Elliptic curves with abelian division fields

Let E be an elliptic curve over Q , and let n ≥ 1 . The central object of study of this article is the division field Q ( E [ n ] ) that results by adjoining to Q the coordinates of all n -torsion points on E ( Q ¯ ) . In particular, we classify all curves E / Q such that Q ( E [ n ] ) is as small a...

Full description

Saved in:
Bibliographic Details
Published in:Mathematische Zeitschrift 2016-08, Vol.283 (3-4), p.835-859
Main Authors: González–Jiménez, Enrique, Lozano-Robledo, Álvaro
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let E be an elliptic curve over Q , and let n ≥ 1 . The central object of study of this article is the division field Q ( E [ n ] ) that results by adjoining to Q the coordinates of all n -torsion points on E ( Q ¯ ) . In particular, we classify all curves E / Q such that Q ( E [ n ] ) is as small as possible, that is, when Q ( E [ n ] ) = Q ( ζ n ) , and we prove that this is only possible for n = 2 , 3 , 4 , or 5. More generally, we classify all curves such that Q ( E [ n ] ) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker–Weber theorem), when Q ( E [ n ] ) / Q is an abelian extension. In particular, we prove that this only happens for n = 2 , 3 , 4 , 5 , 6 , or 8, and we classify the possible Galois groups that occur for each value of n .
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-016-1623-z