Binary Quadratic Forms

Transcription

1 Binary Quadratic Forms

2 Duncan A. Buell Binary Quadratic Forms Classical Theory and Modern Computations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

3 Duncan A. Buell Supercomputing Research Center Bowie, MD , USA Mathematical Subject Classification Codes: 11-02, IIRII, IIR29 Library of Congress Cataloging-in-Publication Data Buell. Duncan A. Binary quadratic forms : classical theory and modern computations I Duncan A. Buell. p. cm. Bibliography: p. I. Forms. Binary. 2. Forms. Quadratic. I. Title. QA20l.B '.5-<1c Printed on acid-free paper by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1989 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag. 175 Fifth Avenue. New York. NY 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 of general descriptive names. trade names. trademarks. etc. in this publication. even if the former are not especially identified. is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone. Camera-ready copy prepared by the author using T EX I ISBN-13: e-isbn-13: DOl: /

4 Preface The first coherent exposition of the theory of binary quadratic forms was given by Gauss in the Disqnisitiones Arithmeticae. During the nineteenth century, as the theory of ideals and the rudiments of algebraic number theory were developed, it became clear that this theory of binary quadratic forms, so elementary and computationally explicit, was indeed just a special case of a much more elega,nt and abstract theory which, unfortunately, is not computationally explicit. In recent years the original theory has been laid aside. Gauss's proofs, which involved brute force computations that can be done in what is essentially a twodimensional vector space, have been dropped in favor of n-dimensional arguments which prove the general theorems of algebraic number theory. In consequence, this elegant, yet pleasantly simple, theory has been neglected even as some of its results have become extremely useful in certain computations. I find this neglect unfortunate, because binary quadratic forms have two distinct attractions. First, the subject involves explicit computation and many of the computer programs can be quite simple. The use of computers in experimenting with examples is both meaningful and enjoyable; one can actually discover interesting results by computing examples, noticing patterns in the "data," and then proving that the patterns result from the conclusion of some provable theorem. The second attraction is that, although the theory of forms is just the quadratic case of results in algebraic number theory, the theorems can be independently proved (as in this book) using elementary methods. As Gauss recognized, it is truly delightful to build an elegant algebraic system from just the clever use of high-school algebra. There have been a few other treatments of binary quadratic forms.

5 VI First, of course, there is Gauss. Then there is the book of G. B. Mathews, TheoTY of Numbe1's, published in 1896 but available in reprint form. Mathews is still the standard work on binary quadratic forms, although it predates the algebraic formulations of the early twentieth century and the invention and use of computers. Also, since the forms which are norm forms of quadratic number fields are written (a, b, c), of discriminant b 2-4ac, whereas Gauss and Mathews discuss forms (a, 2b, c) of determinant b 2 - ac, the notation of Gauss and Mathews must be considered "nonstandard"; theirs are not the forms with which the correspondence between the theory of forms and the theory of ideals can be directly made. There are also treatments in the Dirichlet Dedekind ZahlentheoTie, in Weber's LehTbuch det Algebra, Volume III, in the texts of Landau and of Hecke, and occasional chapters in introductory texts in number theory. My experience as a graduate student in mathematics, however, was that of frustration at every turn in trying to collect the computational theory of binary quadratic forms. Gauss, by modern standards, is difficult to read. Mathews is readable, but suffers from the notation problems-my own copy of the Chelsea reprint is liberally annotated with the insertion or removal of those annoying little 2's. It is my hope with this monograph to provide an update to Mathews' book (one book each century doesn't seem unnecessarily frequent), with modern notations and the "correct" definition of forms, an update which is a complete discussion of the classical theory of binary quadratic forms and also a survey of modern computations and applications. Since computations in any quantity are no longer done by hand but by machine, and since one of the major features of this theory is that clear and explicit examples of all theoretical constructs can be found, the emphasis, where possible, will be on explicit, algorithmically sensible proofs and examples. Constructive proofs will be preferred over existence proofs, and poor algorithms will be replaced by better algorithms w here possible. In spite of the notational complications, I must freely acknowledge my debt to the structure and in some places to the proofs of Mathews. I know of no other elementary treatment of this topic, so that of Mathews has become in my mind not just the only known way but the only possible way to proceed. I am also greatly indebted to Oliver Atkin,

6 VB both for his inspiration and for the class notes I took from which this book in part derives. This monograph requires some number theory as a prerequisite, perhaps half of a normal senior-level semester-long course, including the usual material on congruences, quadratic residues, and the reciprocity laws. In addition, "some" abstract algebra is necessary, again about half of a normal senior-level semester-long course. I use some basic group theory, homomorphisms, and describe some material for which the basics of ring theory, ideals, and fields would be useful but are not necessary. I state but do not prove the decomposition theorem for finite abelian groups so as to discuss Sylow subgroups and the special nature of class groups. Although a number of algebraic ideas are used, many of the more advanced or more difficult ideas are not necessary since the structures in number theory are, for the most part, abelian. This is not intended as a classroom text book since the material is not the canonical material of an introductory course in number theory or in algebra. It is intended, however, to be understandable by advanced undergraduates in mathematics or in computer science (and thus usable, perhaps, in a special "readings" course). Judging from the letters and questions I have received over the years from number theorists who understood the theoretical results but had not been exposed to the computational algorithms, I expect that my colleagues in number theory will also find it useful for its computational direction. My goal is a monograph that is readable without being pedantic, that covers completely the basic theory of binary quadratic forms, emphasizing computational and algorithmic aspects, and that adequately summarizes ongoing computations with forms, including both applications and the as-yet-unproved heuristics gained from the extensive computations of the last few years. One of the more difficult parts of writing this book was to find the appropriate level of audience. I have tried not to demand too much background in algebra, and yet to mention the background where extremely relevant, so that naive readers will be given an appreciation of the connections with more advanced algebra and sophisticated readers who will recognize the existence of the connections will be satisfied that the precise theorems have been stated. I hope I have fulfilled that goal. It is incumbent on me to thank those on whom early versions of this

7 V III book were inflicted-my students at Louisiana State University, some of my colleagues at the Supercomputing Research Center, Marvin Wunderlich, Gary Cornell, Mary Ann Grandjean, and Walt Rudd. I am also indebted to Jerome Solinas and Robert 1. Ward for detecting and correcting an error in the proof of Proposition 4.5 and to them and Michael J. Kascic, Jr., for reading this book with probably more care than I took in writing it. What I don't know about mathematics is my fault. This book is dedicated to the three people who are largely responsible for that much smaller subset of what I do know about mathematics. They are David Buell, who first taught me the beauty of mathematics, John Brillhart, who first taught me modern algebra, and Oliver Atkin, who first taught me about binary quadratic forms.

8 Contents Preface 1 Elementary Concepts 2 Reduction of Positive Definite Forms 3 Indefinite Forms 3.1 Reduction, Cycles A utomorphs, Pell's Equation Continued Fractions and Indefinite Forms 4 The Class Group 4.1 Representation and Genera. 4.2 Composition Algorithms Generic Characters Revisited 4.4 Representation of Integers 5 Miscellaneous Facts 5.1 Class Number Computations Extreme Cases and Asymptotic Results. 6 Quadratic Number Fields 6.1 Basic Algebraic Definitions Algebraic Numbers and Quadratic Fields 6.3 Ideals in Quadratic Fields Binary Quadratic Forms and Classes of Ideals 6.5 History... v

9 x 7 Composition of Forms 7.1 Nonfundamental Discriminants 7.2 The General Problem of Composition 7.3 Composition in Different Orders 8 Miscellaneous Facts II 8.1 The Cohen-Lenstra Heuristics Decomposing Class Groups Specifying Subgroups of Class Groups Congruence Conditions Exact and Exotic Groups 9 The 2-Sylow Subgroup 9.1 Classical Results on the Pell Equation 9.2 Modern Results Reciprocity Laws Special References for Chapter Factoring with Binary Quadratic Forms 10.1 Classical Methods SQUFOF 10.3 CLASNO.... loa SPAR Pollard p SPAR CFRAC A General Analysis Bibliography Appendix l:tables, Negative Discriminants Appendix 2:Tables, Positive Discriminants Index CONTENTS