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

TL;DR

This paper proves that for primes $p$ with $p\equiv 13\pmod{24}$ there exist geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The authors translate the problem into a cohomological criterion: a nontrivial class $x\in H^1(G,E^{\times}/\{\pm 1\})$ with specified local vanishing and a trivial connecting map yields such a twist, and they construct infinitely many examples using CM by $E=\mathbb{Q}(\zeta_{3p})$. They combine Galois cohomology with class field theory to verify the required conditions and to control local behavior at all places; they also relate these strong local twists to Grunwald-Wang type phenomena. The results deliver the first geometrically simple counterexamples in arbitrary large dimension and establish minimality statements within the CM framework, highlighting the utility of the strong local notion for global twisting questions.

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.

Theorems & Definitions (20)

• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Proposition 2.1
• Corollary 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• Lemma 3.2