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

Abstract

Let A and A′ be abelian varieties defined over a number field k of dimension g≥ 1. For g≤ 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 p of k of good reduction for A and A′, the reductions Ap and Ap′ 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. © 2022, The Author(s).