Arithmetic of quadratic forms.

*(English)*Zbl 0785.11021
Cambridge Tracts in Mathematics. 106. Cambridge: Cambridge University Press. x, 268 p. (1993).

This is an introduction to the arithmetic theory of quadratic forms over the rational integers, starting from scratch except for some basic knowledge in algebra and some use of algebraic number theory in the last chapter.

Chapter 1 gives a concise introduction to basic notions, Chapter 2 describes reduction theory following Siegel and Blichfeldt’s estimate of Hermite’s constant. Chapter 3 classifies regular quadratic spaces over \(\mathbb{Q}_ p\), Chapter 4 contains a proof of the theorem of Hasse- Minkowski. In Chapter 5 quadratic forms over \(\mathbb{Z}_ p\) are studied, with emphasis on computation of local densities. Chapters 6 and 7 finally concern quadratic forms over \(\mathbb{Z}\): In Chapter 6 representations of lattices by lattices are considered, using approximation theorems and the notions of genus and spinor genus, ending with a proof of the Minkowski- Siegel formula (without proof of the analytic part). In Chapter 7 functorial properties of positive definite quadratic forms are investigated, especially tensor products and scalar extension of positive lattices, describing in textbook form some of the author’s research results in this area.

The book ends with “Notes”, which give hints to further results and the literature and which contain a list of unsolved problems (10 for Chapter 6, 14 for Chapter 7) with some ideas how to handle them. Problem 1 of the Notes to Chapter 7 has a partial answer by J.-F. Burnol [C. R. Acad. Sci., Paris, Sér. I 312, 367-368 (1991; Zbl 0714.11035)], who uses Rogers’ bound to improve the bound 43 to 45.

This is a well-written textbook with many new aspects and recent results. It can be recommended very much for courses and seminars and for self- study.

Chapter 1 gives a concise introduction to basic notions, Chapter 2 describes reduction theory following Siegel and Blichfeldt’s estimate of Hermite’s constant. Chapter 3 classifies regular quadratic spaces over \(\mathbb{Q}_ p\), Chapter 4 contains a proof of the theorem of Hasse- Minkowski. In Chapter 5 quadratic forms over \(\mathbb{Z}_ p\) are studied, with emphasis on computation of local densities. Chapters 6 and 7 finally concern quadratic forms over \(\mathbb{Z}\): In Chapter 6 representations of lattices by lattices are considered, using approximation theorems and the notions of genus and spinor genus, ending with a proof of the Minkowski- Siegel formula (without proof of the analytic part). In Chapter 7 functorial properties of positive definite quadratic forms are investigated, especially tensor products and scalar extension of positive lattices, describing in textbook form some of the author’s research results in this area.

The book ends with “Notes”, which give hints to further results and the literature and which contain a list of unsolved problems (10 for Chapter 6, 14 for Chapter 7) with some ideas how to handle them. Problem 1 of the Notes to Chapter 7 has a partial answer by J.-F. Burnol [C. R. Acad. Sci., Paris, Sér. I 312, 367-368 (1991; Zbl 0714.11035)], who uses Rogers’ bound to improve the bound 43 to 45.

This is a well-written textbook with many new aspects and recent results. It can be recommended very much for courses and seminars and for self- study.

Reviewer: M.Peters (Münster)

##### MSC:

11E12 | Quadratic forms over global rings and fields |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11E08 | Quadratic forms over local rings and fields |

11E10 | Forms over real fields |

11E88 | Quadratic spaces; Clifford algebras |