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arXiv · 2201.11839

The local-global principle for divisibility in CM elliptic curves

Brendan Creutz·Sheng Lu
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Abstract

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of $p$. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with $p > 2d+1$. We also deduce that the local-global principle for divisibility by powers of $7$ holds over quadratic fields.

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Brendan Creutz, Sheng Lu. 2022-01-27. The local-global principle for divisibility in CM elliptic curves. https://arxiv.org/abs/2201.11839

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