Geometrically simple counterexamples to a local-global principle for quadratic twists

Abstract

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic twists over $K$, the only known counterexamples (necessarily of dimension $\ge 4$) are not geometrically simple. We show that, for every prime $p\equiv 13(\mathrm{mod}24)$, there exists a pair of geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.

15 pages. Comments are welcome! To appear in Math. Zeit