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 exists 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 $\mathcal{E}(k)$.

20 pages