Parity of ranks of Jacobians of curves

TL;DR

This paper studies the parity of ranks of Jacobians of curves with automorphisms, linking local invariants to global rank parity, and provides new insights and proofs related to the parity conjecture for elliptic curves.

Contribution

It introduces a method to express the parity of Jacobian ranks using local invariants and develops the arithmetic theory of Jacobians for Galois covers, with applications to the parity conjecture.

Findings

Derived an expression for Mordell-Weil rank parity under the Shafarevich--Tate conjecture.

Provided a new proof of the parity conjecture for elliptic curves.

Developed the theory of Jacobians for Galois covers, including L-function decomposition.

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.