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arXiv · 1705.01880

Counterexamples to the local-global divisibility over elliptic curves

Gabriele Ranieri
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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 ${\mathcal{E}} ( k )$.

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Gabriele Ranieri. 2017-05-04. Counterexamples to the local-global divisibility over elliptic curves. https://arxiv.org/abs/1705.01880

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