Brauer relations, isogenies and parities of ranks

Abstract

In this paper, we present applications of pseudo Brauer relations and their regulator constants in the study of isogenies and parities of Selmer ranks of Jacobians. In particular, we revisit and reconstruct a diverse array of classical isogenies in a uniform way and derive local formulae for Selmer rank parities, drawing from an extensive body of literature. These include local expressions found in the works of Birch–Cassels (isogenies between elliptic curves), Mazur–Rubin (dihedral extensions), Coates–Fukaya–Kato–Sujatha (p-power degree isogenies) and Kramer (quadratic twists of elliptic curves).