Details
| Original language | English |
|---|---|
| Pages (from-to) | 21-66 |
| Number of pages | 46 |
| Journal | Pacific journal of mathematics |
| Volume | 317 |
| Issue number | 1 |
| Publication status | Published - 2022 |
| Externally published | Yes |
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∞-division fields of E, with p ∈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
- Complex multiplication, Division fields, Elliptic curves, Entanglement
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Pacific journal of mathematics, Vol. 317, No. 1, 2022, p. 21-66.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Entanglement in the family of division fields of elliptic curves with complex multiplication
AU - Campagna, Francesco
AU - Pengo, Riccardo
N1 - Funding Information: Ce travail a été réalisé au sein du LABEX MILYON (ANR-10-LABX-0070) de l’Université de Lyon, dans le cadre du programme “Investissements d’Avenir” (ANR-11-IDEX-0007) géré par l’Agence Nationale de la Recherche (ANR). Funding Information: This project has received funding from the European Union Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 801199. Publisher Copyright: © 2022. Mathematical Sciences Publishers
PY - 2022
Y1 - 2022
N2 - 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∞-division fields of E, with p ∈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.
AB - 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∞-division fields of E, with p ∈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.
KW - Complex multiplication
KW - Division fields
KW - Elliptic curves
KW - Entanglement
UR - http://www.scopus.com/inward/record.url?scp=85132788055&partnerID=8YFLogxK
U2 - 10.2140/pjm.2022.317.21
DO - 10.2140/pjm.2022.317.21
M3 - Article
AN - SCOPUS:85132788055
VL - 317
SP - 21
EP - 66
JO - Pacific journal of mathematics
JF - Pacific journal of mathematics
SN - 0030-8730
IS - 1
ER -