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arXiv · 2006.00883

Entanglement in the family of division fields of elliptic curves with complex multiplication

Francesco Campagna·Riccardo Pengo
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Abstract

For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.

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Francesco Campagna, Riccardo Pengo. 2020-06-01. Entanglement in the family of division fields of elliptic curves with complex multiplication. https://doi.org/10.2140/pjm.2022.317.21

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