Jeanine Van Order (Einstein Institute of Mathematics)
Average values of automorphic L-functions via spectral decomposition of shifted convolution sums.


Abstract: I will explain a technique for estimating average values Rankin-Selberg L-functions for GL(2) via spectral decompositions of certain shifted convolution sums. Such a technique can be used to determine the nonvanishing of certain moments of central values of interest to the Iwasawa main conjectures (along the lines of previous works of Greenberg, Rohrlich, Vatsal, and Cornut), allowing e.g. for an analytic study of the heights of CM points on Shimura curves. The technique applies rather more generally than this though. For instance, it also suggests some interesting avenues for the study of central values of automorphic L-functions of certain higher-rank groups, which I will describe if time permits.