Near coincidences and nilpotent division fields
Abstract
Let E/ޑ be an elliptic curve.We say that E has a near coincidence of levelWe classify near coincidences of prime power level and use this result to give a classification of values of n for which Gal((ޑE[n])/)ޑ is a nilpotent group.Along the way we prove a Gauss-Wantzel analog for the elliptic curve E : y 2 = x 3 -x, showing that (ޑ E[n])/ޑ is constructible if and only if ϕ(n) is a power of 2. Assuming that there are no non-CM rational points on the modular curves X + ns ( p) for primes p > 11, we show that Gal((ޑE[n])/)ޑ nilpotent implies that n is a power of 2 or n ∈ {3, 5, 6, 7, 15, 21}.
Fields of Study & Subjects
Rings
Modules
and Algebras
Meromorphic and Entire Functions
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Pacific Journal of Mathematics
- Volume
- 342
- Issue
- 1
- Pages
- 79–113
- Publication date
- 23 March 2026
- Language
- EN