On Galois covers of curves and arithmetic of Jacobians

TL;DR

This paper investigates the arithmetic properties of curves and Jacobians with finite group actions, focusing on G-modules, Selmer groups, and rational points, and refines existing constructions for isogenies related to permutation representations.

Contribution

It introduces new G-descent results for Selmer groups and rational points, and refines a construction linking permutation representations to isogenies, extending to non-connected curves.

Findings

p^0-Selmer groups are self-dual G-modules

Refined G-descent results for Selmer groups and rational points

Extended properties of Jacobians to non-connected curves

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.