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

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For every CM elliptic curve $E$ 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. If $F = K$ and $E$ is obtained as the base-change of an elliptic curve defined over $\mathbb{Q}$, we prove that this finite subfamily is never linearly disjoint over $K$ as soon as it contains more than one element.
Original languageEnglish
PublisherarXiv preprint
Publication statusSubmitted - 31 Jul 2020

Bibliographical note

32 pages. This revision fixes minor issues, updates the references and includes new results (Corollary 4.6 and Theorem 4.9). Comments are more than welcome!

    Research areas

  • Elliptic curves, Complex Multiplication, Entanglement, Division fields

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ID: 244330039