#### Vol. 317, No. 1, 2022

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

### Francesco Campagna and Riccardo Pengo

Vol. 317 (2022), No. 1, 21–66
##### 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 $p\in ℕ$ 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 $ℚ$, we describe in detail the entanglement in the family of division fields of $E$.

##### Keywords
elliptic curves, complex multiplication, division fields, entanglement
##### Mathematical Subject Classification
Primary: 11G05, 11G15, 14K22
Secondary: 11F80, 11S15