Algorithmic number theory and the allied theory of theta functions Christophe Ritzenthaler Institut de Mathématiques de Luminy, CNRS Edinburgh 10-10 e-mail: ritzenth@iml.univ-mrs.fr web: http://iml.univ-mrs.fr/ ∼ ritzenth/ Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 1 / 37
Outline Link with number theory, cryptography and coding theory 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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 2 / 37
Link with number theory, cryptography and coding theory Diffie-Hellman key exchange Let ( G = < g >, × ) be a cyclic group of order N . Alice Bob a random k A a random k B h A = g k A h B = g k B h A − → h B ← − secret = h Bk A secret = h Ak B g k A k B is the common secret A priori, the difficulty for an adversary is to compute g k A k B knowing g k A et g k B . Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 3 / 37
Link with number theory, cryptography and coding theory DLP and Jacobians In many cases, it is known to be equivalent to the Discrete Logarithm Problem: giving g and g a find a . Two constraints: the operations in G are fast; the best attack to solve the DLP is the ‘generic attack’ which requires ≈ √ # G operations. Currently, the best G are the groups of rational points on the Jacobians of curves over finite fields with prime order. Problem: how to construct/find such curves ? No brute force method: the finite field is typically F 2 127 − 1 for a genus 2 curve. Many methods have been developed to get ‘polynomial time’ algorithms: ℓ -adic cohomology, p -adic cohomology, deformation, CM,. . . Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 4 / 37
Link with number theory, cryptography and coding theory The algorithms 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. Important points: the theory must be developed over any field (however the intuition comes from C ); the theory must be explicit; computations should be fast. Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 5 / 37
Link with number theory, cryptography and coding theory Coding theory origin Context: to construct good error-correcting codes, one needs curves over finite fields with many rational points. Problem: find a closed formula for the maximal number of points of a curve of genus g over a finite field k . � For g = 1 , 2 , 3 prove that a certain ( A , a ) / k is a Jacobian. Proposition (Precise Torelli theorem) Let ( A , a ) / K be a principally polarized abelian variety which is the Jacobian √ of a curve C over ¯ K, then it is the Jacobian of a curve over L = K ( d ) for a unique d ∈ K ∗ / ( K ∗ ) 2 . Moreover if C is hyperelliptic then we can take L = K. Serre’s strategy for g = 3: d is the product of the 36 Thetanullwerte (correctly normalized). Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 6 / 37
Period matrices and Thetanullwerte Period matrices Link with number theory, cryptography and coding theory 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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 7 / 37
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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 8 / 37
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. Remark: it would be nice to have a free implementation (in SAGE). Christophe Ritzenthaler (IML) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 9 / 37
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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 10 / 37
Period matrices and Thetanullwerte Thetanullwerte Link with number theory, cryptography and coding theory 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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 11 / 37
Period matrices and Thetanullwerte Thetanullwerte Projective embedding The intersection pairing on C induces a principal polarization j on Jac ( C ) . ⇒ The map Sym g − 1 C → Jac ( C ) defines an ample divisor D on ⇐ Jac ( C ) (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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 12 / 37
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 � ǫ � � i π ( n + ǫ 2 ) τ t ( n + ǫ 2 ) + 2 i π ( n + ǫ 2 ) t ( z + 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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 13 / 37
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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 14 / 37
Period matrices and Thetanullwerte From the Thetanullwerte to the Riemann matrix Link with number theory, cryptography and coding theory 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) Algorithmic number theory and the allied theory of theta functions Edinburgh 10-10 15 / 37
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