Level p Paramodular congruences of Harder type

In this thesis we will produce and investigate certain congruences, as predicted by Harder, between Hecke eigenvalues of Siegel and elliptic modular forms. Such congruences form a natural generalisation of the famous 691 congruence of Ramanujan. In direct comparison the moduli of our congruences will come from critical values of L-functions of elliptic modular forms.

In particular we will be interested in congruences between level p paramodular Siegel forms and Gamma0(p) elliptic forms. Evidence for such congruences in these cases is rare (the only known examples being of level 2, due to Bergstrom et al).

In order to simplify matters on the Siegel side we move into spaces of algebraic modular forms for the group GU2(D) for a quaternion algebra D/Q ramified at p,∞. Here we can use a web of conjectures and results due to Ibukiyama along with trace formulae of Dummigan to produce Hecke eigenvalues
of level p paramodular forms (allowing the congruences to be tested with ease). I provide new algorithms for finding explicit descriptions of these spaces of algebraic forms.

In order to provide justification for the paramodular nature of the congruence we will also consider the interplay between associated Galois representations and
automorphic representations.