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

Abstract

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

arXiv comment21 pages. Version to appear in Math. Ann. Some details and background material from v1 have been omitted