Dokchitser, V; Green, H; Konstantinou, A; Morgan, A; (2025) Parity of ranks of Jacobians of curves. Proceedings of the London Mathematical Society , 131 (3) , Article e70083. 10.1112/plms.70083.

Preview

Text Dokchitser_Proceedings of London Math Soc - 2025 - Dokchitser - Parity of ranks of Jacobians of curves.pdf Download (701kB) | Preview

Abstract

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 (Formula presented.) -functions, and the interplay between Brauer relations and Selmer groups.

Download activity - last month

Export as XMLCSVJSON

Download activity - last 12 months

Export as CSVXMLJSON

Downloads by country - last 12 months

Export as XMLCSVJSON

Archive Staff Only

View Item