On the top-degree cohomology groups of congruence subgroups of $\text{Sp}_{2n}(\mathbb{Z})$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916056577081344 |
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| author | Capovilla-Searle, Fabio |
| author_facet | Capovilla-Searle, Fabio |
| contents | Let $Γ_{2n}^ω(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel--Serre demonstrated that the cohomology of $Γ_{2n}^ω(p)$ vanishes above degree $n^2$. We prove that $\text{H}^{n^2}(Γ_{2n}^ω(p); \mathbb{Q})$ surjects onto the homology of the quotient of the symplectic Tits building for $\mathbb{Q}$ by $Γ_{2n}^ω(p)$ and we compute the homology of this quotient. We conclude that $\text{H}^{n^2}(Γ_{2n}^ω(p);\mathbb{Q})$ is nontrivial and provide a lower bound of its rank. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29010 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the top-degree cohomology groups of congruence subgroups of $\text{Sp}_{2n}(\mathbb{Z})$ Capovilla-Searle, Fabio Algebraic Topology Let $Γ_{2n}^ω(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel--Serre demonstrated that the cohomology of $Γ_{2n}^ω(p)$ vanishes above degree $n^2$. We prove that $\text{H}^{n^2}(Γ_{2n}^ω(p); \mathbb{Q})$ surjects onto the homology of the quotient of the symplectic Tits building for $\mathbb{Q}$ by $Γ_{2n}^ω(p)$ and we compute the homology of this quotient. We conclude that $\text{H}^{n^2}(Γ_{2n}^ω(p);\mathbb{Q})$ is nontrivial and provide a lower bound of its rank. |
| title | On the top-degree cohomology groups of congruence subgroups of $\text{Sp}_{2n}(\mathbb{Z})$ |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2605.29010 |