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A note on Sarnak's density hypothesis for $\textrm{Sp}_4$ (2305.04791v2)
Published 8 May 2023 in math.NT
Abstract: In this note we prove Sarnak's (spherical) density hypothesis for the full discrete spectrum of the quotients $\Gamma_{\textrm{pa}}(q)\backslash \textrm{Sp}4(\mathbb{R})$, where $\Gamma{\textrm{pa}}(q)$ are paramodular groups with square-free level $q$. 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.
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