A note on the parity conjecture and base change

TL;DR

This paper explores conditions under which the parity conjecture for abelian varieties over number fields can be deduced from its validity over smaller fields, linking root numbers and ranks.

Contribution

It provides a criterion showing how the parity conjecture over larger fields can be inferred from its validity over the base field and certain extensions.

Findings

If an abelian variety satisfies the parity conjecture over Q and quadratic fields, it does so over all number fields.

A general criterion is established for when the parity conjecture over an extended field follows from smaller fields.

Results depend on the Shafarevich-Tate conjecture and properties of root numbers.

Abstract

The parity conjecture predicts that the parity of the rank of an abelian variety is determined by its global root number, that is by the sign in the conjectural functional equation of its L-function. Assuming the Shafarevich-Tate conjecture, we show that if a semistable principally polarised abelian variety $A/\mathbb{Q}$ satisfies the parity conjecture over $\mathbb{Q}$ and over all quadratic fields, then it satisfies it over all number fields. More generally, we establish a criterion for when the parity conjecture for the base change of an abelian variety to a larger number field is already implied by the parity conjecture over the ground field and over other small extensions.