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
SeriesarXiv

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