Computing modular Galois representations - the modulo p approach (after Jinxiang Zeng) Maarten Derickx 1 Universiteit Leiden and Universit´ e Bordeaux 1 Sage Days 51 22-26 July 2013 1 Original slides by Jinxiang Zeng, modified by D. Computing modular Galois representations
Computing Coefficients of modular forms Introduction/Main Results 1 How fast can τ ( p ) be computed? An algorithm work with finite fields Complexity analysis A lower bound on the number of generators of m ⊂ T A First Description of the Algorithm 2 Congruence of Modular Forms Galois Representations and Modular Forms Computing The Ramanujan subspace Future work 3 Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra The discriminant modular form Discriminant Modular Form Let q := e 2 π iz , the discriminant modular form is defined by ∞ ∞ ( 1 − q n ) 24 = τ ( n ) q n ∈ S 12 ( SL 2 ( Z )) � � ∆( q ) = q n = 1 n = 1 where τ : Z → Z is called Ramanujan tau function. ∆( q ) plays a crucial role during the developments of theory of modular forms. In this lecture we focus on the computational aspects of ∆( q ) . Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra The discriminant modular form Arithmetic of the Ramanujan tau function τ ( mn ) = τ ( m ) τ ( n ) for any integers satisfying ( m , n ) = 1. τ ( p n + 1 ) = τ ( p ) τ ( p n ) − p 11 τ ( p n − 1 ) for any prime p , n ≥ 1. | τ ( p ) | ≤ 2 p 11 / 2 , Deligne’s bound. τ ( p ) ≡ p ( 1 + p 9 ) mod 25 , τ ( p ) ≡ p ( 1 + p 3 ) mod 7, τ ( p ) ≡ 1 + p 11 mod 691 Lehmer’s Conjecture τ ( n ) � = 0 for any n ≥ 1. � � Serre: if τ ( p ) = 0 then p = hM − 1 with M = 2 14 3 7 5 3 691, h + 1 = 1 23 and some h mod 49 ∈ { 0 , 30 , 48 } . Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra How fast can τ ( p ) be computed? A question that Schoof asked to Edixhoven in 1995 Can we compute τ ( p ) for prime p in time polynomial in log p ? Theorem (Edixhoven, Couveignes, etc.) For prime p , there exist algorithms to compute τ ( p ) in time polynomial in log p . work with complex number field, using numerical approximation. work with finite fields, using CRT. | τ ( p ) | ≤ 2 p 11 / 2 so τ ( p ) can be computed by computing τ ( p ) mod ℓ for sufficiently many small primes ℓ (where small means O ( log p ) .) Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra How fast can τ ( p ) be computed? Generalization and explicit calculation Bruin generalized the methods to modular forms for the groups of the form Γ 1 ( n ) . Bosman implemented an algorithm using numerical approximation C and computed ρ proj : Gal ¯ Q / Q → PGL ( V l ) l for ℓ ∈ { 13 , 17 , 19 } . This allows one to calculate ± τ ( p ) mod l which he used to prove τ ( n ) � = 0 , ∀ n < 2 · 10 19 . Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra A probabilistic algorithm Algorithm(Zeng 2012) Following Couveignes’s idea, working with finite fields, we give a probabilistic algorithm, which is rather simple and well suited for implementation. The following calculation was done using a personal computer. level time (projective representation) time (entire representation) ℓ =13 several minutes one hour ℓ =17 several hours one day ℓ =19 several days less than four days ℓ = 29 waiting waiting ℓ = 31 several days several days Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra A probabilistic algorithm Exact value of τ ( p ) mod ℓ Since we can compute the entire representation, the exact values of τ ( p ) mod ℓ for ℓ ∈ { 13 , 17 , 19 } can be computed. Nonvanishing of tau function Since we can compute the projective representation for ℓ = 31, we can prove a τ ( n ) � = 0 , for all n < 982149821766199295999 ≈ 9 · 10 20 a Bosman proved the nonvanishing holds for n < 22798241520242687999 ≈ 2 · 10 19 Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra Complexity of the algorithm Theorem(Zeng 2012) For prime p , τ ( p ) can be computed in time O ( log 6 + 2 ω + δ + ǫ p ) . ω is a constant in [2,4], refers to that addition in Jacobian can be done in time O ( g ω ) , δ is a constant, measuring the heights of the points of the Ramanujan subspace V ℓ , ǫ is any real positive number. ω depends on the complexity of calculations in J 1 ( l )( F p e ) . Using Khuri-Makdisi’s algorithm, the constant ω is 2.376. Our computation suggests δ ≈ 3, although this is based on a very small sample ( l = 13 , 17 , 19 ) Computing modular Galois representations
Computing τ ( p ) Introduction/Main Results A probabilistic algorithm Description of the Algorithm Complexity analysis Future work Generators of maximal ideal of Hecke algebra On the generators of the maximal ideal Theorem(Zeng 2012) If ℓ ≥ 13 is prime and m = ( l , T 1 − τ ( 1 ) , T 2 − τ ( 2 ) , T 3 − τ ( 3 ) , . . . ) ⊂ T , then m can be generated by ℓ and T n − τ ( n ) with n ≤ 2 ℓ + 1 12 . Remarks It makes the algorithm faster. The previous known upper-bound was ( ℓ 2 − 1 ) / 6, making step 5 very slow. In practice the upper bound is even much better. m = ( ℓ, T 2 − τ ( 2 )) for ℓ ∈ { 13 , 17 , 19 , 29 , 37 , 41 , 43 } m = ( ℓ, T 3 − τ ( 3 )) for ℓ = 31 Computing modular Galois representations
Introduction/Main Results Congruence of Modular Forms Description of the Algorithm Galois Representations Future work Computing The Ramanujan subspace Congruence of Modular Forms Theorem (Mazur, Ribet, Gross, Edixhoven etc.) Let n , k ∈ Z + , F / F ℓ finite extension, and f : T ( n , k ) → F a surjective ring morpism. Assume 2 < k ≤ ℓ + 1 and the associated Galois representation ρ f : Gal ( Q / Q ) → GL 2 ( F ) is absolutely irreducible. Then there is a unique ring morphism f 2 : T ( n ℓ, 2 ) → F such that: f 2 is surjective, f 2 ( T i ) = f ( T i ) , f 2 ( < a > ) = f ( < a > ) a k − 2 for all i ≥ 1 and any a satisfying ( a , n ℓ ) = 1. V f := J 1 ( n ℓ )[ ker f 2 ] realizes ρ f . Remark For the rest of this talk: f = ∆( q ) mod ℓ , so F = F ℓ , ker f 2 = < ℓ, T i − τ ( i ) : i ≥ 1 > and V ℓ := V ∆ ,ℓ = J 1 ( ℓ )[ ker f 2 ] . Computing modular Galois representations
Introduction/Main Results Congruence of Modular Forms Description of the Algorithm Galois Representations Future work Computing The Ramanujan subspace Galois Representation Galois representation associated to ∆( q ) Let ρ ℓ be the Galois representation associated to the newform ∆( q ) ρ ℓ : Gal ( Q / Q ) → GL 2 ( F ℓ ) then For prime p � = ℓ : Tr ( ρ ℓ ( Frob p )) ≡ τ ( p ) mod ℓ and det ( ρ ℓ ( Frob p )) ≡ p 11 mod ℓ . The representation space (called Ramanujan subspace denoted by V ℓ ) is � V ℓ = ker ( T k − τ ( k ) , J 1 ( ℓ )[ ℓ ]) 1 ≤ k ≤ ℓ 2 − 1 6 Computing modular Galois representations
Introduction/Main Results Congruence of Modular Forms Description of the Algorithm Galois Representations Future work Computing The Ramanujan subspace Computing V ℓ mod p : the strategy 1) Find an e s.t. V ℓ (¯ F p ) = V ℓ ( F p e ) 2) Compute n := # J 1 ( ℓ )( F p e ) 3) Pick P ∈ J 1 ( ℓ )( F p e ) random. 4) Multiply P by n ℓ − v ℓ ( n ) , and then repeatedly by ℓ until P ∈ J 1 ( ℓ )[ ℓ ] 5) Compute Q := f ( P ) for some surjection J 1 ( ℓ )[ ℓ ] → V ℓ . 6) Repeat 3 ) , 4 ) and 5 ) till you find linearly independent Q 1 , Q 2 ∈ V ℓ . Computing modular Galois representations
Introduction/Main Results Congruence of Modular Forms Description of the Algorithm Galois Representations Future work Computing The Ramanujan subspace Step 1: find e s.t.: V ℓ (¯ F p ) = V ℓ ( F p e ) The characteristic polynomial of Frob p on V ℓ is X 2 − τ ( p ) X + p 11 We need Frob p = Id V ℓ so we can take: e := min { t | t ≥ 1 , X t = 1 ∈ F ℓ [ X ] / ( X 2 − τ ( p ) X + p 11 ) } Remark Step 4 is very expensive if e is big. So we only compute V ℓ mod p for the p s.t. e is small. Computing modular Galois representations
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