On Galois covers of curves and arithmetic of Jacobians

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 -adic representations, Selmer groups, rational points and Shafarevich-Tate groups. In particular, we show that p^-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.