Class numbers and invariant characters of ${\mathfrak{sl}}_2({\mathbb{F}}_p)$

Abstract

Let $p$ be a prime and let $S_2(\Gamma (p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma (p)$. Then ${\mathrm{SL}}_2({\mathbb{F}}_p)$ acts on $S_2(\Gamma (p))$ in a natural way. Around 1928, Hecke proved that if $p>3$ and $p\equiv 3\mathrm{mod}4$, then the class number of $\mathbb{Q}(\sqrt{-p})$ is equal to the difference between the multiplicities of two particular irreducible representations of ${\mathrm{SL}}_2({\mathbb{F}}_p)$ in $S_2(\Gamma (p))$. In this paper we prove a Lie algebra analogue of this result. As an application we extend Hecke's result to ${\mathrm{SL}}_2(\mathbb{Z}/p^r)$ (acting on $S_2(\Gamma (p^r))$) for any $r\ge 2$.

11 pages