The distance between ideals in the orders of a real quadratic field. (English) Zbl 0726.11024

At the 1972 Boulder Number Theory conference 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 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.
Reviewer: R.Schoof (Povo)


11E16 General binary quadratic forms
11R11 Quadratic extensions
11R54 Other algebras and orders, and their zeta and \(L\)-functions