On the computation of the coefficients of modular forms

Abstract

An overview for a non-specialised audience is given of joint work with Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman. This joint work concerns fast computation of coefficients of modular forms, via the computation of associated Galois representations. For example, for p prime, Ramanujan’s τ(p) can be computed in time polynomial in log p. The overview focuses on the main results and ideas. Developments since 2006 are included: more examples by Johan Bosman, generalisation to forms of level one and of arbitrary weight, and an application to theta functions of lattices. Some future developments are mentioned.