The distance between ideals in the orders of a real quadratic field
type: Item
instance of: scholarly article
title: The distance between ideals in the orders of a real quadratic field
zbMATH Open document ID: 0726.11024
author: Pierre Kaplan
author: Kenneth S. Williams
published in: L'Enseignement Mathématique. 2e Série
publication date: 1990-01-01T00:00:00Z
review text: At the 1972 Boulder Number Theory conference \textit{D. Shanks} [Proc. 1972 Number Theory Conf., Boulder 1972, 217-224 (1972; Zbl 0334.12005)] observed a group-like behaviour of the reduced binary quadratic forms of discriminant \(\Delta >0\). He defined a distance d(f,g)\(\in {\mathbb{R}}\) between forms f,g in the socalled principal cycle, i.e. between forms that are \(SL_ 2({\mathbb{Z}})\)-equivalent to the principal form. Shanks observed, for instance, that \(d(f,g+h)\approx d(f,g)+d(f,h)\) modulo the regulator R of the quadratic order of discriminant \(\Delta\). Here \(g+h\) denotes the quadratic form obtained from a composition à la Gauss of f and g followed by a reduction. At the Journées arithmétiques in Exeter in 1981, things were clarified by \textit{H. W. Lenstra} [Lond. Math. Soc. Lect. Notes Ser. 56, 123-150 (1982; Zbl 0487.12003)] who introduced a certain topological group isomorphic to a double circle group \(F=S^ 1\times {\mathbb{Z}}/2{\mathbb{Z}}\) with the usual metric. The quadratic forms in the principal cycle can, in a natural way, be identified with a dense subset of F; the reduced ones form a finite subset. Shanks's distance appears to be very close to the natural distance on the circle group of Lenstra. In the present expository paper this theory is explained. The relations to the arithmetic of real quadratic orders are indicated, but everything is formulated in terms of quadratic forms.
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R54
zbMATH DE Number: 4198083
zbMATH Keywords: binary quadratic forms
zbMATH Keywords: distance
zbMATH Keywords: principal cycle
zbMATH Keywords: double circle group
zbMATH Keywords: expository
zbMATH Keywords: arithmetic of real quadratic orders
reviewed by: René Schoof
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4198083