Spectral Reciprocity: A Fourier--Analytic Approach

Abstract

We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking ${\mathrm{GL}}_3$ and ${\mathrm{GL}}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal ${\mathrm{GL}}_3$ representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for ${\mathrm{GL}}_3\times {\mathrm{GL}}_2$ $L$-functions over number fields, an explicit twisted fourth moment for ${\mathrm{GL}}_2$ $L$-functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product $L$-functions, and new simultaneous nonvanishing results.

158 pages