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On the local-global principle for twists of abelian varieties

Nirvana Coppola, Lorenzo La Porta, Matteo Longo

TL;DR

The paper develops a cohomological framework for local-global questions about twists of abelian varieties over number fields, introducing Selmer and Tate--Shafarevich sets $ ext{Sel}_m(G_K, D_L^ imes)$ and $ ext{X}_m(G_K, D_L^ imes)$ to measure obstructions. It proves finiteness of $ ext{X}_m(G_K, D_L^ imes)$ under mild assumptions leveraging Faltings-type results, and provides explicit criteria ensuring $ ext{X}_m(G_K, D_L^ imes)=1$, notably in cases where $A$ is geometrically simple with commutative endomorphism algebra or when $ ext{μ}_m subseteq D_K^ imes$ together with coprimality conditions involving $d= rac{2 ext{dim}(A)}{[Z:bQ]}$. The work yields broad positive results for odd $m eq 1$, including CM scenarios and rational roots of unity, and applies to concrete families such as CM elliptic curves and Jacobians, clarifying when local twists glue to global twists. Overall, the results offer a robust cohomological toolkit for determining when a locally prescribed twisting behavior arises globally, with implications for the arithmetic of abelian varieties and their endomorphism structures.

Abstract

This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for $m$ a positive integer, locally $m$-atic twists of an abelian variety $A$ over a number field $K$ are globally $m$-atic. We define and study a "Tate-Shafarevich cohomology set" that governs the obstruction to the local-global principle for $m$-atic twists. We prove that, under some mild assumptions, this set is finite, and give criteria for it to be trivial, i.e. for the local-global principle to be satisfied.

On the local-global principle for twists of abelian varieties

TL;DR

The paper develops a cohomological framework for local-global questions about twists of abelian varieties over number fields, introducing Selmer and Tate--Shafarevich sets and to measure obstructions. It proves finiteness of under mild assumptions leveraging Faltings-type results, and provides explicit criteria ensuring , notably in cases where is geometrically simple with commutative endomorphism algebra or when together with coprimality conditions involving . The work yields broad positive results for odd , including CM scenarios and rational roots of unity, and applies to concrete families such as CM elliptic curves and Jacobians, clarifying when local twists glue to global twists. Overall, the results offer a robust cohomological toolkit for determining when a locally prescribed twisting behavior arises globally, with implications for the arithmetic of abelian varieties and their endomorphism structures.

Abstract

This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for a positive integer, locally -atic twists of an abelian variety over a number field are globally -atic. We define and study a "Tate-Shafarevich cohomology set" that governs the obstruction to the local-global principle for -atic twists. We prove that, under some mild assumptions, this set is finite, and give criteria for it to be trivial, i.e. for the local-global principle to be satisfied.
Paper Structure (10 sections, 16 theorems, 18 equations)

This paper contains 10 sections, 16 theorems, 18 equations.

Table of Contents

  1. Setup and main definitions
  2. Selmer and Shafarevich--Tate sets
  3. Representation-theoretic description of twists
  4. Results on the local-global principle
  5. Reduction to finite Galois extensions
  6. Finiteness of the Shafarevich--Tate group
  7. Normal subgroups and coprime order
  8. The geometrically simple abelian case
  9. The case of rational m-roots of unity
  10. Applications

Key Result

Lemma 2.1

Assume that $D_L = D$, that is, all geometric endomorphisms of $A$ are defined over $K$. Then, the local-global principle holds for $(A,m)$ for all $m$.

Theorems & Definitions (40)

  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • ...and 30 more