Congruences of Local Origin for Higher Levels

In this thesis we extend the work of Dummigan and Fretwell on congruences of ``local origin''. Such a congruence is one whose modulus is a divisor of a missing Euler factor of an L-function. The main congruences we will investigate are between the Hecke eigenvalues of a level N Eisenstein series of weight k and the Hecke eigenvalues of a level Np cusp form of weight k.

We first prove the existence of a congruence for weights k greater than or equal to 2. The proof will be an adaptation of the one used by Dummigan and Fretwell. We then show how the result can be further extended to the case of weight 1. The same method of proof cannot be used here and so we utilise the theory of Galois representations and make use of class field theory in order to prove the existence of a congruence in this case.

Inspired by an analogy with the weight 1 case, we prove the existence of a congruence between the Hecke eigenvalues of a weight k, level N cusp form and the Hecke eigenvalues of a paramodular Siegel newform of a particular level and weight. We will show how when k=2 we end up with a scalar valued Siegel modular form and when k is greater than 2 we end up with a vector valued Siegel modular form.

We will also consider the link with the Bloch-Kato conjecture in each case.