1. On Galois covers of curves and arithmetic of Jacobians
- Author
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Konstantinou, Alexandros and Morgan, Adam
- Subjects
Mathematics - Number Theory ,11G30 (Primary) 11G10, 11G20, 14H25, 14H30, 14H40, 14K02 (Secondary) - Abstract
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., Comment: A significant part of this paper has been split out of the original version of arXiv:2211.06357. 27 pages, comments welcome
- Published
- 2024