# Counterexamples to the local-global divisibility over elliptic curves

## TL;DR

This paper classifies certain subgroup actions on elliptic curve points over number fields, providing counterexamples to the local-global divisibility principle for prime powers.

## Contribution

It characterizes all subgroup configurations of GL_2(Z/pZ) that lead to counterexamples to local-global divisibility on elliptic curves over number fields.

## Key findings

- Identifies all subgroups G of GL_2(Z/pZ) associated with counterexamples.
- Constructs explicit examples of elliptic curves where local-global divisibility fails.
- Provides a complete classification of G that admit such counterexamples.

## Abstract

Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal} ( k ( {\mathcal{E}}[p] ) / k )$-module ${\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \mathbb{Z} / p \mathbb{Z} )^2$ and there exists $n \in \mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\mathcal{E}} ( k )$.