Saito-Kurokawa lifts, L-values for GL2, and congruences between Siegel modular forms.

Abstract

Let <italic>f</italic> be a newform of weight 2<italic>k </italic> - 2 and level 1. There is a conjecture of Bloch and Kato that states that the special values of the <italic>L</italic>-function attached to <italic> f</italic> should be the size of the Selmer group attached to <italic>f</italic>. In this thesis we provide evidence for this conjecture by demonstrating an implication that if o | <italic>L</italic>_alg(<italic>k</italic>, <italic>f</italic>), then o must divide the order of the Selmer group as well where o is the uniformizer of a finite extension of <blkbd>Qp</blkbd> . The method we employ is the following. Let <italic>F_f</italic> be the Saito-Kurokawa lift of <italic>f</italic>. Given a prime <italic>p </italic> so that o | <italic>L</italic>_alg(<italic>k</italic>, <italic>f</italic>), we construct a cuspidal Siegel eigenform <italic>G</italic> that is not a CAP form so that Hecke eigenvalues of <italic>G</italic> are congruent to the Hecke eigenvalues of <italic>F_f</italic> modulo o. Associated to a Siegel eigenform we have a 4-dimensional Galois representation. Once we have such an eigenform <italic>G</italic>, we use the explicit nature of the Saito-Kurokawa correspondence to make deductions about the Galois representation associated to <italic>G</italic>. These deductions allow us to construct a non-zero element of <italic>p</italic>-power order in the Selmer group, thereby providing the divisibility we seek.