Number Theory Seminar

Relative Trace Formula and Uniform Non-vanishing of Hilbert Modular $L$-values

Liyang Yang, Department of Mathematics, Princeton Univ.,

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 talk, we will use regularized relative trace formulas to establish a uniform positive proportion of $#{\pi \in \mathcal{F}(\mathbf{k}, \mathfrak{q}) : L(1/2, \pi) \neq 0}$ as $#\mathcal{F}(\mathbf{k}, \mathfrak{q}) \to +\infty$. Our result matches the best-known outcomes in both the level and weight aspects. This is joint work with Zhining Wei and Shifan Zhao.

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