English

A note on Sarnak's density hypothesis for $\textrm{Sp}_4$

Number Theory 2024-07-01 v2

Abstract

In this note we prove Sarnak's (spherical) density hypothesis for the full discrete spectrum of the quotients Γpa(q)\Sp4(R)\Gamma_{\textrm{pa}}(q)\backslash \textrm{Sp}_4(\mathbb{R}), where Γpa(q)\Gamma_{\textrm{pa}}(q) are paramodular groups with square-free level qq. To derive this estimate we upgrade a density estimate established via the (pre)-Kuznetsov formula, which only accounts for the generic part, using Arthur's parametrization of the discrete spectrum.

Keywords

Cite

@article{arxiv.2305.04791,
  title  = {A note on Sarnak's density hypothesis for $\textrm{Sp}_4$},
  author = {Edgar Assing},
  journal= {arXiv preprint arXiv:2305.04791},
  year   = {2024}
}
R2 v1 2026-06-28T10:28:49.689Z