On counting cuspidal automorphic representations for $\mathrm{GSp}(4)$

TL;DR

This paper counts specific cuspidal automorphic representations of GSp(4) with prescribed local and archimedean properties, providing formulas and generalizations using automorphic Plancherel density theorem.

Contribution

It derives explicit formulas for counting cuspidal automorphic representations of GSp(4) with fixed local components and extends these results to vector-valued cases and multiple ramified places.

Findings

Derived explicit formulas for $s_k(p,Omega)$

Extended formulas to vector-valued and ramified cases

Applied automorphic Plancherel density theorem for generalization

Abstract

We find the number $s_k(p,\Omega)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $\Omega$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,\Omega)$ generalizes to the vector-valued case and a finite number of ramified places.