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

Let f be a newform of weight 2k - 2 and level 1. There is a conjecture of Bloch and Kato that states that the special values of the L-function attached to f should be the size of the Selmer group attached to f. In this thesis we provide evidence for this conjecture by demonstrating an implication that if o | L alg(k, f), then o must divide the order of the Selmer group as well where o is the uniformizer of a finite extension of Qp . The method we employ is the following. Let Ff be the Saito-Kurokawa lift of f. Given a prime p so that o | Lalg (k, f), we construct a cuspidal Siegel eigenform G that is not a CAP form so that Hecke eigenvalues of G are congruent to the Hecke eigenvalues of F f modulo o. Associated to a Siegel eigenform we have a 4-dimensional Galois representation. Once we have such an eigenform G, we use the explicit nature of the Saito-Kurokawa correspondence to make deductions about the Galois representation associated to G. These deductions allow us to construct a non-zero element of p-power order in the Selmer group, thereby providing the divisibility we seek.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/125307/2/3192587.pd