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

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 $\mathrm{\#}\{\pi \in \mathcal{F}(\mathbf{k},\mathfrak{q}):L(1\mathrm{/}2,\pi )\mathrm{\ne }0\}$ as $\mathrm{\#}\mathcal{F}(\mathbf{k},\mathfrak{q})\to +\mathrm{\infty }$. Moreover, our result matches the strength of the best known results in both the level and weight aspects.