Abstract

The Fourier coefficients of classical modular forms essentially coincide (in the case of Hecke eigenforms) with their Hecke eigenvalues, or equivalently, with the coefficients of the associated L-function. The situation is very different for Siegel modular forms of degree 2. The Fourier expansion now contains substantial information beyond the Hecke eigenvalues. Indeed, a remarkable conjecture of Bocherer predicts that certain averages of these Fourier coefficients are essentially linked to twisted central values of spinor L-functions. In this talk, I will discuss some precise refinements of this conjecture and its relation with the global Gan-Gross-Prasad conjecture as refined by Ichino-Ikeda and Liu. I will also describe several applications of these refined results to non-vanishing, algebraicity and integrality of central L-values of cohomological automorphic forms on GL(2), via a lifting argument from GL(2) to GSp(4). Part of this is joint work with Martin Dickson, Ameya Pitale and Ralf Schmidt.