Saito-kurokawa Lifts and Applications to Arithmetic

In this short survey paper we present an outline for using the Saito-Kurokawa correspondence to provide evidence for the Bloch-Kato conjecture for modular forms. Specific results will be stated, but the aim is to provide the framework for such results with an aim towards future research. 1. The Bloch-Kato conjecture for modular forms In this section we will review the Bloch-Kato conjecture for modular forms. The conjecture is presented in the general framework of using L-functions and automorphic data to obtain arithmetic information. As such, we begin by reviewing Dirichlet’s class number formula and the Birch Swinnerton-Dyer conjecture in this framework. Let K be a finite abelian Galois extension of Q. Of particular arithmetic interest is the size of the class group of K, denoted by hK . In the spirit of the general framework we are trying to establish, we associate automorphic objects to K in the form of the group of characters X of K. Dirichlet’s class number formula gives a precise relationship between the special values of the L-functions of these characters and the size of the class group of K. In particular, we have