A Note on Elliptic Curves and the Monogeneity of Rings of Integers

Abstract

In [5] M.-N. Gras has shown that for a prime p > 5, a Galois extension N of Q of degree p can have a monogenic ring of integers only if N is a real cyclotomic field: N = (cos (2n/l)) with / = 2p + 1 a prime. More recently, results of J. Cougnard (see [4]) have suggested that an analogous result might obtain when the base field is a quadratic imaginary number field, K say, instead of : namely, that the only possible monogenic, Galois extensions of degree p over K are derived from certain class fields of K. In the opposite direction, the authors have shown in [1] (see also [2]) that many such class fields are indeed monogenic, with an explicit generator being provided by a division value of a certain elliptic function (see Theorem 2). These results have led us to believe that the rings of integers of all ray class fields of K are monogenic over the ring of integers of the Hilbert class field of K. The aims of this note are twofold: on the one hand we shall use an elliptic curve with complex multiplication and prescribed good reduction properties in order to construct monogenic generators for rings of integers of division fields of the curve. The disadvantage of our result here (see Theorem 1) is that, at the level of ray class fields, our base field will, in general, have a large conductor. It is therefore of interest to see when we can obtain a monogeneity result where the base field is the Hilbert class field itself: this we do in Theorem 3, in the case where 2 splits in K. Given an OK ideal g, we denote the AT-ray class field with conductor g by K(Q). For the sake of simplicity we shall suppose throughout that K has discriminant less than —4. We let E denote an elliptic curve defined over a number field M containing K; we suppose that E admits complex multiplication by OK, and we let s denote the conductor of E/M. Next we choose a function x on E with valency 2 and with a pole of order 2 at the origin, which is defined over M. We then suppose that there is a torsion point Q of E with OK annihilator a, with x(Q) e M. Throughout this paper we shall always assume that a satisfies the following two conditions. (Cl) The ideal a(2, a) is composite, that is to say, it is divisible by two distinct primes of OK. (C2) There exists reOK such that a(r, a) 1 is composite, with