Arithmetic of Quadratic Forms - Goro Shimura

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Résumé :
This book is divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case. To make the book concise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are stated with an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.

Sommaire:
The Quadratic Reciprocity Law.- Arithmetic in an Algebraic Number Field.- Various Basic Theorems.- Algebras Over a Field.- Quadratic Forms.- Deeper Arithmetic of Quadratic Forms.- Quadratic Diophantine Equations.

From the reviews:

Goro Shimura is one of the world's premier arithmeticians, with his name attached to a number of marvelous things. ... it presents a lot more than its title suggests and does so magnificently. ... Shimura's Arithmetic of Quadratic Forms is another very important monograph by this fine scholar and, to use a hackneyed but apt phrase, will richly repay the reader who invests his time in a careful study of its pages. (Michael Berg, The Mathematical Association of America, August, 2010)

Beginners and experts ... have much to gain from the author's insights and perspectives presented in this book on the arithmetic theory of quadratic forms. ... In order to make the treatment of these topics accessible to readers with a general background in abstract algebra, the author includes preliminary material on algebraic number theory and the theory of semisimple algebras. Throughout the book, the presentation is concise and elegant. (A. G. Earnest, Mathematical Reviews, Issue 2011 m)

In this book Goro Shimura presents some very interesting material in a clear and readable style. ... There is plenty to learn from an introduction to a highly technical subject by one of its leaders. ... The book will serve well as a basis for a couple of postgraduate courses. ... the material is essentially self-contained the exposition is often very technical ... . (Peter Shiu, The Mathematical Gazette, Vol. 96 (536), July, 2012)