Book contents
- Frontmatter
- Contents
- Preface
- Notation
- Chapter 1 General theory of quadratic forms
- Chapter 2 Positive definite quadratic forms over ℝ
- Chapter 3 Quadratic forms over local fields
- Chapter 4 Quadratic forms over ℚ
- Chapter 5 Quadratic forms over the p-adic integer ring
- Chapter 6 Quadratic forms over ℤ
- Chapter 7 Some functorial properties of positive definite quadratic forms
- Notes
- References
- Index
Chapter 2 - Positive definite quadratic forms over ℝ
Published online by Cambridge University Press: 20 March 2010
- Frontmatter
- Contents
- Preface
- Notation
- Chapter 1 General theory of quadratic forms
- Chapter 2 Positive definite quadratic forms over ℝ
- Chapter 3 Quadratic forms over local fields
- Chapter 4 Quadratic forms over ℚ
- Chapter 5 Quadratic forms over the p-adic integer ring
- Chapter 6 Quadratic forms over ℤ
- Chapter 7 Some functorial properties of positive definite quadratic forms
- Notes
- References
- Index
Summary
The aim of this chapter is to give a reduction theory of quadratic forms over ℝ. Let P be the set of all positive definite matrices of degree m and denote GLm(ℤ) by Γ. Then Γ acts discontinuously on P by p → p[g] (p ∈ P, g ∈ Γ). An important problem is to determine the fundamental domain P/Γ, that is to give a “canonical” matrix in the orbit p[g] (g ∈ Γ). This was done by Minkowski for general m and the set of canonical matrices is called Minkowski's domain. Unless there is a need for an exact fundamental domain, Siegel's domain is more useful, especially for analysis on P. So we introduce Siegel's domain instead of Minkowski's. As an application, we prove the finiteness of the class number of (not necessarily positive definite) quadratic forms. In Section 2.2, we give a better estimate due to Blichfeldt of Hermite's constant introduced in Section 2.1. The proof will be an introduction to the theory of geometry of numbers. We need it in the last chapter.
Reduction theory
In this section, we describe a reduction theory of positive definite symmetric matrices.
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- Arithmetic of Quadratic Forms , pp. 33 - 46Publisher: Cambridge University PressPrint publication year: 1993