Résume  $$ I will present a recent proof of Waldspurger's formula relating the square of a period integral of a PGL(2)automorphic form over a torus to the central value of an Lfunction. The feature of this proof is a nonstandard comparison of relative trace formulas, where scalar transfer factors are replaced by transfer operators. This proof is motivated by very general conjectures about periods and the spectral decomposition of relative trace formulas, which suggest the existence of "relative functoriality" between the trace formulas.
