TY - JOUR

T1 - Entanglement in the family of division fields of elliptic curves with complex multiplication

AU - Campagna, Francesco

AU - Pengo, Riccardo

N1 - 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!

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - math.NT

KW - Primary: 11G05, 14K22, 11G15, Secondary: 11S15, 11F80

KW - Elliptic curves

KW - Complex Multiplication

KW - Entanglement

KW - Division fields

UR - https://arxiv.org/abs/2006.00883

U2 - 10.2140/pjm.2022.317.21

DO - 10.2140/pjm.2022.317.21

M3 - Journal article

VL - 317

SP - 21

EP - 66

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -