On Galois covers of curves and arithmetic of Jacobians
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911968461324288 |
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| author | Konstantinou, Alexandros Morgan, Adam |
| author_facet | Konstantinou, Alexandros Morgan, Adam |
| contents | We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and Shafarevich-Tate groups. In particular, we show that $p^\infty$-Selmer groups are self-dual $G$-modules, and give various `$G$-descent' results for Selmer groups and rational points. Along the way we revisit, and slightly refine, a construction going back to Kani and Rosen for associating isogenies to homomorphisms between permutation representations.
With a view to future applications, it is convenient to work throughout with curves that are not assumed to be geometrically connected (or even connected); such curves arise naturally when taking Galois closures of covers of curves. For lack of a suitable reference, we carefully detail how to deduce the relevant properties of such curves and their Jacobians from the more standard geometrically connected case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18258 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Galois covers of curves and arithmetic of Jacobians Konstantinou, Alexandros Morgan, Adam Number Theory 11G30 (Primary) 11G10, 11G20, 14H25, 14H30, 14H40, 14K02 (Secondary) We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and Shafarevich-Tate groups. In particular, we show that $p^\infty$-Selmer groups are self-dual $G$-modules, and give various `$G$-descent' results for Selmer groups and rational points. Along the way we revisit, and slightly refine, a construction going back to Kani and Rosen for associating isogenies to homomorphisms between permutation representations. With a view to future applications, it is convenient to work throughout with curves that are not assumed to be geometrically connected (or even connected); such curves arise naturally when taking Galois closures of covers of curves. For lack of a suitable reference, we carefully detail how to deduce the relevant properties of such curves and their Jacobians from the more standard geometrically connected case. |
| title | On Galois covers of curves and arithmetic of Jacobians |
| topic | Number Theory 11G30 (Primary) 11G10, 11G20, 14H25, 14H30, 14H40, 14K02 (Secondary) |
| url | https://arxiv.org/abs/2407.18258 |