Existence of Hilbert cusp forms with non-vanishing $L$-values

TL;DR

This paper develops a new trace formula to study derivatives of automorphic L-functions for Hilbert cusp forms, proving the existence of forms with non-vanishing central values and large Hecke field degrees.

Contribution

It introduces a derivative version of the relative trace formula on PGL(2) and applies it to establish the existence of Hilbert cusp forms with non-vanishing L-values and large Hecke fields.

Findings

Established a formula for averages of derivatives of L-values.

Proved existence of cusp forms with non-vanishing central derivatives.

Demonstrated forms with large Hecke field degrees.

Abstract

We give a derivative version of the relative trace formula on PGL(2) studied in our previous work, and obtain a formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are sufficiently large.