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

Abstract

In this note we prove Sarnak's (spherical) density hypothesis for the full discrete spectrum of the quotients $\hat{I}{“}_{\text{pa}}(q)\backslash {\text{Sp}}_4(\mathbb{R})$, where $\hat{I}{“}_{\text{pa}}(q)$ 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.