On the Computation of the Coefficients of a Modular Form

Abstract

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 X ₁(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 l ² 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.