On a local-global principle for quadratic twists of abelian varieties

TL;DR

This paper investigates a local-global principle for quadratic twists of abelian varieties over number fields, confirming it for dimensions up to 3 and providing a counterexample in dimension 4.

Contribution

It proves the local-global principle for quadratic twists of abelian varieties of dimension up to 3 and presents a counterexample in dimension 4, extending known results.

Findings

The principle holds for g ≤ 3.

Counterexample exists for g = 4.

Extension of known results from elliptic curves to higher dimensions.

Abstract

Let $A$ and $A'$ be abelian varieties defined over a number field $k$ of dimension $g\geq 1$. For $g\leq 3$, we show that the following local-global principle holds: $A$ and $A'$ are quadratic twists of each other if and only if, for almost all primes $\mathfrak p$ of $k$ of good reduction for $A$ and $A'$, the reductions $A_{\mathfrak p}$ and $A'_{\mathfrak p}$ are quadratic twists of each other. This result is known when $g=1$, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $g=4$.