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

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.