A local-global principle for polyquadratic twists of abelian surfaces

Francesc Fité; Antonella Perucca

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arXiv · 2210.06317

A local-global principle for polyquadratic twists of abelian surfaces

Francesc Fité·Antonella Perucca

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Abstract

We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fit\'e, relying itself on the works by Rajan and Ramakrishnan.

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Francesc Fité, Antonella Perucca. 2022-10-12. A local-global principle for polyquadratic twists of abelian surfaces. https://arxiv.org/abs/2210.06317

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