Relative Trace Formula and Uniform non-vanishing of Central $L$-values of Hilbert Modular Forms

TL;DR

This paper uses regularized relative trace formulas to prove that a positive proportion of Hilbert modular forms have non-vanishing central L-values, matching the best known results in level and weight aspects.

Contribution

It establishes a positive proportion of non-vanishing central L-values for Hilbert modular forms using a novel application of regularized relative trace formulas.

Findings

Proves positive proportion of non-vanishing L-values as form count grows

Matches strength of best known results in level and weight aspects

Utilizes regularized relative trace formulas effectively

Abstract

Let $\mathcal{F}(\mathbf{k},\mathfrak{q})$ be the set of normalized Hilbert newforms of weight $\mathbf{k}$ and prime level $\mathfrak{q}$. In this paper, utilizing regularized relative trace formulas, we establish a positive proportion of $\#\{\pi\in\mathcal{F}(\mathbf{k},\mathfrak{q}):L(1/2,\pi)\neq 0\}$ as $\#\mathcal{F}(\mathbf{k},\mathfrak{q})\to+\infty$. Moreover, our result matches the strength of the best known results in both the level and weight aspects.