A note on Sarnak’s density hypothesis for $\operatorname{Sp}_{4}$
Abstract
In this note, we prove Sarnak’s (spherical) density hypothesis for the full discrete spectrum of the quotients \Gamma_{\mathrm{pa}}(q) \backslash \operatorname{Sp}_{4}(\mathbb{R}) , where \Gamma_{\mathrm{pa}}(q) are paramodular groups with square-free level q . To derive this estimate we upgrade a density estimate established via the Kuznetsov formula, which only accounts for the generic part, using Arthur’s parametrization of the discrete spectrum.
Fields of Study & Subjects
Analytic Number Theory Research
Random Matrices and Applications
Advanced Algebra and Geometry
Publication details
- Journal / source
- Commentarii Mathematici Helvetici
- Volume
- 101
- Issue
- 3
- Pages
- 393–428
- Publication date
- 8 January 2026