Relative Galois module structure of rings of integers and elliptic functions. II
This paper is one of a series by the author [for part I, see Math. Proc. Camb. Philos. Soc. 94, 389-397 (1983; Zbl 0532.12009)] which represents a breakthrough in a subject which has been considered in vain by a number of mathematicians: the aim is to find elliptic analogues of the classical cyclotomic results on Galois module generators. In this paper an analogue is offered for the following classical result of Leopoldt: ''Let \(n>1\) and let \(\zeta\) denote a primitive \(n^ 2\)-th root of unity. We set \(M={\mathbb{Q}}(\zeta)\), \(H={\mathbb{Q}}(\zeta^ n)\), and we let \({\mathfrak M}\) denote the maximal order in the group ring \(H\Gamma\). Then \({\mathcal O}_ M\), the ring of algebraic integers in M, is a free, rank one \({\mathfrak M}\)- module on \((1-\zeta^ n)/(1-\zeta)''.\) The elliptic analogue is ''Let \(K\to {\mathbb{C}}\) be a quadratic number field with discriminant \(<-4\) such that 2 is split in K/\({\mathbb{Q}}\), let \(\pi\in {\mathcal O}_ K\) with \(\pi\) \(\equiv 1 (4)\), all primes in K dividing \(\pi\) are split over \({\mathbb{Q}}\). Let N (resp. L) denote the ray class field of K with conductor \(4\pi^ 2 {\mathcal O}_ K\) (resp. \(4\pi\) \({\mathcal O}_ K)\) and \(\Gamma =Gal(N/K)\). Let \({\mathfrak N}\) denote the \({\mathcal O}_ L\)-order in \(L\Gamma\) which is 'outside (\(\pi)\)' equal to \({\mathcal O}_ L \Gamma\) and 'at (\(\pi)\)' equal to the maximal order. The ring \({\mathcal O}_ N\) is then a free, rank one \({\mathfrak N}\)-module on \((D(\alpha)/D(\pi \alpha))(1+\gamma /2)\). Here D is a specific elliptic function on \({\mathbb{C}}/n {\mathcal O}_ K\) \((n=\pi {\tilde \pi})\), \(\alpha\) is a particular choice of a \(4\pi^ 2\)-division point for \({\mathbb{C}}/n {\mathcal O}_ K\), \(\gamma\) is a generator of the cyclic group \(\Gamma\).'' The source which the author uses for the elliptic functions in his constructions is \textit{R. Fueter}'s'' Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen'' from 1924.
scientific article; zbMATH DE number 3955048