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arXiv · 1104.4762

On local-global divisibility by $p^n$ in elliptic curves

Laura Paladino·Gabriele Ranieri·Evelina Viada
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Abstract

Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (\zeta_p + \bar{\zeta_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\E$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for elliptic curves defined over a number field of degree at most 5 over $\mathbb{Q}.

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Laura Paladino, Gabriele Ranieri, Evelina Viada. 2011-04-25. On local-global divisibility by $p^n$ in elliptic curves. https://arxiv.org/abs/1104.4762

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