Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms

TL;DR

This paper develops an explicit relative trace formula for Hilbert modular forms over totally real fields, leading to new results on equidistribution of Satake parameters and subconvex bounds for quadratic base change L-functions.

Contribution

It introduces an explicit version of the relative trace formula for Hilbert modular forms and applies it to obtain equidistribution and subconvexity results.

Findings

Spectral equidistribution of Satake parameters in the level aspect.

Subconvex bounds for quadratic base change L-functions.

Explicit trace formula tailored for Hilbert modular forms.

Abstract

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.