Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers.

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. With two integral polynomials we associate an integer which we call the congruence number. It has the virtue that it can be very quickly computed and that – in many cases – it is the product of all prime powers modulo which the polynomials have roots in common. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising will be presented.