Explicit root numbers of abelian varieties

TL;DR

This paper derives explicit formulas for local and global root numbers of abelian varieties, facilitating computations related to the Birch and Swinnerton-Dyer conjecture and demonstrating invariance properties under twists.

Contribution

It provides explicit Jacobi symbol formulas for local root numbers of abelian varieties and extends these to twisted cases, generalizing previous elliptic curve results.

Findings

Explicit formulas for local root numbers as Jacobi symbols

Global root number computation for abelian varieties

Invariance of root number under quadratic twist for a specific genus two curve

Abstract

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist.