Relative Trace Formula, Periods, L-functions and Harmonic Analysis – FERPLAY

Automorphic forms and Langlands functoriality is a very active area of contemporary international mathematical research at the crossroad of number theory, representation theory and arithmetic and algebraic geometry.

Endoscopy, a technique that allows to study certain instances of functoriality, was initiated by Langlands and Shelstad almost fourty years ago, and is now at a mature state. Recent highlights are the Stabilization of the (twisted) Arthur-Selberg Trace Formula by J. Arthur (which relies on the proof of
the fundamental lemma by Ngô B.-C and results by Waldspurger) and the classification of the automorphic spectrum of orthogonal and symplectic groups that can be extracted from it.

New technics and methods are needed for further study of functoriality. The common motivation of the project FERPLAY is to study the "periods" of automorphic forms. More precisely, we hope to characterize the nonvanishing of certain periods via Langlands functoriality and possibly to link these periods to special values of L-functions (a typical case is the global Gross-Prasad conjecture).

Our approach, follows an idea due to Jacquet, namely to establish a general "relative" trace formula that expresses the periods of the automorphic spectrum in geometric terms and to compare such formulas between various groups. With this objective we plan to develop the necessary tools (relative harmonic analysis, transfers,fundamental lemmas, distinguished representations, spectral decomposition on symmetric spaces, etc.)