Topics in Number Theory

Transcription

1 Topics in Number Theory

2 THE UNIVERSITY SERIES IN MATHEMATICS Series Editor: Joseph J. Kohn Princeton University THE CLASSIFICATION OF FINITE SIMPLE GROUPS Daniel Gorenstein VOLUME 1: GROUPS OF NONCHARACTERISTIC 2 TYPE ELLIPTIC DIFFERENTIAL EQUATIONS AND OBSTACLE PROBLEMS Giovanni Maria Troianiello FINITE SIMPLE GROUPS: An Introduction to Their Classification Daniel Gorenstein INTRODUCTION TO PSEUDODIFFERENTIAL AND FOURIER INTEGRAL OPERATORS Francois Treves VOLUME 1: PSEUDODIFFERENTIAL OPERATORS VOLUME 2: FOURIER INTEGRAL OPERATORS MATRIX THEORY: A Second Course James M. Ortega A SCRAPBOOK OF COMPLEX CURVE THEORY C. Herbert Clemens TOPICS IN NUMBER THEORY J. S. Chahal

3 Topics in Number Theory J. S. Chahal Brigham Young University Provo, Utah Springer Science+Business Media, LLC

4 Library of Congress Cataloging in Publication Data Chahal, J. S. Topics in number theory / J. S. Chahal. p. cm. (The University series in mathematics) 1. Numbers, Theory of. I. Title. II. Series: University series in mathematics (Plenum Press) QA241.C '.7-dcl CIP ISBN DOI / ISBN (ebook) Springer Science+Business Media New York 1988 Originally published by Plenum Press, New York in 1988 Softcover reprint of the hardcover 1st edition 1988 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

5 To Professor John T. Tate

6 Preface This book reproduces, with minor changes, the notes prepared for a course given at Brigham Young University during the academic year It is intended to be an introduction to the theory of numbers. The audience consisted largely of undergraduate students with no more background than high school mathematics. The presentation was thus kept as elementary and self-contained as possible. However, because the discussion was, generally, carried far enough to introduce the audience to some areas of current research, the book should also be useful to graduate students. The only prerequisite to reading the book is an interest in and aptitude for mathematics. Though the topics may seem unrelated, the study of diophantine equations has been our main goal. I am indebted to several mathematicians whose published as well as unpublished work has been freely used throughout this book. In particular, the Phillips Lectures at Haverford College given by Professor John T. Tate have been an important source of material for the book. Some parts of Chapter 5 on algebraic curves are, for example, based on these lectures. The chapter on the computation of Mordell-Weil groups is borrowed from his lectures without any changes. Siegel's proof of Dirichlet's theorem on the group of units of an algebraic number field is from a course given by Professor Takashi Ono at the Johns Hopkins University (with the exception that we have avoided using finiteness of the number of ideals of bounded norm). The proof of the Mordell-Weil theorem is from Weil's paper of 1930, "Sur un Theoreme de Mordell." The elementary proof of the "Riemann hypothesis" is due to Yuri I. Manin. An important but not well explained argument in Manin's original paper has been clarified. The main and excellent source of our information on finite fields has been the lectures on "Equations over Finite Fields" by Professor Wolfgang M. Schmidt. I would like to thank Professor Schmidt and Professor Tate for suggesting several improvements in the manuscript. vii

7 viii Preface Finally, I would like to express my gratitude to Professor Stephen P. Humphries for his help in proof reading and to Lonette Stoddard and Jill Fielding for the excellent job of typing the manuscript. Theorems marked with an asterisk have not been proved in this book. The interested reader can find those proofs in the references we have cited. After the book had been completed, several texts appeared that provide excellent material for further reading: D. Husemol\er, Elliptic Curoes, GTM 111, Springer Verlag, New York (1987). N. Koblilz, Elliptic Curoes and Modular Forms, GTM 97, Springer Verlag, New York (1985). J. H. Silverman, The Arithmetic of Elliptic Curoes, GTM 106, Springer Verlag, New York (1986). Provo, Utah J. S. Chahal

8 Contents Notation xi t. Basic Properties of the Integers 1.1. Divisibility The Division Algorithm 1.3. Primes Congruences Diophantine Equations 1.6. Congruent Numbers Algebraic Methods 2.1. Groups 2.2. Subgroups Quotient Groups 2.4. Order of Elements 2.5. Direct Product of Groups 2.6. Generators of a Group 2.7. Homomorphisms of Groups 2.8. Rings Ring Homomorphisms Fields Finite Fields Polynomials over Rings Representation of Integers by Forms 3.1. Introduction ix

9 x Contents 3.2. Quadratic Reciprocity Some Special Quadratic Forms 3.4. Equivalence of Quadratic Forms 3.5. Minima of Positive Quadratic Forms 3.6. Reduction of Positive Quadratic Forms Algebraic Number Fields 4.1. Introduction 4.2. Number Fields 4.3. Discriminant of a Polynomial 4.4. Conjugate Fields 4.5. Algebraic Integers 4.6. Integral Bases 4.7. Group of Units 4.8. Quadratic and Cyclotomic Fields s. Algebraic Curves 5.1. Introduction 5.2. Preliminaries 5.3. Homogeneous Polynomials and Projective Spaces 5.4. Plane Algebraic Curves 5.5. Singularities of a Curve Birational Geometry Some Results from Algebraic Geometry 5.8. The Genus of a Curve 5.9. Elliptic Curves The Mordell-Weil Theorem 6.1. Introduction Heights of Rational Points 6.3. Abscissas of Collinear Points 6.4. Review of Linear Algebra 6.5. Descent The Mordell-Weil Theorem Computation of the Mordell-Weil Group 7.1. Introduction Factorization of the Duplication Map 7.3. A Formula for the Rank

10 Contents 7.4. Computation of a (f) 7.5. Examples Points of Finite Order 7.7. Examples Application to Congruent Numbers xi Equations over Finite Fields Riemann Hypothesis Manin's Proof of Hasse's Theorem 8.3. Proof of the Basic Identity 8.4. Analytic Methods Application to Congruent Numbers 8.6. Remarks on Curves of Higher Genus Appendix. Weierstrass Theory A.1. Review of Complex Analysis A.2. Elliptic Functions.... A.3. The Weierstrass Equation A.4. Addition Theorems A.5. Isomorphic Classes of Elliptic Curves A.6. Endomorphisms of an Elliptic Curve A.7. Points of Finite Order Some Great Number Theorists 185 Index

11 Notation o XuY XnY X s;; Yor Y;;;1 X X Yor ~ Y ~ X X E X or X 3X x ~ X or X tj x Ixi {x I P{x)} X-y I:X -+ Y gol X3X-+YEY N = {l, 2, 3,00 o} l = {O, ±1, ±2,.. o} Q = {m/nlm,n E l, n O} Fq IR C H [x] exp{x) Izi Rez Imz The empty set The union of two sets X and Y The intersection of X and Y X is a subset of Y X is a proper subset of Y x is in X x is not in X The number of elements in X The set of all x having the property P{x) { x E x l x ~ Y } I is a (map or) function from X into Y Composition of functions A function taking x in X to y in Y The natural numbers The integers The rational numbers A finite field of q elements The real numbers The complex numbers The Hamiltonians The integer part of a real number x The exponential function ex Absolute value of a complex number z Real part of z Imaginary part of z xiii

12 xiv M(n, A) AX P' Ipi or det(p) GL(n, A) or GLn(A) SL(n, A) or SLn(A) IE(L) Notation The polynomials in n variables Xt,,X n with coefficients in a ring A = Z, C, f q' iii, C, etc. The ring of n x n matrices over a ring A The group of units of a ring A 3 1 the transpose of a matrix P The determinant of P {P E M(n, A) Idet(P) E AX} {P E GL(n, A)ldet(P) = I} Elliptic functions with period lattice L Implies Is implied by If and only if The sum at an n aj j=t The product at... an