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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

We investigate the order of the Tate--Shafarevich group of abelian varieties modulo rational squares. Our main result shows that every square-free natural number appears as the non square-free part of the Tate--Shafarevich group of some abelian variety, thereby validating a conjecture of W. Stein., Comment: 8 pages

The present paper illustrates the utility of Brauer relations, Galois covers of curves and the theory of regulator constants in the context of studying isogenies between Jacobians and their relevance to the parity conjecture. This framework presents a unified approach, enabling the reconstruction of a diverse array of classical isogenies and the derivation of local expressions for Selmer rank parities, drawing from an extensive body of existing literature. These include the local expressions found in the works of Mazur--Rubin (dihedral extensions), Coates--Fukaya--Kato--Sujatha ($p^g$ isogenies), Kramer--Tunnell (quadratic twists of elliptic curves), Dokchitser--Maistret (Richelot isogenies), and Docking (prym construction)., Comment: v1, 24 pages

We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich--Tate conjecture, we give an expression for the parity of the Mordell--Weil rank of an arbitrary Jacobian in terms of purely local invariants; the latter can be seen as an arithmetic analogue of local root numbers, which, under the Birch--Swinnerton-Dyer conjecture, similarly control parities of ranks of abelian varieties. As an application, we give a new proof of the parity conjecture for elliptic curves. The core of the paper is devoted to developing the arithmetic theory of Jacobians for Galois covers of curves, including decomposition of their L-functions, and the interplay between Brauer relations and Selmer groups., Comment: Major changes. Parts of this paper have been split out into separate submissions, which will appear shortly ("On Galois covers of curves and arithmetic of Jacobians" and "A note on the parity conjecture and base change"). Added a section on the decomposition of L-functions of curves with automorphisms. Added a new proof of the parity conjecture for elliptic curves, assuming the finiteness of Sha