From the curve to its Jacobian and back Christophe Ritzenthaler Institut de Mathématiques de Luminy, CNRS Montréal 04-10 e-mail: ritzenth@iml.univ-mrs.fr web: http://iml.univ-mrs.fr/ ∼ ritzenth/ Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 1 / 40
Link with the conference 1 Period matrices and ThetaNullwerte 2 Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte From the curve to its Jacobian 3 Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte From the Jacobian to its curve 4 Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 2 / 40
Link with the conference Why do we care ? CM method: CM-type + fundamental unit � lattice + polarization � � the curve over C � curve / F q . period matrix � ThetaNullwerte � 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 04-10 3 / 40
Period matrices and ThetaNullwerte Period matrices Link with the conference 1 Period matrices and ThetaNullwerte 2 Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte From the curve to its Jacobian 3 Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte From the Jacobian to its curve 4 Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 4 / 40
Period matrices and ThetaNullwerte Period matrices Definitions 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 = Ω Z 2 g , the matrix Ω = [Ω 1 , Ω 2 ] ∈ M g , 2 g ( 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 04-10 5 / 40
Period matrices and ThetaNullwerte Period matrices Construction v 1 , . . . , v g be a k -basis of H 0 ( C , Ω 1 ) , δ 1 , . . . , δ 2 g be generators of H 1 ( C , Z ) such that ( δ i ) 1 ... 2 g form a symplectic basis for the intersection pairing on C . �� � Ω := [Ω 1 , Ω 2 ] = v i . δ j i = 1 , . . . , g j = 1 , . . . , 2 g 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 � − 1 � � � 0 1 ¯ t Ω 2 i Ω . − 1 0 Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 6 / 40
Period matrices and ThetaNullwerte Period matrices Example Ex: E : y 2 = x 3 − 35 x − 98 = ( x − 7 )( x − a )( x − ¯ a ) which has complex multiplication by Z [ α ] with α = − 1 −√− 7 √− 7 and a = − 7 2 − 2 . 2 � ¯ � 7 a � dx dx � Ω = 2 2 y , 2 = c · [ α, 1 ] . 2 y a a (Chowla, Selberg 67) formula gives 1 · Γ( 1 7 ) · Γ( 2 7 ) · Γ( 4 c = √ 7 ) 8 π 7 with � ∞ t z − 1 exp ( − t ) dt . Γ( x ) = 0 Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 7 / 40
Period matrices and ThetaNullwerte ThetaNullwerte Link with the conference 1 Period matrices and ThetaNullwerte 2 Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte From the curve to its Jacobian 3 Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte From the Jacobian to its curve 4 Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 8 / 40
Period matrices and ThetaNullwerte ThetaNullwerte Projective embedding 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 n g − 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 04-10 9 / 40
Period matrices and ThetaNullwerte ThetaNullwerte ThetaNullwert A basis of sections of L ( 4 D ) is given by theta functions θ [ ε ]( 2 z , τ ) with integer characteristics [ ε ] = ( ǫ, ǫ ′ ) ∈ { 0 , 1 } 2 g where 2 ) t ( z + ǫ ′ � ǫ � � i π ( n + ǫ 2 ) τ t ( n + ǫ 2 ) + 2 i π ( n + ǫ � � θ ( z , τ ) = exp 2 ) . ǫ ′ n ∈ Z g When ǫ t ǫ ′ ≡ 0 ( mod 2 ) , [ ε ] is said even and one calls ThetaNullwert � ǫ � � ǫ � θ ( 0 , τ ) = θ ( τ ) = θ [ ε ]( τ ) = θ ab ǫ ′ ǫ ′ where the binary representations of a and b are ǫ, ǫ ′ . Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 10 / 40
Period matrices and ThetaNullwerte ThetaNullwerte Example Let q = exp ( π i τ ) . There are 3 genus 1 ThetaNullwerte: � 0 � q n 2 , � θ 00 = θ ( 0 , τ ) = 0 n ∈ Z � 1 � 2 q ( n + 1 2 ) � θ 10 = θ ( 0 , τ ) = , 0 n ∈ Z � 0 � ( − 1 ) n q n 2 . � θ 01 = θ ( 0 , τ ) = 1 n ∈ Z Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 11 / 40
Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Link with the conference 1 Period matrices and ThetaNullwerte 2 Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte From the curve to its Jacobian 3 Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte From the Jacobian to its curve 4 Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 12 / 40
Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Case g = 1 Gauss, Cox 84, Dupont 07 Let z = θ 01 ( τ ) 2 /θ 00 ( τ ) 2 . Duplication formulae vs AGM formulae : θ 00 ( τ ) 2 + θ 01 ( τ ) 2 a n − 1 + b n − 1 θ 00 ( 2 τ ) 2 = a n = , 2 2 θ 01 ( 2 τ ) 2 � = θ 00 ( τ ) · θ 01 ( τ ) b n = a n − 1 · b n − 1 , θ 00 ( τ ) 2 − θ 01 ( τ ) 2 θ 10 ( 2 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 04-10 13 / 40
Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Transformation formula : � 2 � 2 θ 00 ( τ ) 2 = i � − 1 θ 10 ( τ ) 2 = i � − 1 τ · θ 00 , τ · θ 01 . τ τ τ · lim θ 00 ( 2 n · − 1 τ ) 2 = i ⇒ AGM ( θ 00 ( τ ) 2 , θ 10 ( τ ) 2 ) = i τ · 1 √ 1 − z 2 ) = i 1 ⇒ AGM ( 1 , τ · θ 00 ( τ ) 2 . Proposition i · AGM ( 1 , z ) √ = τ. AGM ( 1 , 1 − z 2 ) Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 14 / 40
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 ] = [ π i π AGM ( a , b ) , AGM ( a + b , a − b ) ] is a period matrix relative to dx / y. 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 04-10 15 / 40
Period matrices and ThetaNullwerte From the ThetaNullwerte to the Riemann matrix Case g ≥ 2 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 04-10 16 / 40
From the Riemann matrix to the (quotients of) Period matrices and ThetaNullwerte ThetaNullwerte Link with the conference 1 Period matrices and ThetaNullwerte 2 Period matrices ThetaNullwerte From the ThetaNullwerte to the Riemann matrix From the Riemann matrix to the (quotients of) ThetaNullwerte From the curve to its Jacobian 3 Hyperelliptic case and the first tool: s ε Non hyperelliptic case and the second tool: Jacobian Nullwerte From the Jacobian to its curve 4 Even characteristics Odd characteristics Christophe Ritzenthaler (IML) From the curve to its Jacobian and back Montréal 04-10 17 / 40
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