Abstract
In its simplest form, reduction theory is the study of the quotient H(ℝ)/H(ℤ), where H denotes a linear algebraic group defined over ℚ. The basic content of this study is the construction of “good” fundamental domain for the operation of H(ℤ) on H(ℝ) by multiplication, or rather the proof of the existence of such a domain; this yields certain finiteness results, among which the finiteness of vol H(ℝ)/H(ℤ) and the finite presentability of H(ℤ) are the most important ones. By the fundamental work of Borel and Harish-Chandra [BH], reduction theory has become a standardized topic in the arithmetic theory of algebraic groups, see [PR]. The case of units of orders had been settled before by Siegel [Si 1] and was treated again, in a somewhat different and more extensive manner, by Weyl [Wey]. The justification of the present chapter (which follows Weyl) lies in the considerable simplifications arising in our case; all one needs is some elementary linear algebra and, of course, Minkowski’s lattice point theorem. This fortunate circumstance is due to the fact that the process of Jacobi transformation degenerates to identity.
Chapter PDF
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Basel AG
About this chapter
Cite this chapter
Kleinert, E. (2000). Siegel-Weyl Reduction Theory. In: Units in Skew Fields. Progress in Mathematics, vol 186. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8409-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8409-9_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9555-2
Online ISBN: 978-3-0348-8409-9
eBook Packages: Springer Book Archive