model ( y 2 = x x [µ ) ( 1, µ 3 ] x [µ ) ( 1, µ 4 ] x [µ ) ( 1, µ 5 ] x [µ ) 1, µ 6 ]. [µ 2, µ 3 ] [µ 2, µ 4 ] [µ 2, µ 5 ] [µ 2, µ 6 ] Remarks: his theory of symmetric models has nice invariants, nice reduction properties. he (also in Shimura s book 98 p.192) shows how to find algebraic differentials: Ω 2 = 1 2iπθ[ε] J(µ 1,..., µ g ) is such that (dz 1,..., dz g )Ω 2 are algebraic over k if τ comes from an abelian variety A defined over k. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

39 From the Jacobian to its curve Odd characteristics Non hyperelliptic curves of genus 3: Guàrdia 09 Refinement of Riemann model: a smooth plane quartic over k is k-isomorphic to v v v u t [b 7b 2 b 3 ][b 7 b 2 b 3 ] [b 1 b 2 b 3 ][b 1 b 2 b 3 ] X 1X 1 + u t [b 1b 7 b 3 ][b 7 b 1 b 3 ] [b 1 b 2 b 3 ][b 1 b 2 b 3 ] X 2X 2 + u t [b 1b 2 b 7 ][b 7 b 1 b 2 ] [b 1 b 2 b 3 ][b 1 b 2 b 3 ] X 3X 3 = 0 where X i, X i are the equations of the bitangents b i, b i. Ex: Take A = E 3 where E has CM by 19 + the unique undecomposable principal polarization. Then A = Jac(C) where C : U 4 + 2U 3 V 2U 3 W + 6 3i 19 U 2 V U 2 VW i 19 U 2 W i 19 UV i 19 UV 2 W i 19 UVW 2 C descends over Q as + 5 3i 19 UW i 19 V i 19 V 3 W 30V 2 W i 19 VW i 19 W 4 = 0. 2 C : x 4 + (1/9)y 4 + (2/3)x 2 y 2 190y 2 570x 2 + (152/9)y 3 152x 2 y 1083 = 0 Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40

40 Summary From the Jacobian to its curve Odd characteristics g = 1 g = 2 g 3 h. g = 3 n.h. g > 3 n.h. θ τ fast fast conj. fast conj. fast conj. fast conj. τ θ algo algo algo algo algo fast quotient fast quot. C Ω fast (free) algo algo algo plane model C θ fast algo algo algo quot. theory θ C fast fast fast fast? θ C fast fast fast fast? algo: there exists an algorithm but slow. fast (conj.): there exists a fast (conjectural) algorithm. quot.: for the quotient of ThetaNullwerte. theory: the theory is done but no implementation has been done.?: nothing is done. Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal / 40