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

TL;DR

This paper explores the relationship between class numbers of imaginary quadratic fields and invariant characters of $ ext{SL}_2( ext{F}_p)$, extending classical results to a Lie algebra setting and to higher powers of primes.

Contribution

It establishes a Lie algebra analogue of Hecke's classical result and extends the relationship to $ ext{SL}_2( ext{Z}/p^r)$ for all $r \\geq 2$, broadening the understanding of modular forms and group actions.

Findings

Proves a Lie algebra analogue of Hecke's class number result.

Extends the classical relation to $ ext{SL}_2( ext{Z}/p^r)$ for all $r \\geq 2$.

Provides new insights into the representation theory of $ ext{SL}_2( ext{F}_p)$ and its connection to number theory.

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 $p\equiv 3\mod 4$, 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 $r\geq 2$.