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 $\ell$. To this end, we design a practical variant of the complex approximations method presented in the book edited by B. Edixhoven and J.-M. Couveignes. 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 $\Delta$ and $E_4 \Delta$, and manage to compute for the first time the associated faithful representations modulo $\ell$ and the values modulo $\ell$ of Ramanujan's $\tau$ function at huge primes for $\ell \in {11,13,17,19,29}$. In particular, we get rid of the sign ambiguity stemming from the use of a non-faithful representation as in J. Bosman's work. As a consequence, we can compute the values of $\tau(p) \bmod 2^11.3^6.5^3.7.11.13.17.19.23.29.691 \approx 2.8.10^19$ for huge primes $p$. These representations lie in the jacobian of modular curves of genus up to 22.