Relative trace formulae and the Gan–Gross–Prasad conjectures

Abstract

This paper reports on some recent progress that have been made on the so-called Gan–Gross–Prasad conjectures through the use of relative trace formulae. In their global aspects, these conjectures, as well as certain refinements first proposed by Ichino–Ikeda, give precise relations between the central values of some higher-rank $L$-functions and automorphic periods. There are also local counterparts describing branching laws between representations of classical groups. In both cases, approaches through relative trace formulae have shown to be very successful and have even lead to complete proofs, at least in the case of unitary groups. However, the works leading to these definite results have also been the occasion to develop further and gain new insights on these fundamental tools of the still emerging relative Langlands program.