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

TL;DR

This paper constructs explicit low-dimensional, geometrically simple abelian varieties over  that are strongly locally quadratic twists but not quadratic twists, challenging previous assumptions and using Galois cohomology and class field theory.

Contribution

It provides the first known low-dimensional, geometrically simple counterexamples to the local-global principle for quadratic twists of abelian varieties.

Findings

Existence of geometrically simple abelian varieties of dimension p-1 for primes per 13 mod 24.

Counterexamples are explicitly constructed over .

The proof employs Galois cohomology and class field theory techniques.

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 $\geq 4$) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod{24}$, 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.