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

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.