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

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 $E$.

arXiv comment35 pages. Final version, to appear in Pacific Journal of Mathematics

References in corpus (6)

• Torsion points and Galois representations on CM elliptic curves
• On Orders in Number Fields: Picard Groups, Ring Class Fields and Applications
• Mahler's measure and elliptic curves with potential complex multiplication
• On Ring Class Fields of Number Rings
• Frobenius lifts and elliptic curves with complex multiplication
• Ramification in Division Fields and Sporadic Points on Modular Curves

Cited by in corpus (2)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Ray class groups and ray class fields for orders of number fields