Computing modular Galois representations

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 Edixhoven and Couveignes (Ann. of Math. Stud., vol. 176, Princeton University Press, Princeton, 2011). 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 E4 . Delta, and manage to compute for the first time the associated faithful representations modulo l and the values modulo l of Ramanujan's tau 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 projective representation as in Bosman (On the computation of Galois representations associated to level one modular forms. arxiv. org/abs/0710.1237, 2007). As a consequence, we can compute the values of t(p) mod 2(11) x 3(6) x 5(3) x 7x 11x 13 x 17 x 19 x 23 x 29x 691 approximate to 2.8x1019 for huge primes p. The representations we computed lie in the jacobian of modular curves of genus up to 22.