Parity of ranks of Jacobians of curves

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 L-functions, and the interplay between Brauer relations and Selmer groups.

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