Coincidences of division fields

2019/12/11 by Harris B. Daniels, Daniels, Harris B., Álvaro Lozano‐Robledo +1
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1912.05618

openalex publication_date 2019/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over ℚ, and let ρ_E\colon \rm Gal(ℚ/ℚ)→ \rm GL(2,\widehat ℤ ) be the adelic representation associated to the natural action of Galois on the torsion points of E(ℚ). By a theorem of Serre, the image of ρ_E is open, but the image is always of index at least 2 in \rm GL(2,\widehatℤ) due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves E/ℚ, and primes p and q such that ℚ(E[p])∩ ℚ(ζ_q^k) is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves E/ℚ and integers m,n such that the m-th and n-th division fields coincide, i.e., when ℚ(E[n])=ℚ(E[m]), when the division field is abelian.

Related

• Coincidences of Division Fields of an elliptic curve defined over a number field
• Towards a classification of entanglements of Galois representations attached to elliptic curves
• Composite images of Galois for elliptic curves over Q & Entanglement fields
• Near coincidences and nilpotent division fields
• Elliptic curves with non-abelian entanglements