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A local-global principle for polyquadratic twists of abelian surfaces

Francesc Fité, Antonella Perucca

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 Fité, relying itself on the works by Rajan and Ramakrishnan.

A local-global principle for polyquadratic twists of abelian surfaces

Abstract

We say that two abelian varieties and defined over a field are polyquadratic twists if they are isogenous over a Galois extension of whose Galois group has exponent dividing . Let and be abelian varieties defined over a number field of dimension . In this article we prove that, if , then and are polyquadratic twists if and only if for almost all primes of their reductions modulo are polyquadratic twists. We exhibit a counterexample to this local-global principle for . This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fité, relying itself on the works by Rajan and Ramakrishnan.
Paper Structure (16 sections, 14 theorems, 43 equations)

This paper contains 16 sections, 14 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theoretic descriptions
  4. Polyquadratic twists of abelian varieties
  5. A theorem by Khare and Larsen.
  6. Squares of elliptic curves and quaternionic multiplication
  7. Non CM or quaternionic multiplication
  8. CM over $K$
  9. CM not over $K$
  10. Proof of Theorem \ref{['theorem: main']}
  11. Background results.
  12. Proof strategy.
  13. Absolute types $\textbf{E}$, $\textbf{F}, \textbf{A}, \textbf{C}$.
  14. Absolute type $\textbf{D}$
  15. Absolute type $\textbf{B}$
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that $A$ and $A'$ are abelian surfaces defined over a number field $K$. Then they are polyquadratic twists if and only if they are locally polyquadratic twists.

Theorems & Definitions (36)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Remark 5
  • Proposition 6
  • proof
  • Remark 7
  • Remark 8
  • ...and 26 more