Computing modular Galois representations

Abstract

We compute modular Galois representations associated with a newform f , and study the related problem of computing the coefficients of f modulo a small prime ℓ. To this end, we design a practical variant of the complex approximations method presented in [EC11]. Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms ∆ and E 4 · ∆, and manage to compute for the first time the associated faithful representations modulo ℓ and the values modulo ℓ of Ramanujan's τ function at huge primes for ℓ ∈ {11, 13, 17, 19, 29}. In particular, we get rid of the sign ambiguity stemming from the use of a non-faithful projective representation as in [Bos07]. As a consequence, we can compute the values of τ (p) mod 2 11 · 3 6 · 5 3 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 691 ≈ 2.8 · 10 19 for huge primes p. These representations lie in the jacobian of modular curves of genus up to 22.

The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

“…These are given as polynomials of degree ℓ + 1 whose splitting field is the corresponding extension (with Galois group a subgroup of PGL 2 (F ℓ )). These computations were extended by Mascot [25], who computed polynomials that allow recovery of τ (p) mod ℓ for primes ℓ up to 31.…”

On the computation of coefficients of modular forms: The reduction modulo $p$ approach

“…As one of the applications, he largely improved the known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function. For the recent progress in this direction see [20]. Following Couveignes's idea [9], we give a probabilistic algorithm, which seems to be more suitable to deal with complexity analysis.…”

Computing functions on Jacobians and their quotients

“…These Alpha functions form a subfamily of Eta functions. Mascot introduced, in [25], an efficient evaluation method that applies to another interesting subfamily. One can also define and evaluate functions on J using determinants, see [2,14,31].…”