Computing modular polynomials in dimension 2 ECC 2015, Bordeaux
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1 Computing modular polynomials in dimension 2 ECC 2015, Bordeaux Enea Milio 29/09/2015 Enea Milio Computing modular polynomials 29/09/ / 49
2 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
3 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
4 Motivation An isogeny between two elliptic curves E 1 and E 2 is a surjective map with finite kernel. The degree of the isogeny is the cardinality of the kernel. Many applications : Theory ; Cryptography : transfert the DLP ; SEA algorithm ; Class polynomials ; Graph of isogenies. Enea Milio Computing modular polynomials 29/09/ / 49
5 Motivation For cryptography, we work over finite fields. Here, we work on C. The theory is easy on C ; Numerical computation ; The modular polynomials can be reduced modulo p. Enea Milio Computing modular polynomials 29/09/ / 49
6 Complex elliptic curves Let H 1 = {a + ıb : b > 0} C be the Poincaré half plane. Proposition Let E/C be an elliptic curve. Then there exists a lattice Λ = Z + τz, where τ H 1 and a complex analytic isomorphism E C/Λ of complex Lie groups. Enea Milio Computing modular polynomials 29/09/ / 49
7 Isomorphism Modular group : SL 2 (Z) = { ( ) a b c d : a, b, c, d Z, ad bc = 1} Group action : SL 2 ( (Z) H 1 H 1 a b ) c d τ = aτ+b cτ+d Proposition Two elliptic curves E 1 and E 2 over C corresponding to the lattices Λ 1 = Z + τ 1 Z and Λ 2 = Z + τ 2 Z are isomorphic if and only if there exists γ SL 2 (Z) such that τ 2 = γτ 1. = change of basis of the lattice. Enea Milio Computing modular polynomials 29/09/ / 49
8 Fundamental domain F 1 H 1 F 1 i Enea Milio Computing modular polynomials 29/09/ / 49
9 Fundamental domain F 1 H 1 F 1 i τ Enea Milio Computing modular polynomials 29/09/ / 49
10 Fundamental domain F 1 F 1 H 1 τ = ( ) a b c d τ i τ Enea Milio Computing modular polynomials 29/09/ / 49
11 Modular function Let p be a prime and Γ 0 (p) = {( ) a b c d SL2 (Z) : c 0 mod p }. Definition Let Γ SL 2 (Z) be a congruence subgroup of finite index. f : H 1 C is a modular function for Γ if 1 f is meromorphic on H 1 (and on the cusps) ; 2 for all γ Γ and τ H 1, f (γτ) = f (τ). Example j(τ) is a modular function for SL 2 (Z) ; j p (τ) := j(pτ) is a modular function for Γ 0 (p). Enea Milio Computing modular polynomials 29/09/ / 49
12 Modular function Let Γ SL 2 (Z) be a congruence subgroup of finite index. Denote by C Γ the field of modular functions for Γ. Then Theorem C SL2 (Z) = C(j) ; C Γ0 (p) = C(j, j p ). Enea Milio Computing modular polynomials 29/09/ / 49
13 Isogeny We are interested in the isogenies of degree p. Let C p be a set of representatives of SL 2 (Z)/Γ 0 (p). The isogenous points of degree p are : pγτ, γ C p. Theorem The field extension C Γ0 (p)/c SL2 (Z) = C(j, j p )/C(j) is algebraic of degree [SL 2 (Z) : Γ 0 (p)] = p + 1. Conjugate functions of j p in C Γ0 (p)/c SL2 (Z) : j γ p (τ) := j(pγτ), γ C p. Enea Milio Computing modular polynomials 29/09/ / 49
14 Modular polynomial The classical modular polynomial of index p is the polynomial Φ p that parameterizes isomorphism classes of elliptic curves together with an isogeny of degree p : Φ p (X, j(e)) = (X j(e )). E p isogenous to E It is also the minimal polynomial of j p for the extension C Γ0 (p)/c SL2 (Z), thus Φ p (X, j) = γ C p (X j γ p ) Z[X, j]. Enea Milio Computing modular polynomials 29/09/ / 49
15 Algorithm Computation of the modular polynomials by evaluation/interpolation (Enge 2009). Φ p (X, j) = (X jp γ ) = X p+1 + γ C p Evaluate : (X j(pγτ)) = X p+1 + γ C p p c i (j)x i. i=0 p c i (j(τ))x i ; Evaluate in deg j (Φ p ) + 1 = (p + 1) + 1 values τ. Interpolate c i. Evaluation of j in Õ(N) at precision N digits (Dupont 2006) ; Algorithm quasi-linear : Õ(p3 ). Enea Milio Computing modular polynomials 29/09/ / 49 i=0
16 Examples p = 2 Φ 2 (X, Y ) = X 3 + ( Y Y )X 2 + (1488Y Y )X + (Y Y Y ) p = 3 Φ 3 (X, Y ) = X 4 + ( Y Y Y )X 3 + (2232Y Y Y )X 2 + ( Y Y Y )X + (Y Y Y Y ) p = 5 Φ 5 (X, Y ) = X 6 +( Y Y Y Y Y )X 5 +(3720Y Y Y Y Y )X 4 + ( Y Y Y Y Y )X 3 + ( Y Y Y Y Y )X 2 + ( Y Y Y Y Y )X + (Y Y Y Y Y Y ) Other invariants : Schläfli, Weber, theta functions. Schläfli 1870, p = 5 : x 6 x 5 y 5 + 4xy + y 6. Enea Milio Computing modular polynomials 29/09/ / 49
17 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
18 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
19 Motivation Dimension 2 : principally polarized abelian surfaces (ppas) = Jacobian of hyperelliptic curves of genus 2 (or product of elliptic curves) ; Cryptography : competitive with elliptic curves ; = we want to do the same thing! Enea Milio Computing modular polynomials 29/09/ / 49
20 Siegel space Siegel upper half-space H 2 the set of 2 2 symmetric matrices over C with positive definite imaginary part. Ppas on C : A C 2 /Λ where Λ = Z 2 + ΩZ 2, with Ω H 2 (period matrix). ( ) Let J = 0 Id2 Id 2 0. Symplectic group : Sp 4 (Z) = {γ GL 4 (Z) : t γjγ = J}. Group action : ( A B C D ) Ω = (AΩ + B)(CΩ + D) 1. We have a fundamental domain F 2. Siegel modular threefold : H 2 /Sp 4 (Z). Enea Milio Computing modular polynomials 29/09/ / 49
21 Modular forms and functions Let Γ be a subgroup of finite index of Sp 4 (Z) and k Z. Definition A Siegel modular form of weight k for Γ is a function f : H 2 C such that : 1 f is holomorphic on H 2 ; 2 f (γω) = det(cω + D) k f (Ω), γ = ( A B C D ) Γ and Ω H2. Definition Siegel modular function for Γ : f = f 1 f2 of same weight. Thus, f (γω) = f (Ω). quotient of Siegel modular forms Enea Milio Computing modular polynomials 29/09/ / 49
22 Theta functions Theta functions (of characteristic 1 2 ) : Define for a = (a 0, a 1 ) and b = (b 0, b 1 ) in {0, 1} 2 : θ b0 +2b 1 +4a 0 +8a 1 (Ω) = n Z 2 exp(ıπ t (n + a 2 )Ω(n + a 2 ) + ıπ t (n + a 2 )b) 16 theta functions ; 6 are identically zero ; P = {0, 1, 2, 3, 4, 6, 8, 9, 12, 15} ; θi 2 = Siegel modular form of weight 1 for Γ(2, 4). Γ(2, 4) = {( ) A B C D Sp4 (Z) : ( ) A B C D Id4 mod 2, B 0 C 0 0 mod 4 }. Enea Milio Computing modular polynomials 29/09/ / 49
23 Theta functions Let h 10 = i P θ2 i, h 4 = i P θ8 i, h 6 = 60 triples (i,j,k) P 3 ±(θ iθ j θ k ) 4, h 12 = 15 tuples (i,j,k,l,m,n) P 6(θ iθ j θ k θ l θ m θ n ) 4. h i is a Siegel modular form of weight i for the group Sp 4 (Z). Enea Milio Computing modular polynomials 29/09/ / 49
24 Generalization of the j-invariant Definition We call Igusa invariants, or j-invariants, the Siegel modular functions j 1, j 2, j 3 for Sp 4 (Z) defined by j 1 := h5 12 h 6 10, j 2 := h 4h 3 12 h 4 10 Theorem (Igusa 1962, Spallek 1994), j 3 := (h 12h 4 2h 6 h 10 )h12 2 3h10 4. The field of Siegel modular functions invariant by Sp 4 (Z) is C(j 1, j 2, j 3 ). Generically, two ppas have the same j-invariants if and only if they are isomorphic. Enea Milio Computing modular polynomials 29/09/ / 49
25 Isogeny The functions j l,p (Ω) := j l (pω), l = 1, 2, 3, are Siegel modular functions for Γ 0 (p), where Γ 0 (p) := {( ) A B C D Sp4 (Z) : C 0 mod p } is of index [Sp 4 (Z) : Γ 0 (p)] = p 3 + p 2 + p + 1. Ppas (p, p)-isogenous to Ω : pγω, where γ C p = Sp 4 (Z)/Γ 0 (p). Theorem (Bröker-Lauter 2009) The field of Siegel modular functions invariant by Γ 0 (p) is C(j 1, j 2, j 3, j 1,p ). We define j γ l,p (Ω) := j l(pγω), l = 1, 2, 3. Enea Milio Computing modular polynomials 29/09/ / 49
26 Modular polynomials in dimension 2 Φ 1,p (X ) = (X j γ 1,p ), γ C p (minimal polynomial of the extension C(j 1, j 2, j 3, j 1,p )/C(j 1, j 2, j 3 )), and for l = 2, 3, Ψ l,p (X ) = j γ l,p γ C p γ C p\{γ} (X j γ 1,p ). Proposition (Bröker-Lauter 2009) Φ 1,p, Ψ 2,p, Ψ 3,p Q(j 1, j 2, j 3 )[X ]. We have j γ l,p (Ω) = Ψ l,p (j γ 1,p (Ω), j 1(Ω), j 2 (Ω), j 3 (Ω))/Φ 1,p (j γ 1,p (Ω), j 1(Ω), j 2 (Ω), j 3 (Ω)). Enea Milio Computing modular polynomials 29/09/ / 49
27 Algorithm How to compute the modular polynomials? Evaluation/interpolation : p 3 +p 2 +p Φ 1,p (X, j 1 (Ω), j 2 (Ω), j 3 (Ω)) = X p3 +p 2 +p+1 + c i (j 1 (Ω), j 2 (Ω), j 3 (Ω))X i. where c i Q(j 1, j 2, j 3 ). Problem : Interpolation of trivariate rational fractions. i=0 Enea Milio Computing modular polynomials 29/09/ / 49
28 Interpolation We can not choose the matrices Ω as we want! j (1,1) 3 j (1) 2 j (1,2) 3 j 1 j (2) 2 j (1,n) 3 j (n) 2 Enea Milio Computing modular polynomials 29/09/ / 49
29 Complexity Inversion : (j 1 (Ω), j 2 (Ω), j 3 (Ω)) Ω in Õ(N) (Dupont 2006). Fast evaluation of the Igusa invariants : Õ(N) (Dupont 2006, Enge Thomé 2014). Complexity of the computation of the modular polynomials : Õ(d j1 d j2 d j3 p 3 N). Enea Milio Computing modular polynomials 29/09/ / 49
30 Streng invariants The modular polynomials have been computed by Dupont for p = 2 only. For p = 3 they are too big. Other invariants? Streng 2010 : while Igusa : i 1 := h 4h 6, i 2 := h2 4 h 12 h 10 h10 2, i 3 := h5 4 h10 2. j 1 := h5 12 h 6 10, j 2 := h 4h 3 12 h 4 10, j 3 := (h 12h 4 2h 6 h 10 )h12 2 3h10 4. Theorem The field of Siegel modular functions is C(j 1, j 2, j 3 ) = C(i 1, i 2, i 3 ). Enea Milio Computing modular polynomials 29/09/ / 49
31 Comparison For p = 3 : Φ 1,3 (X, f 1, f 2, f 3 ) = X i=0 c i(f 1, f 2, f 3 )X i. Memory space : i j 1 i 1 j 2 i 2 j 3 i p = 2 : 2.1 MB against 57 MB. p = 3 : 890 MB. Enea Milio Computing modular polynomials 29/09/ / 49
32 Denominators Denominators for Igusa invariants for p = 2 : j1 α D 2 (j 1, j 2, j 3 ) 6 (α ranging between 5 and 21) Denominators for Streng invariants for p = 2 : i3 α D 2 (i 1, i 2, i 3 ) and i3 α D 2 (i 1, i 2, i 3 ) 2 (α varies from 0 to 3) Denominators for Streng invariants for p = 3 : i3 α D 3 (i 1, i 2, i 3 ) 2 and i3 α D 3 (i 1, i 2, i 3 ) 4 (α varies from 0 to 4) Enea Milio Computing modular polynomials 29/09/ / 49
33 Examples D 2 (Ig) = j ( 972j (5832j )j 2 + ( 8748j j ))j (j4 2 + ( 12j )j (54j j )j ( 108j j j 3)j 2 + (81j j 3 3 ))j3 1 + (78j ( 1332j )j (8910j j 3 )j ( 29376j j2 3 )j j 4 3 j j 5 3 )j ( 159j (1728j )j j 2 3 j j3 3 j3 2 )j 1 + (80j j 3j 6 2 ). D 2 (Str) = (24576i 3 i (96i i 3i 2 )i ( i 3i i 2 3 )i ( 23328i i 3i i 3i i 2 3 i i 2 3 )i (93312i 3 i i 3i i 2 3 i 2 + (1536i i 2 3 ))i 1 + ( i i 3i (6i i 3)i i 2 3 i ( 288i i 2 3 )i 2 + ( i i 2 3 ))). Enea Milio Computing modular polynomials 29/09/ / 49
34 D3(Str) = i 13 1 i2i i 12 1 i i 12 1 i 2 2i i 12 1 i i 11 1 i 2 2i i 11 1 i2i i 11 1 i i 10 1 i i 10 1 i 4 2i i 10 1 i 3 2i i 10 1 i 2 2i i 10 1 i2i i 10 1 i i 10 1 i i 9 1i 4 2i i 9 1i 3 2i i 9 1i 2 2i i 9 1i2i i 9 1i2i i 9 1i i 8 1i i 8 1i 5 2i i 8 1i 4 2i i 8 1i 4 2i i 8 1i 3 2i i 8 1i 2 2i i 8 1i 2 2i i 8 1i2i i 8 1i2i i 8 1i i 8 1i i 7 1i 5 2i i 7 1i 4 2i i 7 1i 4 2i i 7 1i 3 2i i 7 1i 2 2i i 7 1i 2 2i i 7 1i2i i 7 1i2i i 7 1i i 7 1i i 6 1i i 6 1i 6 2i i 6 1i 5 2i i 6 1i 5 2i i 6 1i 4 2i i 6 1i 4 2i i 6 1i 3 2i i 6 1i i 6 1i 2 2i i 6 1i 2 2i i 6 1i 2 2i i 6 1i2i i 6 1i2i i 6 1i i 6 1i i 6 1i i 5 1i 6 2i i 5 1i 5 2i i 5 1i 5 2i i 5 1i 4 2i i 5 1i 4 2i i 5 1i 3 2i i 5 1i 3 2i i 5 1i 2 2i i 5 1i 2 2i i 5 1i2i i 5 1i2i i 5 1i2i i 5 1i i 5 1i i 5 1i i 4 1i i 4 1i 7 2i i 4 1i 6 2i i 4 1i 6 2i i 4 1i 5 2i i 4 1i 5 2i i 4 1i 4 2i i 4 1i i 4 1i 4 2i i 4 1i 3 2i i 4 1i 3 2i i 4 1i 3 2i i 4 1i 2 2i i 4 1i 2 2i i 4 1i 2 2i i 4 1i2i i 4 1i2i i 4 1i2i i 4 1i i 4 1i i 4 1i i 3 1i 7 2i i 3 1i 6 2i i 3 1i 6 2i i 3 1i 5 2i i 3 1i 5 2i i 3 1i 4 2i i 3 1i 4 2i i 3 1i 4 2i i 3 1i 3 2i i 3 1i 3 2i i 3 1i 2 2i i 3 1i 2 2i i 3 1i 2 2i i 3 1i2i i 3 1i2i i 3 1i2i i 3 1i2i i 3 1i i 3 1i i 3 1i i 2 1i i 2 1i 8 2i i 2 1i 7 2i i 2 1i 7 2i i 2 1i 6 2i i 2 1i 6 2i i 2 1i 5 2i i 2 1i 5 2i i 2 1i 5 2i i 2 1i 4 2i i 2 1i 4 2i i 2 1i 4 2i i 2 1i 4 2i i 2 1i 3 2i i 2 1i 3 2i i 2 1i 3 2i i 2 1i 2 2i i 2 1i 2 2i i 2 1i 2 2i i 2 1i 2 2i i 2 1i2i i 2 1i2i i 2 1i2i i 2 1i i 2 1i i 2 1i i 2 1i i1i 8 2i i1i 7 2i i1i 7 2i i1i 6 2i i1i 6 2i i1i 5 2i i1i 5 2i i1i 5 2i i1i 4 2i i1i 4 2i i1i 4 2i i1i 3 2i i1i 3 2i i1i 2 2i i1i 2 2i i1i 2 2i i1i 2 2i i1i2i i1i2i i1i2i i1i2i i1i i1i i1i i1i i i 9 2i i 8 2i i i 7 2i i 7 2i i 6 2i i 6 2i i 6 2i i 5 2i i 5 2i i 5 2i i 5 2i i 4 2i i 4 2i i 4 2i i 4 2i i 3 2i i 3 2i i 3 2i i 3 2i i 2 2i i 2 2i i 2 2i i 2 2i i2i i2i i2i i2i i i i i i 4 3 Enea Milio Computing modular polynomials 29/09/ / 49
35 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
36 Alternative invariants look at modular functions for another group. b i (Ω) := θ i θ 0 (Ω/2), i = 1, 2, 3. Modular functions for Γ(2, 4). Enea Milio Computing modular polynomials 29/09/ / 49
37 Modular polynomials with b 1, b 2, b 3 Theorem (Mumford, Manni) The field of modular functions invariant by Γ(2, 4) is C(b 1, b 2, b 3 ). We look at C p = Γ(2, 4)/(Γ 0 (p) Γ(2, 4)), p > 2. The index is still p 3 + p 2 + p + 1. Proposition The field of modular functions invariant by Γ 0 (p) Γ(2, 4) is C(b 1, b 2, b 3, b 1,p ). We compute Φ 1,p (X, b 1, b 2, b 3 ) = γ C p (X b γ 1,p ) and Ψ l,p (X, b 1, b 2, b 3 ) = γ C p b γ l,p γ C p\{γ} (X bγ 1,p ). They are in Q(b 1, b 2, b 3 )[X ]. Enea Milio Computing modular polynomials 29/09/ / 49
38 Algorithm Evaluation of the b i in Õ(N) (Dupont 2006, Enge Thomé 2014). Inversion : (b 1, b 2, b 3 )(Ω) Ω? (b 1, b 2, b 3 )(Ω) (j 1, j 2, j 3 )(Ω) Ω mod Sp 4 (Z). Problem : we want Ω mod Γ(2, 4)! Solutions : Compute b i (γω) for γ Sp 4 (Z)/Γ(2, 4). But index 11520! Use of functional equation of the theta functions. Enea Milio Computing modular polynomials 29/09/ / 49
39 Denominators with the theta functions Polynomials computed for p = 3, 5, 7. Always D p in the denominator. D 3 (b 1, b 2, b 3 ) = 64(b1 2b2 2 b2 3 )(16b4 1 b4 2 b )(b4 1 + b4 2 + b4 3 ) 32(48b1 4b4 2 b b2 1 b2 2 b )(b4 1 b4 2 + b4 1 b4 3 + b4 2 b4 3 )+ 256(b1 8b8 2 + b8 1 b8 3 + b8 2 b8 3 ) + 32(b4 1 b4 2 b4 3 )( 24b4 1 b4 2 b b2 1 b2 2 b ) + 1. It is symmetric and there are relations modulo 2 and 4 between the exponents. Enea Milio Computing modular polynomials 29/09/ / 49
40 Symmetries Theorem (M. 2014) For all prime p, we have Φ 1,p (X, b 1, b 2, b 3 ) = Φ 1,p (X, b 1, b 3, b 2 ) and Ψ 2,p (X, b 1, b 3, b 2 ) = Ψ 3,p (X, b 1, b 2, b 3 ). Proof : there always exist γ Sp 4 (Z)/Γ(2, 4) such that for all Ω H 2 : b 1 (γω) = b 1 (Ω) b 1,p (γω) = b 1,p (Ω) b 2 (γω) = b 3 (Ω) and b 2,p (γω) = b 3,p (Ω) b 3 (γω) = b 2 (Ω) b 3,p (γω) = b 2,p (Ω) Φ 1,p is a minimal polynomial. Ψ l,p (b 1,p ) = b l,p Φ 1,p (b 1,p) for l = 2, 3. Action on Ψ 2,p (X ) : Ψ 2,p (b 1,p, b 1, b 3, b 2 ) = b 3,p Φ 1,p(b 1,p, b 1, b 2, b 3 ) := Ψ 3,p (b 1,p, b 1, b 2, b 3 ). Enea Milio Computing modular polynomials 29/09/ / 49
41 Relations modulo 2 and 4 We look at matrices γ such that b 1 (γω) = i α 1 b 1 (Ω) b 1,p (γω) = i α 4 b 1,p (Ω) b 2 (γω) = i α 2 b 2 (Ω) and b 2,p (γω) = i α 5 b 2,p (Ω) b 3 (γω) = i α 3 b 3 (Ω) b 3,p (γω) = i α 6 b 3,p (Ω) Enea Milio Computing modular polynomials 29/09/ / 49
42 Comparison For p = 3 : i j 1 i 1 b 1 j 2 i 2 b 2 j 3 i 3 b p = 3 : 175 KB = 5000 smaller than Streng ; p = 5 : 200 MB ; p = 7 : 30 GB ; Enea Milio Computing modular polynomials 29/09/ / 49
43 Computing modular polynomials 1 Dimension 1 : elliptic curves 2 Dimension 2 : abelian surfaces Computation of the modular polynomials Smaller invariants 3 Real Multiplication : cyclic isogenies Enea Milio Computing modular polynomials 29/09/ / 49
44 Hilbert space Let D Z >0 and K = Q( D) a real quadratic field. We take D {2, 5} for simplicity. The group SL 2 (O K ) acts on H 2 1. Proposition The Hilbert modular surface H 2 1 /SL 2(O K ) is a moduli space for isomorphism classes of ppas with real multiplication by O K. Let p a prime number such that p = ββ, β O + K. β-isogenous surfaces : βγz, z H 2 1 and γ C p = SL 2 (O K )/Γ 0 (β) ; #C p = p + 1. Enea Milio Computing modular polynomials 29/09/ / 49
45 Hilbert and Humbert The following diagram is commutative : H 2 1 φ H 2 H 2 1 /SL 2(O K ) ρ H 2 /Sp 4 (Z) where ρ is generically of degree 2 onto the Humbert surface H K. Enea Milio Computing modular polynomials 29/09/ / 49
46 Hilbert modular function Gundlach invariants : J 1 and J 2 for D = 2 and 5 only. Two modular polynomials Φ β and Ψ β in Q(J 1, J 2 )[X ]. Enea Milio Computing modular polynomials 29/09/ / 49
47 Algorithm For D = 5, we have (Resnikoff 1974, Lauter Yang 2011) j 1 φ = 8J 1 (3J 2 2 /J 1 2) 5 ; j 2 φ = 1 2 J 1(3J 2 2 /J 1 2) 3 ; j 3 φ = 2 3 J 1 (3J 2 2 /J 1 2) 2 (4J 2 2 /J J 2 /J 1 3). These equations can be inverted by Gröbner basis. Fast evaluation of the Gundlach invariants : z φ(z) = Ω (j 1 (Ω), j 2 (Ω), j 3 (Ω)) (J 1 (z), J 2 (z)). Inversion of the Gundlach invariants : (J 1 (z), J 2 (z)) (j 1 (φ(z)), j 2 (φ(z)), j 3 (φ(z))) φ(z) z. Enea Milio Computing modular polynomials 29/09/ / 49
48 Results D=2 p Memory space 8.5KB 172KB 5.8MB 21MB 70MB 225MB D=5 p Memory space 22KB 3.5MB 33MB 188MB 248MB Enea Milio Computing modular polynomials 29/09/ / 49
49 Theta functions Other invariants? or j i = j i φ, i = 1, 2, 3 b i = b i φ, i = 1, 2, 3. Works for any D. Three invariants for a space of dimension 2 : need the equation P of the Humbert component (Gruenewald 2008). Interpolation : Q( b 1, b 2, b 3 )/(P) = Q( b 1, b 2 )[ b 3 ]/(P). Enea Milio Computing modular polynomials 29/09/ / 49
50 Conclusion Implementation and generalization of the algorithm of Dupont ; Used smaller invariants and proved properties with them ; Definition and computation of modular polynomials with cyclic isogenies. Enea Milio Computing modular polynomials 29/09/ / 49
51 Perspectives Compute more modular polynomials ; Release the code ; Applications of the polynomials. Enea Milio Computing modular polynomials 29/09/ / 49
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