A local-global principle for polyquadratic twists of abelian surfaces

We say that two abelian varieties $A$ and $A^{\mathrm{\prime }}$ 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 $A^{\mathrm{\prime }}$ be abelian varieties defined over a number field $K$ of dimension $g\ge 1$. In this article we prove that, if $g\le 2$, then $A$ and $A^{\mathrm{\prime }}$ are polyquadratic twists if and only if for almost all primes $\text{p}$ of $K$ their reductions modulo $\text{p}$ 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 Fité, relying itself on the works by Rajan and Ramakrishnan.