Abstract: The comparison of period integrals and special values of L-functions has been a major theme in the study of automorphic representations. I will discuss a conjecture relating periods associated to certain symmetric spaces of unitary groups to central values of standard L-functions on linear groups, which generalizes a theorem of Waldspurger for GL(2).
I will then discuss the strategy to prove this conjecture via a comparison of a new family of relative trace formulas. This is based on joint work with Jingwei Xiao and Wei Zhang. A new feature of this comparison is the presence of relative endoscopy. I will also mention joint work with Thomas Hameister in establishing the endoscopic fundamental lemmas required in the twisted cases.