Class numbers and invariant characters of $\mathfrak{sl}_2(\mathbb{F}_p)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916914180128768 |
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| author | Chen, Zhe Feng, Yongqi |
| author_facet | Chen, Zhe Feng, Yongqi |
| contents | Let $p$ be a prime and let $S_2(Γ(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $Γ(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(Γ(p))$ 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 $S_2(Γ(p))$. 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 $S_2(Γ(p^r))$) for any $r\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17214 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Class numbers and invariant characters of $\mathfrak{sl}_2(\mathbb{F}_p)$ Chen, Zhe Feng, Yongqi Representation Theory Number Theory Let $p$ be a prime and let $S_2(Γ(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $Γ(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(Γ(p))$ 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 $S_2(Γ(p))$. 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 $S_2(Γ(p^r))$) for any $r\geq 2$. |
| title | Class numbers and invariant characters of $\mathfrak{sl}_2(\mathbb{F}_p)$ |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2508.17214 |