Arithmetic of Quadratic Forms / Edition 1

by Goro Shimura

Overview

This book can be divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. The raison d’ˆ etre of the book is in the second part, and so let us first explain the contents of the second part. There are two principal topics: (A) Classification of quadratic forms; (B) Quadratic Diophantine equations. Topic (A) can be further divided into two types of theories: (a1) Classification over an algebraic number field; (a2) Classification over the ring of algebraic integers. To classify a quadratic form— over an algebraic number field F, almost all previous authors followed the methods of Helmut Hasse. Namely, one first takes— in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely it was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to— at each prime spot, which is the same as Eichler’s, but we define it in a different and more direct way, using Clifford algebras. In Sections 27 and 28 we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number field, which is included in class field theory. We prove it when the base field is the rational number field to make the book self-contained in that case.

Product Details

ISBN-13:	9781461426189
Publisher:	Springer New York
Publication date:	09/05/2012
Series:	Springer Monographs in Mathematics
Edition description:	2010
Pages:	238
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

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.

Editorial Reviews

From the reviews:

“This impressive monograph is devoted to some aspects of the arithmetic theory of quadratic forms to which the author has made important contributions in recent years. … Anyone with an interest in the book’s topic will find this volume well worth his or her time.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 169 (1), January, 2013)

“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)

“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)

“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)

From the Publisher