2024/07/11 by Alexandros Konstantinou, Adam Morgan, Konstantinou, Alexandros +1
Computer Science · Engineering · Mathematics · #11G20 #11G30 (Primary) 11G10 #14H25 #14H30 #14H40 #14K02 (Secondary) #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2407.18258
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ-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.