Arithmetic of Quadratic Forms
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 se
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Goro Shimura
Arithmetic of Quadratic Forms
123
Goro Shimura Department of Mathematics Princeton University Princeton, NJ 08544 USA [email protected]
ISSN 1439-7382 ISBN 978-1-4419-1731-7 DOI 10.1007/978-1-4419-1732-4
e-ISBN 978-1-4419-1732-4
Library of Congress Control Number: 2010927446 Mathematics Subject Classification (2010): 11D09, 11E08, 11E12, 11E41, 11Rxx, 11Sxx, 17C20 c 2010 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media, LLC (springer.com)
PREFACE
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’ˆetre 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 ϕ over an algebraic number field F, almost all previous authors followed the methods of Helmut Hasse. Namely, one first takes ϕ 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 ϕ 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. The advantage of our method is that it enables us to discuss (a2) in a clearcut way. The main problem is to determine the genera of quadratic forms with integer coefficients that have given local invariants. A quaratic form of n varin ables with integer coefficients can be given in the form ϕ[x] = i, j=1 cij xi xj with a sym
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