Spectral Means of Central Values of Automorphic L-Functions for GL(2)

Spectral Means of Central Values of Automorphic L-Functions for GL(2) Synopsis

Starting with Green's functions on adele points of $\mathrm{GL}(2)$ considered over a totally real number field, the author elaborates an explicit version of the relative trace formula, whose spectral side encodes the informaton on period integrals of cuspidal waveforms along a maximal split torus. As an application, he proves two kinds of asymptotic mean formula for certain central $L$-values attached to cuspidal waveforms with square-free level.