The goal of the proposal is to relate period integrals defined on spaces of automorphic forms to special values of L-functions. Specifically the co-PI expects to generalize results of Waldspurger to higher rank by relating period integrals to central values of quadratic base change L-functions. The main tool to be used in this work is the relative trace formula as initiated by Jacquet. The co-PI also plans to explore the use of the relative trace formula in the study of families of L-functions with a view towards understanding how the relative trace formula can be used to attack the subconvexity problem.
L-functions provide a connection between the world of automorphic forms and number theory. Special values of L-functions frequently encode important arithmetic information; for example the Birch and Swinnerton-Dyer conjecture asserts that the L-function of an elliptic curve determines important information about the structure of the elliptic curve. Elliptic curves have become a focal point of much research, from Wiles' proof of Fermat's Last Theorem to cryptography.
