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

Abstract

AbstractLet A and $$A'$$ A ′ be abelian varieties defined over a number field k of dimension $$g\ge 1$$ g ≥ 1 . For $$g\le 3$$ g ≤ 3 , we show that the following local-global principle holds: A and $$A'$$ A ′ are quadratic twists of each other if and only if, for almost all primes $$\mathfrak {p}$$ p of k of good reduction for A and $$A'$$ A ′ , the reductions $$A_\mathfrak {p}$$ A p and $$A_\mathfrak {p}'$$ A p ′ are quadratic twists of each other. This result is known when $$g=1$$ 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 $$g=4$$ g = 4 .