A Density Theorem for $\operatorname{Sp}(4)$

TL;DR

This paper establishes strong bounds on the number of automorphic forms for Sp(4) that violate the Ramanujan conjecture, surpassing previous density hypotheses using trace formulas and Kloosterman sum bounds.

Contribution

It introduces new bounds for automorphic forms violating Ramanujan conjecture for Sp(4), utilizing a Kuznetsov-type trace formula and improved Kloosterman sum estimates.

Findings

Bounds exceed Sarnak's density hypothesis

Utilizes a Kuznetsov-type trace formula

Achieves best-possible bounds for Kloosterman sums

Abstract

Strong bounds are obtained for the number of automorphic forms for the group $\Gamma_0(q) \subseteq \operatorname{Sp}(4,\mathbb{Z})$ violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak's density hypothesis. The proof is based on a relative trace formula of Kuznetsov type, and best-possible bounds for certain Kloosterman sums for $\operatorname{Sp}(4)$.