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| Formato: | Preprint |
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2026
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| Acceso en línea: | https://arxiv.org/abs/2603.13984 |
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| _version_ | 1866908886296952832 |
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| author | Horozov, Ivan |
| author_facet | Horozov, Ivan |
| contents | The congruence subgroups $Γ_1(m,p)$ that we consider here are subgroups of $GL_m(\Z)$ that fix the vector $(0,\dots,0,1) \mod p$, where $p\geq 5$ is a prime. We present a method and many computations of homological Euler characteristics of $GL_m(\Z)$ and $Γ_1(m,p)$ with coefficients in any highest weight representation $V$. By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in $V$. We compute the homological Euler characteristics for $Γ_1(2,p)$, and $Γ_1(3,p)$ with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of $Γ_1(4,p)$ and $Γ_1(5,p)$ with coefficients in the trivial and the determinant representations. We give application to cohomology of $Γ_1(3,p)$ with trivial and with determinant representation. We also give an alternative method for computing the cohomology of $GL_4(\Z)$ compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Euler Characteristics of a Family of Congruence Subgroups of $GL_m(\Z)$ Horozov, Ivan Number Theory 11F75, 11F70, 11F22, 11F06 The congruence subgroups $Γ_1(m,p)$ that we consider here are subgroups of $GL_m(\Z)$ that fix the vector $(0,\dots,0,1) \mod p$, where $p\geq 5$ is a prime. We present a method and many computations of homological Euler characteristics of $GL_m(\Z)$ and $Γ_1(m,p)$ with coefficients in any highest weight representation $V$. By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in $V$. We compute the homological Euler characteristics for $Γ_1(2,p)$, and $Γ_1(3,p)$ with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of $Γ_1(4,p)$ and $Γ_1(5,p)$ with coefficients in the trivial and the determinant representations. We give application to cohomology of $Γ_1(3,p)$ with trivial and with determinant representation. We also give an alternative method for computing the cohomology of $GL_4(\Z)$ compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}. |
| title | Euler Characteristics of a Family of Congruence Subgroups of $GL_m(\Z)$ |
| topic | Number Theory 11F75, 11F70, 11F22, 11F06 |
| url | https://arxiv.org/abs/2603.13984 |