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

Abstract

We find the number $s_k(p,\hat{I}\text{c◯})$ 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 $\hat{I}\text{c◯}$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,\hat{I}\text{c◯})$ generalizes to the vector-valued case and a finite number of ramified places.

Some minor changes are made. To appear in Forum Mathematicum