A local-global principle for polyquadratic twists of abelian surfaces

TL;DR

This paper establishes a local-global principle for polyquadratic twists of abelian surfaces over number fields, showing it holds for dimensions up to 2 but fails for dimension 3, thus advancing understanding of abelian variety twists.

Contribution

It proves a local-global criterion for polyquadratic twists of abelian surfaces and provides a counterexample in dimension 3, extending previous quadratic twist results.

Findings

Local-global principle holds for abelian surfaces (g ≤ 2).

Counterexample shows failure of principle for abelian threefolds (g=3).

Builds on geometric and algebraic criteria for twists.

Abstract

We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fit\'e, relying itself on the works by Rajan and Ramakrishnan.