A note on elliptic curves and the monogeneity of rings of integers. (English) Zbl 0639.12001

Cassou-Noguès, Ph.; Taylor, M. J.

J. Lond. Math. Soc., II. Ser. 37, No. 1, 63-72 (1988).

Call an extension \(L| M\) of number fields “monogenic” if \({\mathfrak O}_ L={\mathfrak O}_ M[x]\) with some \(x\in {\mathfrak O}_ L\). By results of M.-N. Gras and Cougnard, monogenic Galois extensions of M of prime degree \(p\geq 5\) are rare if \(M={\mathbb{Q}}\) or M imaginary quadratic, i.e., the possible extensions of this type are those constructed from certain class fields of M.
Let now K be imaginary quadratic with Hilbert class field H. The main result (theorem 1) of the article states that \(L| M\) is indeed monogenic (a generator of \({\mathfrak O}_ L\) over \({\mathfrak O}_ M\) being constructed by means of elliptic functions) if M contains H, L is a certain class field obtained from K, and the whole situation is subject to certain technical assumptions which will not be repeated here. In theorems 2 and 3, the authors descend their result to a statement on the monogeneity of certain extensions over the Hilbert class field H.
As an application, they mention the following beautiful example: Let \(K={\mathbb{Q}}(\sqrt{-7})\), \(\ell =2p+1=\bar {\mathfrak l}\cdot {\mathfrak l}^ a \)prime that splits in K. Then the ray class fields \(K({\mathfrak l})\) and \(K(\bar{\mathfrak l})\) are monogenic over K.

Reviewer: E.-U.Gekeler

MSC:

11R99	Algebraic number theory: global fields
14H52	Elliptic curves
11R37	Class field theory
14H25	Arithmetic ground fields for curves
14H45	Special algebraic curves and curves of low genus
11R11	Quadratic extensions

Keywords:

elliptic curves; monogenic Galois extensions; elliptic functions; Hilbert class field