On the Computation of the Coefficients of a Modular Form

We give an overview of the recent result by Jean-Marc Couveignes, Bas Edixhoven and Robin de Jong that says that for l prime the mod l Galois representation associated to the discriminant modular form Δ can be computed in time polynomial in l. As a consequence, Ramanujan’s τ(p) for prime numbers p can be computed in time polynomial in logp. The mod l Galois representation occurs in the Jacobian of the modular curve X1(l), whose genus grows quadratically with l. The challenge therefore is to do the necessary computations in time polynomial in the dimension of this Jacobian. The field of definition of the l2 torsion points of which the representation consists is found via a height estimate, obtained from Arakelov theory, combined with numerical approximation. The height estimate implies that the required precision for the approximation grows at most polynomially in l.