Counterexamples to the local–global divisibility over elliptic curves

Abstract

Let $$p \ge 5$$ be a prime number. We find all the possible subgroups G of $$\mathrm{GL}_2 ({\mathbb Z}/ p {\mathbb Z})$$ such that there exist a number field k and an elliptic curve $${\mathcal {E}}$$ defined over k such that the $$\mathrm{Gal}(k ({\mathcal {E}}[p])/k)$$ -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 $${\mathcal {E}}(k)$$ .