Jacobian varieties with group algebra decomposition not affordable by Prym varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912037700894720 |
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| author | Moraga, Benjamín |
| author_facet | Moraga, Benjamín |
| contents | The action of a finite group $G$ on a compact Riemann surface $X$ naturally induces another action of $G$ on its Jacobian variety $\operatorname{J}(X)$. In many cases, each component of the group algebra decomposition of $\operatorname{J}(X)$ is isogenous to a Prym varieties of an intermediate covering of the Galois covering $π_G\colon X \to X/G$; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of $\operatorname{J}(X)$ is not affordable by Prym varieties; namely, affine groups $\operatorname{Aff}(\mathbb{F}_q)$ with some exceptions: $q = 2$, $q = 9$, $q$ a Fermat prime, $q = 2^n$ with $2^n-1$ a Mersenne prime and some particular cases when $X/G$ has genus $0$ or $1$. In each one of this exceptional cases, we give the group algebra decomposition of $\operatorname{J}(X)$ by Prym varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jacobian varieties with group algebra decomposition not affordable by Prym varieties Moraga, Benjamín Algebraic Geometry Primary 14H40, Secondary 14H30 The action of a finite group $G$ on a compact Riemann surface $X$ naturally induces another action of $G$ on its Jacobian variety $\operatorname{J}(X)$. In many cases, each component of the group algebra decomposition of $\operatorname{J}(X)$ is isogenous to a Prym varieties of an intermediate covering of the Galois covering $π_G\colon X \to X/G$; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of $\operatorname{J}(X)$ is not affordable by Prym varieties; namely, affine groups $\operatorname{Aff}(\mathbb{F}_q)$ with some exceptions: $q = 2$, $q = 9$, $q$ a Fermat prime, $q = 2^n$ with $2^n-1$ a Mersenne prime and some particular cases when $X/G$ has genus $0$ or $1$. In each one of this exceptional cases, we give the group algebra decomposition of $\operatorname{J}(X)$ by Prym varieties. |
| title | Jacobian varieties with group algebra decomposition not affordable by Prym varieties |
| topic | Algebraic Geometry Primary 14H40, Secondary 14H30 |
| url | https://arxiv.org/abs/2402.17998 |