On counting cuspidal automorphic representations for GSp(4)

Abstract

We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \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 ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

Dimension formulas for Siegel modular forms of level 4

“…We note that the cases of $$s_k(\Omega )$$ for $$\Omega \in \lbrace \text{IIb, Vb, VIb, VIc}\rbrace$$ can be found in [30, (3.6) and Section 3.2]. Corollary The dimension formulas for Saito–Kurokawa cusp forms given in Table B.4 hold.…”

“…Proof For types (Q) or (B) , the proof is analogous to that of [30, Proposition 2.1]. If $$\pi \cong \otimes \pi _v$$ lies in an Arthur packet of type (Q) or (B) , then the characters parametrizing the packet are ramified at least at one prime p .…”

“…Then every eigenform (for the Hecke operators at all odd primes) 𝑓 ∈ 𝑆 𝑘 (Γ) arises from a vector in 𝜋 𝐶 2 , for some 𝜋 ∈ 𝑆 𝑘 (Ω), by a procedure similar to the one explained in [30,Section 2.1]. Thus, we obtain the formula…”

“…More precisely, $$$$\begin{equation} \Gamma ={\rm Sp}(4,{\mathbb {Q}})\cap {\left(C\times \prod _{\substack{p<\infty \\ p\ne 2}}{\rm GSp}(4,{\mathbb {Z}}_p)\right)}. \end{equation}$$$$Then every eigenform (for the Hecke operators at all odd primes) $$f\in S_k(\Gamma )$$ arises from a vector in $$\pi _2^C$$, for some $$\pi \in S_k(\Omega )$$, by a procedure similar to the one explained in [30, Section 2.1]. Thus, we obtain the formula $$$$\begin{equation} \dim S_k(\Gamma )=\sum _\Omega \sum _{\pi \in S_k(\Omega )}\dim \pi _2^C=\sum _\Omega s_k(\Omega )d_{C,\Omega }, \end{equation}$$$$where $$d_{C,\Omega }$$ is the common dimension of the space of C ‐fixed vectors of the representations π 2 of type Ω with $$r_K(\pi _2)\ne 0$$.…”