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

Zhining Wei, Liyang Yang, Shifan Zhao

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 $\#\{π\in\mathcal{F}(\mathbf{k},\mathfrak{q}):L(1/2,π)\neq 0\}$ 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.

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

Abstract

Let be the set of normalized Hilbert newforms of weight and prime level . In this paper, utilizing regularized relative trace formulas, we establish a positive proportion of as . Moreover, our result matches the strength of the best known results in both the level and weight aspects.

Paper Structure

This paper contains 84 sections, 60 theorems, 443 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Uniform non-vanishing
  4. The weight and level aspects
  5. Strategy of the Proof
  6. The weighted second moment via relative trace formula
  7. The weighted first moment via relative trace formula
  8. Contributions from oldforms
  9. Removing the harmonic weight
  10. Outline of the Paper
  11. The second moment via the regularized $\mathbf{RTF}$
  12. The first moment via the $\mathbf{RTF}$
  13. Uniform non-vanishing in harmonic average
  14. Uniform non-vanishing in natural average
  15. Appendix: $\mathbf{RTF}$'s of different types
  16. ...and 69 more sections

Key Result

Theorem A

Let notation be as before. Suppose that $\mathbf{k}\geq\mathbf{4}.$ Let $0<\varepsilon<10^{-3}$, $(\log N(\mathfrak{q})\|\mathbf{k}\|)^{\varepsilon}\leq \xi\leq N(\mathfrak{q})^{1/2-\varepsilon}\|\mathbf{k}\|^{1/4-\varepsilon}$, and $(\log\xi)^{3/2+\varepsilon}<A\leq +\infty$. Then when $\|\mathbf{k}\|N(\mathfrak{q})$ is sufficiently large, where with $\delta_{\mathbf{k}}:=\textbf{1}_{\sum_{v\mi

Theorems & Definitions (111)

  • Theorem A
  • Corollary A
  • Corollary B
  • Theorem B
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Lemma 3.1
  • ...and 101 more