Elliptic curves with abelian division fields

Abstract

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 ntorsion 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(ζn), 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.

Torsion of rational elliptic curves over the maximal abelian extension of ℚ

“…In fact, if E(Q ab ) tors ∼ = Z/mZ×Z/mnZ, both values m and n are controlled primarily by isogenies. For instance, in the proof of Theorem 2.4, González-Jiménez and Lozano-Robledo make use of a key corollary relating full-p-torsion over Q ab to Q-rational p-isogenies: Corollary 2.5 (González-Jiménez, Lozano-Robledo; [8], Corollary 3.9). Let E/Q be an elliptic curve, let p > 2 be a prime, and suppose that Q(E[p])/Q is abelian.…”

“…In particular, the proof of Corollary 2.4 in [8] shows that for all p > 2 if Q(E[p])/Q is abelian for some E/Q, then E has two independent p-isogenies over Q. Note that the converse is also true.…”

“…In fact, [16]. For elliptic curves with CM we examine the finitely many j-invariants over Q (see [8] Table 1). The table is broken up into quadratic twist families by j-invariant.…”

“…Finally, we consider the case E : y 2 = x 3 + sx, with s = ±t 2 for any t ∈ Q. We see from [8] Table 1 that the largest division field that is abelian is Q(E [2]). Suppose that E has a point of order 4 over Q ab , then Lemma 3.4 gives that s is a square in Q or −4s 2 is a square in Q, contradicting our assumption that s = ±t 2 for any t ∈ Q. Thus,…”

“…However, the only elliptic curves over Q that have a 27-isogeny are CM curves (those with j-invariant −215 · 3 · 5 3 ). By Table 1 in [8] we see that such a curve does not have Q(E[n]) abelian for any n ≥ 2.…”