Classical Diophantine Equations

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen Subseries: LOMI and Euler International Mathematical Institute, St. Petersburg Adviser: L. D. Faddeev

1559

Vladimir G. Sprind~uk

Classical Diophantine Equations

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Vladimir G. Sprind~uk t Translation Editors Ross Talent t A l l van der Poorten Centre for Number Theory Research Macquarie University NSW 2109, Australia

Title of the original Russian edition: Klassicheskie diofantovy uravneniya ot dvukh neizvestnykh. "Nauka". Moscow 1982.

Mathematics Subject Classification (1991): I 1J86, 11D00, 11J68, 12E25

ISBN 3-540-57359-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57359-3 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Sprindzhuk, V. G. (Vladimir Gennadievich) [Klassicheskie diofantovy uravneniia ot dvukh neizvestnykh. English] Classical diophantine equations / Vladimir G. Sprindzhuk. p. cm. - (Lecture notes in mathematics; 1559) Includes bibliographical references and index. ISBN 3-540-57359-3 (Berlin: acid-free) - ISBN 0-387-57359-3 (New York: acidfree) 1. Diophantine equations. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1559. QA3.L28 no. 1559 [QA242] 512'.74-dc20 9333443 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper

Foreword

The author had initiated a revision and translation of this volume prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present monograph, which is essentially a translation of a work originally published in the then USSR in 1982, is mostly superseded. That is not so. There is in any event a certain amount of updating inserted by the author. However, the author's emphasis remains original and almost unique, and well warrants study now that this work appears in the mathematical lingua franca* thus making it easily accessible to the majority of mathematicians. Most research mathematicians will be familiar with the eccentricities of Russian style - - in this case I should correct that to Byelorussian style - in mathematical writing. There is quite an amount of repetitive detail and little assumption about notation, exemplified by a great deal more 'letting' in enunciations of lemmata and theorems than now seems customary; and the natural logarithm remains ln, just as on the engineer's calculator. Notwithstanding that, Sprind~uk maintains a pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant emulation. I had the pleasure of meeting the author at several Oberwolfach meetings. Indeed, it was his instruction 'You will walk with me,' that led to the one and only time that I have allowed myself to be subjected to the post-breakfast perambulation all the way down and then, worse, back up the drive. I was a little surprised to find that Sprindguk's spoken English was rather better than I had been led to expect given his apparent reticence at tea and dinner. But that may have been a function of the bad old days. Nonetheless, the translation from which the present volume is derived was just from Russian to 'Russlish'. I am indebted, in the first instance to the late Ross Talent who commenced TEXing and 'translating' the translation prior to his death in a car accident in September 1991, and then to Sam Williams and Dr Deryn Griffiths who assisted with preliminary typing of the remainder of the manuscript. I owe special and extensive thanks to Dr Chris Pinner who carefully read all that preliminary typescript and carefully annotated it with corrections both to its TEX and to its mathematics. Incidentally, Chris Pinner's efforts make it clear that at least some of the detail provided must be * I cannot resist using this phrase and irritating my French colleagues.

Yl taken flexibly. What is presented here is entirely correct in spirit; that is, in its principal parameters. In applying it one should, as always, rework the details to the purpose at hand. That will be all the more so given the errors I will inadvertently have introduced, notwithstanding all the efforts of my minders. I have gone to some pains to translate from the Russlish to English, but restrainedly, if only so as not to hide Sprind~uk's style and personality. That may mean the retention of some eccentric phrasing. I hope that I have not done so to such an extent as to hide important meaning. However, once or twice, I should confess, I had no idea what was intended, even after retreating to the original Russian. So it goes. I began by saying that much of this monograph remains fresh, interesting and useful. The reader should notice the unusual emphases in the first seven chapters; I am confident that there is much yet to be usefully done along the lines there delineated. I am not aware of any other place that a reader can find a congenial entry to the ideas of the final two chapters and am certain that the present volume will spark a great deal of useful thought and fascinating work. Alfred J. van der Poorten ceNTRe for Number Theory Research Macquarie University alf@mpce, mq. edu. au Sydney, Australia May 1993 Afterword: In mid-1993, a volume on diophantine equations seems incomplete if it fails to allude to the surprising announcement by Andrew Wiles of his proof of the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves, and its spectacular consequence. As it happens, Fermat's Last Theorem gets barely a mention in the present volume; the one oasis is the concluding remarks of Chapter VII. Thus to bring this volume up to date in this respect it suffices just to eliminate mention of a paper of Inkeri and mine! Of course it is no longer totally out of the question that the work on elliptic curves be extended to prove the abc conjecture; that will warrant a rather more significant revision. July 1993

Preface

The theory of diophantine equations has a long history, and like human cultm'e as a whole, has had its ups and downs. This monograph aims to show that the last 10 to 15 years were a period of uplift, at least in the field of diophantine equations in two integral unknowns, a part of the subject which has intrigued and attracted researchers throughout its history. Even a cursory acquaintance with the work preceding the papers of Runge [166] in 1887 and Thue [229] in 1909 will impress with the dramatic search for general laws for the behaviour of solutions of diophantine equations, and the realisation of the peculiar difficulties of attaining this aim (see, for example, [56], vol. 2). It was Runge who obtained the first general theorem on the finiteness of the number of integer points on a wide variety of algebraic curves. After nearly a century it is difficult to judge the influence of Runge's work on his contemporaries. Certainly it is evident in Hilbert's proof of his irreducibility theorem [98], which initiated research on the inverse problem of Galois theory. It is possible that Thue was stimulated by Runge's arguments to investigate the representation of numbers by irreducible binary forms, a closely related problem not covered by Runge's theorem. However, the peculiar virtue of Runge's methods - the possibility of making them effective and obtaining explicit bounds for the solutions - was lost in both cases. Thue's work initiated a most fruitful period of development of the theory of diophantine equations in two unknowns - the golden age of ineffective methods! Two monumental results of that period are widely known: Siegel [193] proved that curves of genus greater than zero have only finitely many integer points, and Roth [165], in the problem of representation of nulnbers by irreducible binary forms (the Thue equation), obtained the best possible exponent estimate for the unknowns in terms of the number represented. Both results were achieved by a thorough development and enrichment of Thue's method, and on the way to these results many specific facts were obtained, special methods were worked out, and phenomena arising from these two general theorems were discovered. The monographs by Skolem [196], Lang [120], Mahler [136] and Mordell [145] give a good idea of the variety of the results obtained. One of the above-mentioned special methods is among the most beautiful in the theory of diophantine equations: Skolem's method. Though Skolem himself and his adherents achieved nmch by this method, and were for a long time

VIII the leaders in questions of number representation by norm forms in three or more variables, the ascendancy was finally won by Thue's method (Schmidt's theory of representation of numbers by norm forms is a fine testament to that [185]). Nevertheless, the fundamental idea of Skolem's method, the reduction of algebraic diophantine equations to exponential equations, has shown exceptional vitality in recent episodes of the theory of diophantine equations. In 1952 Gelfond [77] suggested that non-trivial effective estimates for linear forms in the logarithms of three or more algebraic nmnbers would make it possible to obtain explicit bounds for the solutions of exponential diophantine equations, in particular those to which Thue's equation reduces, thereby yielding an effective bound for the solutions of this equation. By that time the necessary estimates were known in the case of two logarithms, but the transition to three logarithms presented considerable difficulty and had not been carried through. In 1966 Baker [8] obtained such estimates for forms in logarithms of any number of algebraic numbers, and later applied them to diophantine equations. Baker's work had a stimulating effect on his close colleagues, and during the next decade the theory of diophantine equations was enriched by results of a qualitatively new type, which will occupy a considerable portion of this monograph. This book covers all the main types of diophantine equations in two unknowns for which the solutions are to be integers or S-integers or rationals or algebraic numbers from a fixed field. Such a broad notion of solution domain makes available a wider arsenal of arithmetic facts than would be possible if only the classical case of rational integer solutions (which, of course, remains the main case here as well) were considered. In particular, by transcending the rational integer domain, we are able to analyse certain classes of diophantine equations in several unknowns (for example, representations of numbers by certain norm forms). Special attention is given to the influence of the paz'ameters of the equation on the magnitude of its solutions, and to the construction (in principle) of best possible bounds for the solutions. Here an interesting general phenomenon is observed which formerly revealed itself in very special cases only: the regulator of some algebraic number field connected with the equation has a preeminent influence on the magnitude of the equation's solutions. (In virtue of the Siegel-Brauer fornmla, this amounts to preeminence of the class number.) We use this phenomenon to describe parametric construction of algebraic number fields with large class number. Further work in this direction may lead to major improvements to known bounds for solutions of diophantine equations in terms of the height of the equation, or to a proof that such an improvement is impossible (which seems more probable). Not all results concerning the value of class numbers are directly connected with the theory of linear forms in logarithms of algebraic numbers, but they were inspired by the above-mentioned relationship between class numbers and the value of solutions of diophantine equations. Chapter IX is altogether independent o f the theory of logarithms. The theory of algebraic units, the theory of ideals in algebraic number fields, and the concepts and techniques of p-adic analysis in both arithmetic

IX and analytic form predominate in this monograph. The informed reader will notice that p-adic analysis makes some quite unexpected appearences. Many of the results can be obtained without the use of p-adic analysis, but there are some which cannot even be formulated without reference to p-adic metrics (See Chap. IX). There is also another approach to the investigation of integer points on algebraic curves which uses parametrisation of curves and the Mordell-Weil theorem on the group of rational points on the curve. We do not touch upon this approach, because the main results obtained in this way are still ineffective. Besides, this topic is treated in a recent monograph by Lang [124]. I have often seen the admiration felt for modern diophantine analysis by older mathematicians who have worked in number theory or taken an interest in its development; for what is done now was in their youth just a pleasant dream. Younger mathematicians will take its achievements for granted, and will feel that its deficiencies shouid be criticised. If this monograph should stinmlate them to creative work or offer clues to new discoveries, its aims will be more than fulfilled. As this work was nearing completion, it became clear that for many readers it will make an impression nmch as the one tourists in Paris feel on seeing the Pompidou Centre: all the main lines, informative and logical, are extremely plain and to the fore. It is, of course, easier to construct a building or write a book in the 'good old style', but then inevitably a great deal will be hidden for the sake of a favourable external impression. Extreme frankness, whether in art or science, imposes nmch more on our time. Central themes of this monograph were the subject of my lectures at the Institut Henri Poincard (Paris, May-June 1980) by invitation of the Universit~ de Paris VI. Namely, (1) generalisations and effective improvements to Liouville's inequality, (2) a connection between bounds for the solutions of diophantine equations and class numbers, and also the manner in which the class number varies, (3) effective versions of Hilbert's irreducibility theorem and rational points on algebraic curves. The audience's interest in and attention to these topics helped to finalise their presentation in this monograph. Michel Waldschmidt and Daniel Bertrand contributed most of all. I am obliged to Alan Baker for the exceptional stinmlus which his works gave me in the late sixties, and to Andrzej Schinzel for information given to me during previous investigations of Hilbert's theorem. I am heartily grateful to all the above-mentioned persons. Minsk September 1980

V. Sprind~uk

Table of Contents

I. Origins 1.1 1.2 1.3 1.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

R u n g e ' s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liouville's Inequality; the Theorem and Method of Thue . . . . Exponential Equations and Skolem's Method . . . . . . . . . . . . . . Hilbert's Seventh Problem and its Development . . . . . . . . . . . .

II. Algebraic Foundations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I L l Height and Size of an Algebraic Number . . . . . . . . . . . . . . . . . . 11.2 Bounds for Units, Regulators and Class Numbers . . . . . . . . . . 11.3 Analytic Functions in p-adic Fields . . . . . . . . . . . . . . . . . . . . . . .

Ill. Linear Forms in the Logarithms of Algebraic Numbers

..........

III.1 Direct Bounds and Connections Between Bounds in Different Metrics . . . . . . . . . . . . . . . . . . . . . III.2 Preliminary Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.3 T h e Auxiliary System of Linear Equations . . . . . . . . . . . . . . . . III.4 Transition to the Auxiliary Function . . . . . . . . . . . . . . . . . . . . . III.5 Analytic-arithmetical Extrapolation . . . . . . . . . . . . . . . . . . . . . . III.6 Completion of the P r o o f of Lemma 1.1 . . . . . . . . . . . . . . . . . . . III.7 Connection Between Bounds in Different Non-archimedean Metrics . . . . . . . . . . . . . . . . . . . . III.8 L e m m a s on Direct Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The T h u e Equation

.........................................

IV.1 IV.2 IV.3 IV.4 IV.5

Existence of a Computable Bound for Solutions . . . . . . . . . . . . Dependence on the Number Represented by the Form . . . . . . Exceptional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimates for the Solutions in terms of the Main P a r a m e t e r s N o r m Forms with Two Dominating Variables . . . . . . . . . . . . . . IV.6 Equations in Relative Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. T h e T h u e - M a h l e r Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.1 Solution of the Thue Equation in S-integers . . . . . . . . . . . . . . .

1

1 3 7 10 14 14 17 22 30 30 32 36 39 43 46 53 58 61 61 64 70 73 75 80 85 85

Xl V.2 Rational Approximation to Algekraic Numbers in Several Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.3 T h e Greatest Prime Factor of a Binary Form . . . . . . . . . . . . . . V.4 T h e Generalised Thue-Mahler Equation . . . . . . . . . . . . . . . . . . V.5 A!cpmximations to Algebraic Numbers by Algebraic Nmnbers of Fixed Field . . . . . . . . . . . . VI. Elliptic and Hyperelliptic Equations VI. 1 VI.2 VI.3 VI.4 VI.5 VI.6 VI.7 VI.8

..........................

T h e Simplest Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . T h e General Hyperelliptic Equation . . . . . . . . . . . . . . . . . . . . . . Linear Dependence of Three Algebraic Units . . . . . . . . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E s t i m a t e for the Number of Solutions . . . . . . . . . . . . . . . . . . . . Linear Dependence of Three S-units . . . . . . . . . . . . . . . . . . . . . . Solutions in S-integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S-integer Points on Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . .

VII. Equations of Hyperelliptic Type

..............................

............................

VHI.1 Influence of the Value of the Class Number on the Size of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.2 Real Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.3 Fields of Degree 3,4 and 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.4 Superposition of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.5 The Ankeny-Brauer-Chowla Fields . . . . . . . . . . . . . . . . . . . . . . . VIII.6 A Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII.7 Conjectures and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Reducibility of Polynomials and Diophantine Equations IX.1 IX.2 IX.3 IX.4 IX.5 IX.6

.........

An Irreducibility Theorem of Hilbert's Type . . . . . . . . . . . . . . . Main argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details and Sharpenings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e o r e m s on Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proofs of the Reducibility Theorems . . . . . . . . . . . . . . . . . . . . . . F~rther Results and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References

.................................................

105 111 111 116 118 120 125 126 131 133

138

VII.1 Equations with Fixed Exponent . . . . . . . . . . . . . . . . . . . . . . . . . VII.2 Equations with indefinite exponent . . . . . . . . . . . . . . . . . . . . . . . VII.3 The C a t a l a n Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. The Class N u m b e r Value Problem

92 94 98

138 144 148 155 155 159 163 167 170 176 184 188 188 191 197 205 208 213 219

Notation The following notation, mainly standard, is fi'equently used. Q C Q N, L , . . . [IL : Q] [L : K] Z Ix IK(x, y , . . . ) K[x, y,...] E~: DR R~ h~: N(a) Nm(o0 Nm~/~:(a) h(o0 I-K] deg a ordp c~

Qp Zp

Qp

~p

~] or HF deg F deg~ F D(F)

R(F, c) Rx(F, G) .

.

.

111 X

the field of rational numbers the field of complex numbers the field of all algebraic numbers algebraic number fields of finite degree over Q the degree of the field L the degree of L over K the ring of rational integers the ring of integers of K the field of rational functions in x, y , . . . over K the ring of polynomials in x, y , . . . over K the group of units of the field K the discriminant of K the regulator of K the number of ideal classes of K the absolute norm of an ideal a the absolute norm of an algebraic number c~ the absolute norm from L to K of an algebraic number the height of an algebraic number ct the size of an algebraic number (the maximum modulus of the conjugates of a) the degree of an algebraic number a, the exponent of the power to which a prime ideal p divides the p-adic metric, normalised so that Iplp = p-1 the field of p-adic numbers the ring of p-adic integers the algebraic closure of Qp the completion in I Ip of an algebraic closure of Qp the height of a polynomial F the degree of a polynomial F the degree of a polynomial F with respect to x the discriminant of a polynomial F the resultant of polynomials F and G the resultant of polynomials F and G with respect to x positive quantities depending only on the indicated parameters the 'natural logarithm' of x, the logarithm to base e the integer part of a real number a.

I. O r i g i n s

This chapter reviews the origin and development of the fundamental principles of the contemporary analysis of diophantine equations, from the perspective of the theory of diophantine approximation.

1. Runge's Theorem Let F(x,y) be an integral polynomial irreducible in Q[x, y]. We suppose as we may without loss of generality that its degree in y is at least its degree in x and set degy F(x, y) = n > 2. We consider solutions in integers x, y of the equation F(x,y) = 0. (1.1) Although Fermat and Euler had analysed special equations of this form (for example, x 2 - Dy 2 = 1 with square-free D), results of a more or less gem eral nature were for a long time elusive. In 1887 Runge devised the general approach whose essence is described below (see also [145], p.262). Equation (1.1) determines an algebraic function y(x) which takes integral values at integer solutions of the equation. Suppose there are infinitely many solutions. One can find y(x) numerically for sufficiently large x by expansion in a power series about the point at infinity. Let F ( x , y ) = I~(x,y) + I ~ - l ( z , y ) + . . . + Io(x,y) where f j ( x , y ) is a binary form of degree j. Put x = t -1 and y = st -1, and write

G(t, s) = tnF(t -1, st -1) = = t'~f,~(t -1, st -1) + t. t'~-lf,~_l(t-1, st -1) + . . . = gn(s) q- gn-l(s)t -t-"" where gn(s), gn-l(s), ... are polynomials in s. Suppose t h a t g~(s) --- f,~(1, s) has no multiple roots. The equation G(t, s) = 0 following from (1.1) defines n power series expansions 8 ~ Or-f- a O t - } - a l t 2 - ~ - . . .

,

2

I. Origins

one for each root a of the polynomial g~ (s), the numbers ai being in the field K = Q(a). Consequently we have n expansions y ---- O~X -~- a 0

-t- O~1 x - 1

nt- 9 9 9

(1.2)

corresponding to the roots a of g,~(s). Let r denote one of the n power series (1.2). The principal idea of Runge consists in a choice of integral polynomials Ai(x), (0 < i < n - 1) of degree not exceeding some bound h, such that a power series expansion of the function n,--1

~(x) = ~

A,(x)r

i=O

about the point at infinity has only negative powers of x: 9 (z) = ~ l x -1 + f~2z -2 + . . .

(1.3)

When can this be done? Writing ~(x) in the form OO

j-=--h-n-t-1

observe t h a t each ~ j is a linear form in the unknown integer coefficients of the polynomials Ao(x),..., A,~_l(X). We will have (1.3) when f~j = 0

(j = - h -

n+ 1,-h-

n,...,0).

(1.4)

Each ~j lies in K, and may therefore be represented by its coordinates with respect to a basis of K as a Q-vector space. Then the system of equations (1.4) becomes a system of d(h+n) linear homogeneous equations with rational coefficients, where d = deg K, in the n(h + 1) unknown integer coefficients of the polynomials Ao(x),... ,A,~_l(x). Provided n(h + 1) > d(h + n) we can guarantee the existence of a non-zero set of integers satisfying the system. If d = n this cannot be done, but for d < n it suffices to take h = n 2 - n + 1. Thus we can find a non-zero set of integral polynomials of degree not exceeding n 2 - n + 1 for which (1.3) will hold, provided that the polynomial f,~(1, s) is reducible in Q[s]. We now substitute in (1.3) the integer values of x for which there exists an integer y satisfying (1.1) and (1.2). For such x, y, with Ix] sufficiently large, we obtain n--1

A , ( x ) r = 0,

(1.5)

i=0

since it follows from (1.3) t h a t I~(x)l < 1 for sufficiently large Ixh and so qS(x), being a rational integer, is zero. We have obtained an equation (1.5) which is independent of (1.1). The polynomial F(x, y) is irreducible in Q[x, y] and its degree with respect to y is n, while the left hand side of (1.5) has degree in y not exceeding n - 1. Writing the resultant of these polynomials

2. Liouville's Inequality; the Theorem and Method of Thue

3

with respect to y, we obtain an equation only for x, which completes the proof of the finiteness of the number of solutions to (1.1) under the assumption that the polynomial f~(1,y) is reducible. Clearly the above argument is effective, and may be used in concrete cases to determine all solutions of (1.1). Its further development yields very strong bounds on the solutions (such as a power of the height of F(x, y)). Certainly the requirement that f= (1, y) be reducible is a serious restriction. Even the case F(x, y) = f,~(x, y) + fo(x, y), with f= irreducible, is of interest, being the problem of representation of numbers by irreducible binary forms. For n = 2 the finiteness or otherwise of the number of solutions is easily resolved, but even for n = 3 significant difficulties arise. The general case was solved by Thue, using a method which has influenced the development of the whole of this branch of number theory.

2. Liouville's Inequality; t h e T h e o r e m and M e t h o d of Thue In 1844 Liouville [128] observed that algebraic numbers do not admit 'very strong' approximation by rational numbers, and was thereby able to give the first construction of transcendental numbers. Since then the approximation estimate he obtained has been so frequently and widely applied that it has acquired a proper name: Liouville's Inequality. Let a be a real algebraic number of degree n _> 2 and let p, q be integers. Then Liouville's inequality is

la-P/ql > clq -'~,

(2.1)

where cl = Cl(O~) > 0 is a value depending explicitly on a. The proof is immediate from the upper bound for the absolute value of N m ( a q - p ) and the observation that it is a non-zero rational number with denominator dividing a s, where a is an integer such that aa is an algebraic integer. For n = 2 it is impossible to improve on (2.1) by replacing cl by some positive function A(q) increasing monotonically to infinity, for it is known from the theory of continued fractions that, for any quadratic irrational a, the reverse of (2.1) has infinitely many solutions in integers p, q when Cl is replaced by v ~ (see [40], Ch. II). For n > 3,however, a sharpening of the (2.1) of the type

[ a - p/q[ > A(q)/q'*,

A(q) i" oo

(2.2)

is of great interest for the study of diophantine equations. Indeed, let f(x, y) be an integral irreducible binary form of degree n > 3, and suppose that A # 0 is an integer. If the inequality (2.1) admits a sharpening of the form (2.2) for some A(q), then the diophantine equation

f(x, y) =- A

(2.3)

4

I. Origins

has only finitely m a n y solutions. If f(x, 1) is a polynomial without real roots, it is obvious t h a t (2.3) has only a finite number of solutions. Suppose instead t h a t a is a real root of f(x, 1) and a ( 0 , i = 1, 2 , . . . n its conjugates. It follows from (2.3) and y # 0 that

i l l a(O - x/yl = A/(lallyr ~)

(2.4)

i=l

where a is the leading coefficient of the polynomial f(x, 1). Assuming the equation (2.3) has integer solutions with arbitrarily large IYl we see t h a t the product on the left of (2.4) takes arbitrarily small values for solutions x, y of (2.3). As all the a(i) are different, x/y must be correspondingly close to one of the real numbers a (i), say a. Thus we obtain - x/yl < c2/ly?

where c2 depends only on a, n, and 1-L#j Ia(i) - a(J)I-1A (see Ch. IV, w Comparison of this inequality with (2.2) shows that lYl cannot be arbitrarily large, and so the number of solutions of (2.3) is finite. It is not difficult to see t h a t the arguments are effective, and t h a t an explicit bound can be constructed for solutions of (2.3) once an effective inequality (2.2) is known. The sharpening of the Liouville inequality (2.1), however, especially in effective form, proved to be very difficult. In 1909 Thue published a proof [229] that

[ a - p/q[ < q-~-l-~

(2.5)

has only finitely many solutions in integers p, q > 0 for all algebraic numbers a of degree n > 3 and any e > 0. In essence, he obtained the inequality (2.2) with A(q) of the form c3q89~ - l - e , where c3 > 0 depends on a and e. But Thue's arguments do not allow one to find a bound for the greatest q satisfying (2.5), so it is impossible to exhibit the dependence of c3 on a and e, and so the bound for the number of solutions to (2.3) cannot be given in explicit form either: it is ineffective. We shall show t h a t the inequality (2.5) has just finitely many solutions following the arguments of Thue himself (see also [51]). Obviously, one may suppose t h a t (p, q) = 1 in (2.5) and that a is an algebraic integer. Suppose t h a t h > 0 is an integer, 5 satisfies 0 < 5 < 1, and m --~- /L2! ( n - 2)(1 +

5)hJ

(2.6)

For each h we will construct auxiliary polynomials P(x), Q(x) of minimal degree and height such t h a t P(x) - aQ(x) is divisible by (x - a) h. In more detail, put P(x)

-- o L Q ( x ) :

( x - o~) h { R 0 -~- R l ( X ) o ~ -[- . . . -~- R n _ l ( X ) o ~ n - 1 }

(2.7)

where the integral polynomials R 0 ( x ) , . . . ,R,~-x(x) are chosen so t h a t their degrees do not exceed m and not all of them are zero. Then we have n(m+ 1)

2. Liouville's Inequality; the Theorem and Method of Thue

5

unknown integer coefficients of these polynomials which are to satisfy the ( n - 2)(h + m + 1) linear homogeneous equations implied by the representation of the right-hand side of (2.7) in the form

So(x) + asl(X) + . . . + a"-ls,,_l(x) and by the vanishing of all coefficients of the polynomials $2 ( x ) , . . . , S,~-1 (x). Condition (2.6) implies

~(~+ 1) ( n - 2)(h + m + l)

>

~(~ -

2)(1 + 6)

n ( n - 2)(1+ 6) - 4$ + 2(n - 2)/h

>1

if h > (n/2-1)5 -1, and so the number of unknowns is greater than the number of equations. It is easy to obtain a bound of the type c4u for coefficients of these equations, where c4 depends only on a. Then for the unknown coefficients of the polynomials R0 ( x ) , . . . , / ~ - l ( X ) one obtains an estimate of the same form c5u with c5 = cs(a, 6). This shows that the heights of the polynomials P(x), Q(x) satisfy m a x ( ~ - - ~ , ~-x-~) < c5h. (2.8) Equation (2.7) shows that the polynomial

W(x) = P(x)Q'(x) - P'(x)Q(x)

(2.9)

cannot b e identically zero, since otherwise P(x) and Q(x) differ only by a numerical factor, and are divisible by (x - a) h and hence by the h-th power of the minimal polynomial of a; so their degrees are not less than nh. But that is impossible, as in fact their degrees do not exceed h+m e -(lna(~))~+~,

(4.1)

where c~1, c~2 are algebraic numbers other than 0 or 1, In c~1/In c~2 is an irrational number, /3 is an algebraic number, and h(/3) is the height of/3 and is taken to exceed some computable value depending on ~1, ~2, the degree of ;9, and e > 0. Later still he improved the exponent 5 + ~ to 2 + E [78]. Let us examine the main ideas which Gelfond applied towards these goals. He begins by constructing an auxiliary transcendental function of a complex variable z: L

f(z) : E

L

~

P(Al"/2)(~'~2~2)z

(4.2)

At:0 12:0

with zero derivatives f(s)(g) for all integers g, s in the range 0 < g < g0, 0 < s < so, where t0, so are arbitrary natural numbers but L must be consistent with to and so. Since L

(inc~1)-~f(~)(t): ~

L

~

--I P('~I,'~2)(z~I~-'[l

2)~'a~a k 1 2~2~ ]

(4.3)

AI----OA2:0

is an algebraicnumber when 7/: In~I/In ~2 is algebraic,we can assurethe equalities f(s) : 0 by choosing integers P()h,/~2) not too large and not all zero, provided that the number of resulting linear constraints on the p ( t l , A2)

4. Hilbert's Seventh Problem and its Development

11

is less than the number of unknowns, that is when (so + 1)(g0 + 1) < ( L + 1) 2. The large number of zeros of the function f(z) and the integral representation f(s)(e)

=

,=R,

t= 2 =

z

with suitably chosen values R1, R2 (R1 _> 2g0, R2 >_ 2R1 + 3So) show that the numbers f(s)(g) are very small in absolute value for values of s, g in an expanded range 0 < g < gl, 0 < s < sl (gl > g0, sl > so). Following from (4.3), algebraicity of the numbers (in a l ) - ~ f ( s ) (g) now implies f(')(g) = 0

(0 < g < gl,0 < s < sl).

The argument may be repeated anew and the range of g, s extended once more, and so on. Gelfond called the procedure analytic-arithmetical extrapolation, and it is the primary mechanism of the method ([77], p.131). To disprove the algebraicity of ~ it suffices to show f(*)(0) = 0

(0 _< s e - ( l n g ) ~ + l + ~ ,

(4.5)

where the/~j are arbitrary algebraic numbers of heights not exceeding H, and H > H0, H0 being effectively determined by a l , . . . , an, together with the bound for the degrees of the/3j and e > 0. His proof works by introducing a new interpolation method into Gelfond's scheme which enables the analyticarithmetical extrapolation to be carried out in the case of an arbitrary number of logarithms of algebraic numbers. In place of the auxiliary function (4.2) Baker uses a function in several complex variables: ~(Zl,

"'''

Zn-l) = ~-~s

Aa+A''B')z'

"''

O~n(X'-a+A'~B"-I)z''-' --1

,

where the summation is over all integer vectors A = ( , ' ~ 1 , . . - , A n ) satisfying 0 ___As < L (i = 1 , . . . , n), and L is an integer parameter chosen in accordance with H. Even for n = 2 this function, being independent of a2, differs from (4.2). Numbers p(A) are chosen as non-trivial integer solutions of the system of linear equations with algebraic coefficients ........ , (e,...,

e) =

= ~p(A)(A1

+ )~n/~l)Sl... (/~n-1 ~- /~n/~n-1) 8n-1 (CgIA1... OLin) / ~- 0

A

(0 < e < CO, Sl + . . . Sn-1 = S ~ SO),

(4.6)

4. Hilbert's Seventh Problem and its Development

13

where (go + 1)(S0 + 1) ~-1 is somewhat less t h a n (L + 1) '~ so that the number of equations is less t h a n then number of unknowns and a suitable estimate for IP(A)I may be obtained. By assuming t h a t the opposite to (4.5) holds, and by choosing L, g0, So judiciously, the system (4.6) implies t h a t the partial derivatives 0s OZ~I ...OZn_ ls,~-lqS(~,''',~')

(0 < ~ < CO, 81q- ...-~- Sn-1 = 8 _< SO)

are small. Application of the Hermite interpolation formula allows this smallness to be demonstrated for a wider range of g under some restriction on the order of the derivatives: 0 < g < gl, 0 < s < $1 (gl > g0, Sx < So). It follows, given the reverse of (4.5), t h a t the corresponding algebraic numbers (4.6) are so small t h a t they must vanish, and this gives the system (4.6) a new range: 0 < g < gl, 0 < s < $1. The procedure is iterated, and eventually yields the system 9 o ..... o ( e ,

.

e) . =

. x

.

.

.

u~ ) = 0

(0 Cll > 0 (d,k) --

(i = 1 , 2 , . . . , r )

where cll depends on g (Lemma 1.1), it follows from (2.13) that

2. Bounds for Units, Regulators and Class Numbers

21

lnlrl~Jl'k)t ... lnlrl~Jq'k)l < e12R for any set ( j l , . . . ,jq). This proves inequality (2.12). From (2.9) we see that the regulator is bounded from below by a value which depends only on the degree of the field. In fact stronger bounds are known (for example, if L is not a purely imaginary quadratic extension of a purely real field, then R > c13 in IDol, where DL is the discriminant of L, and c13 depends only on the degree of L; see [163]), but we do not need them. On the other hand, we will often make use of upper bounds for the regulator. L e m m a 2.4 The class number h, the regulator R and the discriminant D of

a field L are connected by the inequality hR < Cl4]D]l/2(ln ]D]) ~-1,

(2.14)

where c14 is determined explicitly by ~. Proof. The proof of this lemma is well-known: it follows from an upper bound for the residue of the Dedekind zeta function of L. It was probably first obtained by Landau [119]. Expressions for c14 are given in various works (e.g.

[125], [195]). Since h > 1 and R is bounded from below by a value depending only on g, it follows that h and R are each independently bounded from above by expressions of the form (2.14). An upper bound for the residue of the zeta function of L implies the inequality hR > c15[D[ 89 (2.15) for any e > 0, where c15 = c15(e,g) depends on e and e, but an explicit form of this dependence is unknown. The proof of (2.15) is considerably more complicated t h a n that of (2.14). The consequence hR < c161D[89+~ of (2.14) combines with (2.15) to give the single assertion known as the Siegel-Brauer theorem ([194], [34]). In concrete cases, to bound the regulator or the class number from above using (2.14) requires a bound for the discriminant of the field in terms of values associated with a generator of the field (its height, size or discriminant). To do so we shall use the theory of differents of algebraic number fields, relying on three fundamental facts: the discriminant of a field as the norm of the field different, the multiplicative property of differents in relative extensions, and divisibility of the differents of all integers of a field by the field different ([245], [96]). Note that if K is a subfield of L, then hz~; 0). Let a, r c g?p, lalp < p, Irlp < p. Then f

g(z)dz=g(a). ~r

Proof. Integrate the series (3.5) term by term and apply (3.4). L e m m a 3.2 Let x, r E s

and let n > 0 be an integer. Then

,~ ( z - x )

~

(-1p/x

n

if Izlp > Ir[p.

Pro@ The series

k=0

converges for Izlp < Izlp and represents a regular function of z (for fixed x), namely the function (z - x) -~. Hence for Izlp > Irlp the integral is ( - z ) -'~ by Lemma 3.1. When Ix[p < Irlp the integrand has the Laurent series expansion

,z k=O

Since 'on the circle of integration' [Zip = ]rlp, the series may be integrated term by term, which gives

k=O

which is zero. L e m m a 3.3 With the same setting as Lemma 3.1, assume further that x E ~2p, Ixlp < p. Then

f~ g(z)(_~-a) dz = { g(x) iflx-al~ < I%, ,~

z

x

0

/flx-al~>

Iris.

3. Analytic Functions in p-adic Fields

25

Proof. On replacing z by z + a and x by x+a, we see that it suffices to consider the case a = 0. First note that for any integer k > 0 fO

zk+l f zk+l -- xk+l -]- xk+l dz = 3[0 dz ,r Z -- X ,r Z -- X ,r

~r Z -- X

Integrating the first integral t e r m by term and applying L e m m a 3.2 to the second one, we obtain

fO

f xk

zk+l d z =

,r z - x

if IXlp < Irlp, if IXlp > Irlp.

1. 0

Since

fO0 g ( z ) Z d z = ~ ,r Z - - X

k=0

bk ~0

zk+l dz,

,r Z - - X

we obtain the assertion of the lemma. Lemma

3.4 Under the hypotheses of the previous lemma we have for x in

the disc Ix - alp < ]rip f~ 9(z)(z - a)

(n = O, 1,2,...).

Proof. We again start by computing some special integrals, namely ~a

z -- a

(z~-x-)kdz

(k = 1 , 2 , . . . ) .

~r

Since Ix - alp < ]rip, the integrand expands in the series

z-a (z x)k - (z -

1 ( a) k-1 - - (x----~a~q~

-

q=o

\z

-

a/

k

1

~

(z - a) k-1

]

e. s=O

x-a s (=)

say. We know t h a t

f~

dz

_{0

,~ ( z - a ) k+s-1

1

ifk+s1 7~0, ifk+s-l=0.

Therefore

/z-a

,r (Z --- X-~ dz -= E es(x - a)s

Now we note that

s~O

~r

(z--a) k+s-1 --

t01

if k > 1, i l k = 1.

(3.6)

26

II. Algebraic Foundations

g(z) = s

~i ~ z - z) k. g(~)(x)"

k=O

Hence

,~ (z - x) ~+1

k=0

k----f-- ,~ (z ---~ - ~ - k ~ z

-

n!

'

where we apply (3.6) if k < n, and Lemma 3.1 if k > n. We now turn to lemmas on residues of functions of the type g(z)/P(z), where g(z) is regular in some disc and P(z) is a polynomial over Y2p. If zo is a pole of such a function, then Res~ o g(z)/P(z) will denote the residue at the point zo. We begin with the simplest case, the quotient of polynomials. Lemma

3.5 Suppose that G(z) and P(z) = (z - Z l ) k ' . . . ( z - z,~) k" are

polynomials over f?p, with z~ r zj (i # j). Let r E s

f

C(z)z~ -~ az

=

"

E

with [Zj[p # [rip. Then

a(z)

Reszj

.

(3.7)

P(z)

Iz~ I,, 1. So by Lemma 3.1 this integral is equal to f ( z ) / F ( z ) . Each integral

3. Analytic Functions in p-adic Fields

I~=

29

i f(O(r zi).. ,,,-P~(O-(C---~ a;

is easily calculated by the substitution

y(r = ~

f(')_(zO (r _ zOS,

s=O

s!

which gives

s =

~ f(s) (Zi) fz (~=z,)~_~ s+l s~O s! i,t F(r162 - z ) - "

because by L e m m a 3.1

f~,,, P(~- ( ~ -zi),+l z)d~

= o

(s = ~ + 1,~ + 2 , . . ) ,

due to the regularity of the function ( ~ - Z ~)n + l / f (~) at the point z - z~. Thus

I - f(z) + ~ ~ f(~)(z,)1 -

r(z---3

i = 0 s=O

~

(~- Zi)s + l

i,t ~-(~~-- z) ~

and we obtain the desired formula. We conclude this section with a mention of three special functions:

(_l)k-i k zk'

log(Z + z)--

Izlp< 1;

k=l

expz=

~zk~.t' k=O

(l§

t= ~

(tk)zk,

Iz[p< p-1/(p-1);

,z[p< p-1/(p-1)

It[p < 1;

k=0

where z, t E Op. We record the main properties of these functions [93]: a) l l o g ( l + Z)[p = Np if IZ]p< p-1/(p-1); b) logzlz2 = logzl + logz2 if IZl - lip < 1 and Iz2 - lip < 1; c) l e x p z - lip = Izl, if Izl, < p-1/(,-1); d) exp(zl + z2) = exp zl .exp z2 if [zl [p < p-1/(p-1) and [z2 Iv < e) explog(1 + z) = 1 + z and logexpz = z, if Izlp < p-1/(p-a); f) (1 + z) t = exp(t log(1 + z)) if Izlp < p-W(p-x) and Itlv -< 1.

p-1/(p-1);

We note in addition that log z = 0 only for those z which are pS-th roots of unity for some s, and t h a t log(1 + z) is not a bounded function in the disc

Izlp< 1.

III. Linear Forms in t h e L o g a r i t h m s of Algebraic Numbers

This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter9

1. D i r e c t B o u n d s Different Metrics

and Connections

Between

Bounds

in

T h r o u g h o u t a l , a 2 , . . . ,c~, denote algebraic numbers different from 0 or 1 and hi, h2,... ,h,,-x are integers 9 In subsequent chapters we will use nontrivial bounds for the difference a ) l . . 9a,~-I h.-1 -- (~- in archimedean and nonarchimedean metrics, that is, bounds for the ordinary absolute value and the p-adie norm of the difference 9 But first let us see what can be achieved with simpler considerations; for example by Liouville's inequality. Suppose the following: the heights of c~l,..., (~,-1 do not exceed B; the height of am does not exceed A; G is an algebraic number field including al,C~2...,an, [G : Q] = g, H = max[hi I (i = 1 , . . . , n 1); bl > 0 are the minimal natural numbers for which bio~i and bia~-1 are algebraic integers (i = 1, 2 , . . . , n - 1); and a~ is the minimal natural number such t h a t a,~a,~ is an algebraic integer. Let us assume that f~ = alhl .. . a ~hn_l _ 1 -am#0. Then

7 = b~hxl

hlh~-lla,~ 7~ 0 9 .

. vn__

1

is an algebraic integer, and 1 < tNm('y)[ < 17[[5-1g-1 < _< IflIV ('~-I)HA

A

~

+ B(n-1)g~-~


2 - g A - g B - n g H .

(1.2)

For our purposes we need lower bounds for ]/31 and ]/3[, in terms of e -6H for some 6 in the range 0 < 5 < 1. Evidently, (1.1) and (1.2) do not imply such bounds, and complicated manipulations will be needed to obtain them. To date there have been two main approaches to securing 'non-trivial' bounds (that is, stronger than (1.1) and (1.2)). In one approach (which we call direct), the derivation assumes that/3 # 0; the other assumes that for some prime ideal q, ]/3]q is 'not small'. Both these approaches have been applied to the analysis of diophantine equations, and in this monograph we make use of both. In view of the copious computations which the bounds for [/3] and ]/51p require, we will give the details of the second approach only, that is, of the connection between these bounds and a hypothetical bound for ]/3]q. Surveys of results for direct bounds are given in Baker's [25] and van der Poorten's [158] works. Suppose that our initial assumptions on a l , a 2 , . . . , an are fulfilled, and that A z 80g, B Z 80g, and 6 is an arbitrary number satisfying 0 < 6 < 1. Lemma

1.1 Put cr = 21~

l n B ) 2,

(1.3)

let q be a prime number such that x/2q g < a, and let 1 Iq be the q-adic valuation of the field G induced by the prime ideal q. Let S = N ( q 2e)

1

N

'

e=ordqq,

la~]q = 1

(i = 1 , 2 , . . . , n -

1).

Then the inequality h

@--6H

hn-1 . . . O~n_

(1.4)

1

implies

(1.5)

H < 4(n+l) inA provided that either [an[q ~ 1, or lan[q = 1 and

]hx log( f) + . . . +

- log(

)lq > q--

Ho.-4n-2

(1.6)

32

III. Linear Forms in the Logarithms of Algebraic Numbers

where log( ) denotes the q-adic logarithm defined in the metric [

[q.

The condition (1.6) is rather stronger than the analogous trivial estimate of the type (1.2) obtained from the assumption /3 r 0, and it carries more information than that assumption. As a result, the derivation of inequalities of the type (1.5) from the assumption (1.4) is easier given (1.6) rather than just /3 • 0. Application of (1.6) is easy in the context of Thue and ThueMahler equations, and in this way we get a 'quick' proof of an improvement of Liouville's inequality on approximation of algebraic numbers by rationals. Clearly one can consider in place of (1.4) an analogous inequality for the linear form hi lnal + ... + hn-1 in c~n_l - l n a , (1.7) in logarithms of the algebraic numbers a l , . . . , c~,~ (taking any fixed branch of the logarithm). Thus the subject of Lemma 1.1 is really a relationship between bounds for linear forms in logarithms of these algebraic numbers in archimedean and non-archimedean metrics. A similar relationship holds between bounds in different non-archimedean metrics: L e m m a 1.2 Suppose that the conditions of Lernrna 1.1 are satisfied, and that p is a prime number, with x/2qgp 2'~+a 0 are integers. L e m m a 2.3 For any integers 7, m, s > O, there exist integers a (m) satisfying 0_ 2 . Then

Let 7, m, a, b be integers, with ( m7 )

Proof.

If 0 _< 7 < m, then

(:)

]7J

< a, 1 m, then

(2.6)

(:)

,,

m!(7 - m)! is

the number of combinations of "/elements taken m at a time, and we have

O [ exp { - c 6 ( a -2~-2 l n a . H + a - 2 n - 4 H / ? ) } , where c6 = (4g + 1)n in B + 9c3. Therefore, 1 ~ 5 - . . . . . . . . ~( g , . . . , e) > ~ ]P] exp { - c6(a -2'~-2 In a . g + a - 2 ' ~ - 4 H g ) }.

(4.13) To o b t a i n the final b o u n d it is necessary to estimate [P] from below. We note t h a t for any c o m p l e x z the inequality [e ~ - 1[ _< ]z[e M holds, as a consequence of the power series expansion for e ~. Therefore, for i = 1, 2 , . . . , n - 1, ]ai-

1[ = [e Inal - 1[ _< [ l n a i l e [Inal[ < [ l n a i [ e ln(gB)+~r,

I lna~l >

la~-

l l e - ' ( g B ) -1 > lai - lIB -2.

Now b o u n d lai - iI from below: Since ai(ai - 1) E IG,

5. Analytic-arithmetical Extrapolation

la~(~and

1)l

_> [ai(ai

43

- 1)]-9+1 > B-3(g-1),

SO

Ilnai[ > B - 3 ( g - 1 ) a ' i l B -2 k B -39

(i = 1 , 2 , . . . , n -

1).

Hence P > B -39'~'~-2"-2H, and the right-hand side of (4.13) exceeds 1

exp { - ( c 6 + 3gn)(a -2'~-2 in a . H

+ a-2n-4He)}.

(4.14)

As a result we have the following assertion concerning the values (4.9): ( I I I ) For arty integer ~ in the range 0 < g < cr2(n+l), either inequality (4.12) holds, or inequality (4.13) holds with (~.14) replacing the right-hand side. Here m l , . . . , m ~ - i take any values in the range 0 < mi 3), I/3ilq _< 1 (i = 1 , 2 , . . . , n - 1), and where i f t > 0 then M > 4 ( n - 1 ) L + 2 t , while i f t < 0 then M > 4 ( n - 1 ) L + 2 ] t [ L ~ . Then the system of equations (6.4) has only the trivial solution p(A) = p ( A 1 , . . . , A,O = 0 (0 < Ai < Li, i = 1 , 2 , . . . , n ) . Proof. If there is a non-trivial solution, then we may assume max~ Ip(A)lq = 1, in view of the homogeneity of the system (6.4). As before, we shall see that this assumption leads to a contradiction. At the start of the proof of the previous lemma, we established t h a t (6.13) and (6.5) imply (6.6) for co~ = ui - g log 6 , where the ui are arbitrary numbers in ~q with lUiIq _< q-eq (i = 1 , 2 , . . . , n - 1) and g is an arbitrary integer in the range 1 < g < L , + 1. Noting that 7i = Ai +/3iA, (1 < i < n - 1), we can reduee (6.6) to

E

A,~=0

0, and if t < 0 then ]T(An)lq
q89

(6.23)

i=l ~Ttt

and for t < 0 L~+I

IAt(z~)lq -= U qtmin(g,i) > q89

(6.24)

/=1

From (6.21) and (6.23) we have, for t > 0,

[T(An)Iq < q-89 and from (6.22) and (6.24) we have, for t < 0,

IT(An)[q < q--89189 that is, in either case it follows from the lemma's hypothesis that

Equation (6.15) shows that the proof of the lemma is now complete, exactly as for the previous lcmma.

52

III. Linear Forms in the Logarithms of Algebraic Numbers

We shall use Lemmas 6.2 and 6.3 to disprove the existence of non-trivial solutions of the system (3.1) using the premises of Lemma 1.1. Suppose t h a t in Lemmas 6.2 and 6.3 the values L1, . . . , L,~ are the same as in L e m m a 3.1, and that M = [ l a - 2 ~ - 2 H J , 0i = a/s 1 < i < n), and ~i = hi (l (B(gB + 1)) - , ' ' - ' ' - ' / ~

exp {_~(o-~r,-2 ln~. H + ~ - ~ - 4 H e ) } .

N o w p u t S = [ ~ - 2 ~ - 5 : H J , M k = ~ - 2 ~ - ~ H - k S , & = /(~/p)kJ (k = 1, 2 , . . . , 2(n + 1))9 As before, we shall prove by induction t h a t the inequality

(7.5)

~,;~, . . . . . . . . , ( e p , . . . , ep) ~ < p-~H

holds for all integers f, mi in the ranges 0 _< g _< gk, and 0 _< mi _< Mk (i = 1, 2 , . . . , n - 1), if k < 2(n + 1). For k = 1, this follows from assertion (II) above, and we suppose it true for k < 2(n + 1). Fix integers m l , mn-1 < Mk+l, and let f ( z ) = g/* (z, z). 9 9 9 ,

--

rt%l

~...jT7%,~_

I

9 9 . ,

Then

f(~)(z)=

0

+...+0__

~*, ....... ,(z~,..., Z~%-- 1

=

E

"

81!

-

.3n--1' ~I(logoq)rni+SiEP(~)

"/i 7isir27iz -i 9

-

"'

Sl+...+Sn_l=S

zi = z

rn'l

" i=1

,k

i=1

mi

By Lemma 2.3, ri=0

rl,sl

mi

where the a ~,~ (rod are integers. Denoting a (m~) rl~81

"'"

-t- r i

'

~(m._~) ~n--llSn--I

by a r,s (-m-), w e have

f(~) (z) = S!

E st+

~

n--i

-" ~(-~_) 81[...Sn_1 !V Z.~ r,s

+sn_1=s

~=0

" H(logo~i) s'- r'k~m,+r, ........ ,+r,-,(Z, " ' ' ' z). i=i

If s p-ek/@-l). Thus the integral in (7.10) is bounded by p-ek(S+l) (2pgk+l)S+lptk(S+l)(l+,JSr-~)Itl~ < (2p~t:+l)s+lp &(s+l)/(p-O. Now for f(jp) we have from (7.6), (7.9) t h a t

7. Connection Between Bounds in Different Non-archimedean Metrics

[f(jp)[p
p~-2.-5/2(~lp)k,

which implies 2 > fk+t, while we chose e < 2k+1. Hence (7.7) holds, completing the proof of (7.5) tbr all integers 2, m i in the ranges

0 < 2 < (a/p) 2~+2,

0 < m~ < l a - 2 " - 2 H .

--

--

--

2

Following the argument at the close of w we obtain a system of equations . .

x

ml

\m,~-i/

where 2, m l , . . . , m , , - i run over the ranges given above. Since v/2pgp 2'~+3 H -clVlnV'

(8.3)

holds, provided that the left-hand side of (8.3) is non-zero. (Principal values of the logarithms are taken.) Van der Poorten [158] obtained the non-archimedean analogue of this theorem, and also proved the following result: L e m m a 8.3 Suppose that the conditions of Lemma 8.2 hold, and that G = Q ( a l , . . . , a~), [G : Q] = g, and that p is a prime number, with [Ip the p-adic valuation in q3. Then h= _ lip

>

e -c2V(lnH)2,

(8.4)

where c2 = (16(n + 1)g)12('~+Dp -q, prvvided that the left-hand side of (8.4) is non-zero.

In Chapter 5, we use yet another result of Baker in w (see Lemma 5.1), and its p-adie version due to van der Poorten in w167 To prove these theorems by the methods of w167 an auxiliary system of linear equations in p(A) is constructed, more complicated than (3.1), and an auxiliary function more complicated than # ( z l , . . . , z,~-l) of w is used. The proof that the resulting system of equations is inconsistent flows not from p-adie methods, as in w but from the theory of Kummer fields, where the following well-known lemma is pivotal:

60

Ill. Linear Forms in the Logarithms of Algebraic Numbers

Suppose that K is a finite extension of the field of rational numbers and that p is a prime number. T h e n either K ( a l l / P , . . . , a ~ / p ) is of de1 / p ), j n - 1 7 p , where -y e K and . or . a n . O~11 an--1 gree p over Kt'a t 1/p 1 ,. . . , Ol.n--1 O < j l , . . . , j , - 1 < p. For instance, in the fundamental work of Baker and Stark [28], which stimulated extensive applications of this lemma, it was used to prove t h a t the inequality h~_ 1

or

an--1

--

a, l
0 is arbitrary. A system of auxiliary equations of the form 9an

0<m~_<M

)

ml

71

=,_,

"''')'n--1

~-~ 0

(i= 1,2,...,n-i)

is constructed, and its analysis is reduced via the lemma to another problem on bounds for linear forms in logar!thms of algebraic numbers, with aN related to C~l,. .,an-I . . .via .a n .

c~11

3.-i 7 p , w h e r e 0 _< jl,...,Jn-1 < P. a,~-i

Work by Stark [226] and Baker [22] preceded and underlay the proofs of Lemmas 8.2 and 8.3. Due to the bulk and intricacy of these proofs, we refer the reader to the original papers cited above, and to the work of Waldschmidt [243]. Many other bounds are known under stronger restrictions, or highlighting the influence of some parameter (such as the A in Lemmas 1.2 and 1.3).

IV.

The

Thue

Equation

At last, we pass to the analysis of diophantine equations, starting with t h e central problem of the representation of numbers by binary forms. We return, as in Chapter I, to the connection between the magnitude of solutions of Thue's equation and rational approximation of algebraic numbers; but now o u r approach is the opposite of Thue's: we obtain bounds for the approximation as a corollary to bounds for the solutions. We arrive at an effective improvement of Liouville's inequality and its generalisations; and we will see how fundamental parameters of the equation, in particular the height of the form and of the number represented by the form, influence the magnitude of the solutions.

1. E x i s t e n c e of a C o m p u t a b l e Bound for Solutions Let f ( x , y) be an integral binary form of degree n > 3, and A ~ 0 an integer. In this chapter we consider solutions x, y of the equation

f ( x , y ) = A,

(1.1)

which is often called Thue's equation, in honour of Axel Thue, who proved in 1909 that the equation had only finitely many solutions (cf. Ch.Iw The investigation of Thue's equation and its generalisations was central to the development of the theory of diophantine equations in the early 20th century, when it was discovered that many diophantine equations in two unknowns could be reduced to it. Ultimately, this led Mordell to his fundamental theorem on the finite rank of the groups of rational points on an elliptic curve, and led Siegel to his theorem on the finitude of the number of integral points on an algebraic curve of genus greater than zero. Until long after Thue's work, no method was known for the construction of bounds for the number of solutions of (1.1) in terms of the parameters of the equation (such as IAI, or the height or degree of the form). Only in 1968 was such a method introduced by Baker [13], based on his theory of bounds for linear forms in the logarithms of algebraic numbers. Modifications and refinements of this method are still the only way to explicitly bound the number of solutions of (1.1) and to determine the influence of the parameters on that number.

62

IV. The Thue Equation

To appreciate the basic principle of the analysis of (1.1), let us consider one of many ways to reduce the equation to a bound from below for the absolute value of numbers ~ of the form

h~

0.2)

where c~i,. 99 at, c~ are algebraic numbers, a n d hi, 9 99 hr are rational integers. It will suffice to k n o w that ~ # 0 implies that

IPl > Cl e-hH,

H = m a x ( I h i ] , . . . , Ih~l)

(1.3)

with any 5 > 0, where the positive quantity cl is determined effectively by 5, a i , ..., a t , a (cf. Ch.II w Let 0 ( i ) , . . . , 0 (") be the roots of the polynomial f ( x , 1). If integers x , y satisfy (1.1), we have for y # 0 that 1


cloIAI/Iyl '~,

(2.1)

where c10 > 0 depends only on the form f. If one knows that the inequality

y - - 0

c111yP - ~

(lYl > yo).

(2.3)

Thus, it follows from Thue's theorem on rational approximation of algebraic numbers (Ch.Iw that all solutions x, y of (1.1) satisfy ma~(Ixl, lyl) < c121A]2/('~-2)+~ ,

(2.4)

while Roth's theorem gives

max(lxl, lyl) < c131AI 1/('~-2)+~,

(2.5)

where c12,c13 depend only on the form f and the arbitrary number e > 0. However, the dependence cannot be made explicit in terms of the coefficients of f, that is, it is ineffective. Our first goal is the proof of an effective bound for the solutions of (1.1): T h e o r e m 2.1 Let f = f ( x , y) be an integral irreducible binary form of degree n > 3, and let A > 0 be an integer. All solutions of (1.1) in integers x,y satisfy X = max(]x], [y]) < Cl4([A]Hf) c1~

(2.6)

where H f is the height of the form f, and c14 and c15 are functions of n and of the regulator of the field K containing the zeros of f ( x , 1).

2. Dependence on the Number Represented by the Form

65

The dependence on IAI in (2.6) is similar to t h a t in inequalities (2.4) and (2.5), though the exponent cls is less explicit; and it depends on the regulator of K as well as on n. However, (2.6) contains information lacking in (2.4) and (2.5) on the dependence on the height H f when the degree n and the field K are fixed. It follows in particular from (2.6) that one has IAI > ei6X 1/c1~, which with (2.1) gives

l yz - 8(1) > c17]y1_,~+(1/c,5)

(ly{ # 0).

This inequality is a general effective improvement, long unattainable, of Liouville's theorem. It was obtained by successive improvements of Baker's first result

where ClS depends on n, f and e only, with e > 0 arbitrary. We will see two main routes to Theorem 2.1. The first is based on direct bounds for linear forms in logarithms of algebraic numbers; the basic idea was given in the previous section. This was the path followed by Baker [13], [14], Stark [227], and Feldman [69]. The second route is via the connection between bounds in different metrics for the form, and was pursued by the author [203], [206], [209]. Neither approach, however, yields inequality (2.6) with an absolute constant c15, nor even with its value depending on n alone. None of the recent progress in bounds for linear forms in logarithms of algebraic numbers is germane here, and the need for radically new ideas in the main arguments is evident. The first method is applied in two different situations in w167 5 below, but first we examine the second method. We rely on the lemmas of Ch.II and on Lemma 1.1 of Chapter III. In addition, we need some simple lemmas introduced as required below. First of all, we introduce the notion of an "exceptional" number. D e f i n i t i o n . An algebraic number 0 of degree _> 4 is said to be exceptional if there exists an enumeration 0 (1), 8(2),..., 0 (~) of its conjugates such t h a t for all i,j(i # j;3 < i,j I~;jlq -> INm(~;j)1-1

> (16n2h4) - S n ' .

2. Dependence on the Number Represented by the Form

67

We now proceed directly to t h e p r o o f of T h e o r e m 2.1. Set f -- f(x,y) K = Q(8). Let G = f , and d e n o t e by R Setting # = x -

= a N m ( x - By), where a > 0 is an integer, and write Q(8 (1), 0(2),... 8 (~)) d e n o t e the splitting field of t h e form = R~ the regulator of the field K. By, we have t h e identity

(8 (~) - 8U))# (0

-

(8 (o - 8U))# (~) - (8 (~) - 8(0)~ u)

=

(2.8)

0

for any distinct i, j, l (i _< i, j, l < n). We suppose t h a t x and y are integers satisfying (1.1) and t h e e n u m e r a t i o n of the conjugates of the field K is such that

I#(1) 1 =

max(~)I#(~)1,

I#(=)1 = 1~i)~ I#(')1

(i

(2.9)

= 1, 2,..., n).

Take out from (2.8) a s u b s y s t e m (8 (1) - 8(2))# (i) -

(8 (i) -

8(2))# (1) - (8 (1) - 8(i))# (2) : 0,

(2.10)

i = 3,4,...,n. Let U be a g r o u p of units of the field K constructed by L e m m a 2.1 of Ch.II. T h e n by L e m m a 2.2, Ch.II t h e r e exists a unit 77 e U such t h a t for u = # 7 -1 we have I~,(')1 = [A/all/'~e ~

(i = 1,2,...,r~),

181
I--~T~[ >_

[]s

(8(1) _ 8(2))p(i)

8(i) _ 8(2)

I/A(1)[ --

(8(1)

8(I)

8(i))#(1)

=

8(i)

18 (1) _ 8(=)1 i,.,(~)1

,,(1)(8(~)

[8(1)

v(i)(O(1)

8(~)l lu(1)[

_ 8(=))

7/(~) _ __ 7(1)

8(2))

-

But in view of (2.11) and the estimates [8 (1) - 8(i)[
a - n + 1

E

[8(i) - 8(J)[-1 >

(2nHf)-("+2~("-~)+2'

1 (

~

,2nHf,-n2e -2e'eR

u(1)(0(~) _ 8(2)) v(/)(8(1 ) 8(2))

~(i)

?7(1)

:

(2.12)

where j = 2, 3 , . . . , n; i -- 3, 4 , . . . , n. To estimate the difference on the right in (2.12) from below, we apply L e m m a 1.1 of Ch.III, noting t h a t 7/-- 77hi . . . 77hr for some integers h i , . . . , h r . T h e n

68

IV. The Thue Equation

=

t,U)

'

Using the inequalities (1.6), (1.7) of Ch.II, we find the estimates for the heights of algebraic numbers

h t,u(i)(0(1 )

0(2))) < (IAIHseR)c2~

(2.13)

(j = 1 , 2 , . . . , r ) .

(2.14)

and, similarly,

h tT]~l) )

<e

c2'R ,

Let q be any prime (for example q = 2), and let I /q be the valuation on the field G induced by a prime ideal of/G over q. Since the ~ (i),/rlj(1) are units, they are relatively prime to q, and in accordance with the statement 1.1 of Ch.III we consider two cases: u(1)(O(~) _ 0(2)) q

-~(--~

0(2))

# 1

(2.15)

or else this norm equals 1. We first suppose that (2.15) holds for some i (3 < i < n). Then, if some 3 (0 < 6 < 1) satisfies the inequality

.{,)(0(1) _

- \U)

\U)


3. Of course t h e field G = Q ( 0 ( 1 ) , . . . , 0 (~)) contains the c o m p l e x c o n j u g a t e 0(~) of each 0 (i) ( t h e y are zeros of t h e s a m e m i n i m a l p o l y n o m i a l over Q). It follows from (2.7) a n d L e m m a 3.1 t h a t ( i , Q E q3. D e n o t e by F t h e Galois g r o u p of the m i n i m a l p o l y n o m i a l of 0; i.e. t h e galois g r o u p of t h e field G. Because F is transitive, it contains an e l e m e n t which we m a y view as a p e r m u t a t i o n r taking 1 into 3. If r ( 2 ) = r2 > 2, t h e n t h e r e is a s y m b o l i, say, m a p p i n g into 2, and a n o t h e r s y m b o l j, say, m a p p i n g into a s y m b o l rj distinct from 1. A p p l y i n g the p e r m u t a t i o n r to (2.7), we obtain 8 (3) - 0 (2) 0 ('~) - 0('J) 1 - (~ (3.2) 0("2) - 8 ( 2 )

0 (3) -

0(~J)

1 - ([

'

where (* denotes t h e i m a g e of ( under the a u t o m o r p h i s m r. Because the lefth a n d side of this relation is of t h e form (3.1) it is a real n u m b e r . Similarly, if T2 = 2 we can find two s y m b o l s i , j _> 3 not m a p p i n g into 1, and if r2 = 1 such s y m b o l s occur for the inverse p e r m u t a t i o n r -1. In all these cases an a p p l i c a t i o n of T, or r -1 as a p p r o p r i a t e , to (2.7) results in a real n u m b e r . However, t h e r i g h t - h a n d side of (3.2) cannot be real, since t h e n u m b e r s (~, ( [ are distinct from 1; and if t h e y do not coincide, t h e n by L e m m a 3.1 t h e

72

IV. The Thue Equation

n u m b e r w = (i - q ) / ( l - (3) must be complex. Moreover, if q

= (f then

= (i. T h e n (2.7) entails 0 (~) = 0 (j), which cannot occur due to the assumed irreducibility of the form. Therefore we see that the relations (2.7) are impossible if n _~ 5. In the case n -- 4, they reduce to just one equality

0( 1 ) - 0 ( 3 )

0( 2 ) - 0 (4)

0(2)

0(1) - 0(4)

-

-

0(3)

1-(4 -

-

-

I

-- (3 '

(3.3)

Obviously not all the 0(0 here can be real; there is at least one pair of complex conjugates among them. It turns out not to be difficult to show that exactly one pair of complex conjugates occurs, and that the right-hand side of (3.3) is a root of unity. Furthermore, one can check that the zeros of the polynomial x 4 - 2x 2 - 1 can be numbered in such a way that (3.3) holds with (4 -- i, (3 = i 3. Hence, exceptional numbers and exceptional forms of the 4-th degree exist. T h e detailed classification of such numbers and forms is given in an article by Avanesov [5]. L e m m a 8.1 of Ch.III may be used for a further reduction of the set of exceptional numbers to show that on the right-hand side of (3.3) one can always put either e ~i/3 or e - ~ / 3 . An application of the Frobenius theorem on primes in the automorphism sections of a Galois group makes it possible to analyse equation (1.1) in the case of exceptional forms of degree 4 as well, but then one must have an estimate for the least prime q for which a certain integral irreducible polynomial is reduced modq into non-linear irreducible factors. Similar arguments may be applied to cubic forms not equivalent to forms for which 0(1) -- 0(2) 0(3) _ 0(2)

-- e ~i/3

or

e -7ri/3

.

T h e latter constitute the set of the forms considered by Gelfond, since they satisfy the relation (4.4) of Ch.I. We will not describe the details of the necessary arguments since those take us far beyond our main topic. However it may t u r n out t h a t these arguments could prove useful in the solution of other problems (cf. [211], [118]). In the next chapter we again deal with exceptional numbers and forms, but t h e n in a more general sense: Let K be an algebraic number field of finite degree over Q, and let 0 be an algebraic number of degree n over K, with 0(1) , . . . , 0 ('~) denoting the zeros of its minimal polynomial over K. We say that 0 is exceptional relative to K if some permutation of 0 0), . . . , 0 ('~) satisfies the equalities (2.7). The arguments we used above to prove the absence of exceptional numbers if n > 5 work in this case as well, provided we replace the field G by the field G* = K ( 0 ( 1 ) , . . . , 0 (~), (), where ( is a primitive root of unity such t h a t K(~) contains all the roots of unity involved in (2.7). One can again construct permutations ~- leading from the equalities (2.7) and obtain a contradiction if n > 5. Thus if [0 : K] > 5, then 0 is not an exceptional number relative to K.

4. Estimates for the Solutions in terms of the Main Parameters

4. E s t i m a t e s Parameters

73

f o r t h e S o l u t i o n s in t e r m s o f t h e M a i n

The arguments described in the last two paragraphs give a proof of Theorem 2.1 for forms of degree n > 5 and non-exceptional forms of degree n = 4. We also obtained estimates for the values c14 and el5 in terms of the regulator of the field K = Q(0) (cf.(2.27)). In this paragraph we apply Lemmas 2.1, 2.2 of Ch.II and L e m m a 8.2, Ch.III to obtain a bound for the solutions of the equations (1.1) which is a little worse t h a n (2.6) in terms of its dependence on [A[ and H I , but which is better than (2.27) in respect of its dependence on R. Moreover, our arguments deal with all irreducible forms of degree n _> 3 (of. [227]). The assertion of Theorem 2.1 for forms of degree 3 and exceptional forms of degree 4 is bettered by the general Theorem 5.1 of the next section. T h e o r e m 4.1 Under the conditions of Theorem 2.1 we have X = m a x ( [ x l , lYl) < exp(c31RlnR*. T ] n T ) , where T = In

[A[ +

(4.1)

l n H / + R*, R* = max(R,e), and cat depends only on n.

Pro@ We return to the arguments emp]oyed in w but bring to t h e m some additional estimates. Let U be a group of units of the field K constructed by Lemma 2.1 Ch.II, and let rl be a unit in U determined by Lemma 2.2, Ch.II and satisfying (2.11). Then because of(1.5), Ch.II we have h(~) < 2nlal max(l, ~ ) n _< 2n]Ale c19R . Since ~] _< Ixt + ~[Yl < 2 n H f X , we find ['~ = [-fi]/[-~ < 2nHfXlalt/'+e cl~R < 2 n H ~ X e cl~R . Hence r

hjlnlvJ>l 0 is arbitrary.

5. N o r m F o r m s w i t h T w o D o m i n a t i n g V a r i a b l e s We now show t h a t the arguments developed above apply to a wide class of diophantine equations in several unknowns which are essential generalisations of the Thue equation, but with sufficiently much of its rough features for our methods to be applicable [216]. Let K be an algebraic number field of degree n _> 3 over Q, and let 0 be an algebraic integer in K of degree n. Further, let A # 0 be an integer and a real number in the interval 0 < 5 < 1. We consider the equation Nm(x+0y+A)=A,

(5.1)

in unknown rational integers x, y with A an integer in K satisfying

['~ ~ X 1-5 where X = max(Ix], lYl).

(5.2)

If A = 0 the equation (5.1) is Thue's equation (the general equation (1.1) is reduced to (5.1) .with A = 0 by multiplying by the n-th power of the leading coefficient of f(x, 1)). But if A r 0 (5.1) has an essentially different structure and the possibility of analyzing it relying, say, on rational approximations to 0 is dubious. Nonetheless we shall show that arguments close to those described above using the theory of linear forms in the logarithms of algebraic numbers lead to the following result: T h e o r e m 5.1 There are effectively computable numbers c42 > 0 and C43 > 0,

depending only on n and on the regulator of K, such that for any 5 in the interval 0 < 5 < c42, (c42 < 1) expressions (5.1) and (5.2) imply that Z < (21AIH) c430/~)1n(1/~) ,

(5.3)

where H denotes the height of O. As a corollary to this theorem one can see that on completing 1, 0 = 01 until one obtains an integral basis 1, 01,. 9 9 0n-1 of the field K, and considering the equation Nm(x0 + 01xl + . . . + 0,~-Ix,~-1) = A (5.4) in which x0, xl, ..., x , - 1 are unknown rational integers subject to the conditions

Ix l < (max(l 2~i~n--I

--

01,1xll)

76

IV. The Thue Equation

all solutions will have bounds of the form (5.3). In particular, if n = 3 and A = 1, the solutions of (5.4) define units of the cubic field, all components of which (with respect to their representation in terms of the integral basis of the field N) have "almost the same" order of magnitude. It also is easy to see t h a t diophantine inequalities Ixo + Otx: + ... + O~-:x,~-:l < X -'~+1+~/(0/~) ln(:/~)),

max(]x0[, [Xl]) "~ X ,

max

2 Xo have no rational integral solutions (xd, x 1,..., x,~-l) 7~ O. Since 6 is not subject to any condition other than 0 < 6 e and 6 ~ In C in In X/(c44 in X) we find from (5.5), (5.6) that the system of inequalities

IX0 -I- 01Xl

+ . . . -}- O n - - 1 3 : n - t l < C X - n + l

max(lz01,1xll) < X, -

-

max

Iz~l < X ( i n X ) -c4~

2 X - ~ + : + c ' ~ , where c46 > 0. Thus, we obtain an improvement of the inhomogeneous 'Liouville inequality'. The proof of Theorem 5.1 relies on the following lemma due to Baker [20]: L e m m a 5.1 Let c~1, a2, . . . , c~,~ be algebraic numbers of degree not greater than d and different from 0 or 1. Suppose that c~,~ has height not greater than A and that u : , . . . ,u,~-I are rational integers. Then for any 5: in the interval 0 < 5: < 1/2, we have iO~ ,

. . . OZn-lU'~-:

_ an[ > ~:~c471nAe-~:U ,

U =

max

l 1~ (1) - ,(2)1 - [,(2)1 _> [8(1) - e(2)llvI - 2[~ - 1 >

> (2nH)-('~-l)n/2+lly ] - 2 X 1-~ - 1, since ]0(1) - 8(2)[ > (2nH)-('~-l)'~/2+1.

In t h e same way we o b t a i n

[8 (2) [[/z(1)[ > (2nil) -(n-1)n/2+l [x[ -- (2X 1-5 + 1 ) n i l . Hence from (5.8) we find 1#(1)]> X(2(2nH)-(n-1)'~/2

_ 3 X -6) > (2nH)-('~-I)'~/2X.

Therefore

(2nH)-('~-l)'~/2X Now

< I#(1)1 < 2 ~ H x .

(5.9)

from (5.7) and (5.2) we obtain

[8( 3 ) -- 8( 2 ) + (8 (1) -- 0(3))~t(2)/~t(1) -- (8( 1) -- 0 ( 2 ) ) # ( 3 ) / # ( 1 ) [

_< 3 ~ ( 2 n H ) [ # ( 1 ) [ -1 < 6 n H X I - ~ ( 2 n H ) ' K ' ~ - I ) / 2 X

_~

-1 =

= 3(2nH)'~('~-1)/2+lX -~. Hence

[0(1) - 0(3)[l#(2)l

+ 3(2nn)'~('~-l)x -a >

F0(a) _ 0(2)ll#0)[

1 - 0(I) - 0(2) 9 #(3)

-

O(a)

_

0(2)

#(1)

(5.10) 9

Next we need an estimate from below for t h e difference 1

0(1) - 0(2) /z(3) 0(3) -- 0(2)'/A(1)

(5.11) "

First of all we note t h a t we m a y assume t h a t the difference is not zero. Indeed, suppose t h a t (0 (3) -- 0 ( 2 ) ) # (1) -- (8 (1) -- 8(2))/./, (3) = 0 . (5.12) M u c h as in the previous w let F be the galois group of the field Q ( 8 ( 1 ) , . . . , 0 (n)) and consider its elements as a group of p e r m u t a t i o n s ~-. Because t h e group F is transitive, it contains a s u b s t i t u t i o n T taking 2 into 1. O n applying ~- to (5.12) we obtain the relation

78

IV. The Thue Equation (0 (k) -- 0 ( 1 ) ) # (0 -- (0 (l) -- 0 ( 1 ) ) # (k) = 0 ,

where k # 1, l ~ 1 and k r I. It follows that

O0))(x + O(k)y)l 0 is a function o f n and the regulator of the field K. Set /~ = flu; thence h(u) < c49[A[. If 77 = r]~1 . . . r/T ~, where u l , . . . , ur are unknown integers, then

0( 1) _ 0(2) 8(3) __ 8(2)

]A(2) 0(1) _ 0(2) #(1-----~~- 0(3) -- 8(2)

t~(3) (' 7]~3) "~ ul f ~3) "~ u" /j(1----~'~ ; ''" ~ ; "

We can bound the logarithm of the height of the number (8(1) _ 0(2)),(a) (0(3) - 0(2))v0) from above by c50 ln(21AIH), and then on applying L e m m a 5.1 we obtain for (5.11) an estimate from below in the form

6~5~ln(21AIH)e-6~U, with U = max [ui[

(i = 1, 2 , . . . , r ) ,

(5.13)

where 51 is any number in the interval 0 < 51 < 1/2. Since ~ < 2 n H X and Nm(p) = A, we have

lu( )l _> (2nHX)-'~+IIA]

(i = 1 , 2 , . . . , n ) ,

and since

c481A11/'~ >_ ~ - ~ >_ ( 2 n H X ) - ' ~ + ~ I A I I ~ ( ~ ) I

-~ ,

we find from (5.8)

I~(i) 1-1 ~>~=

(0(1) _ 0(2))#(~)

0(~) _ 0(2)

(0(:) ~

~:o-:: -

r,,;,)l -,

- IO(1) o(')l Ip(ol)l (-7777-_ ol",

I]

Because of the estimate (1.14), the logarithm of the height of the number (0(0 - 0(2))p~1)/(0 (1) - 0(2))p 0) is bounded by a quantity similar to (1.19), and by (1.11) the logarithm of the heights of the numbers pj(i),/pj(1) is bounded by cT(R+ h in P), whilst the logarithm of the heights of the numbers r~j(0-/rlj(:l) is bounded by csR. Without repeating the arguments described in w Ch.IV (see also w below) we note at once that for non-exceptional forms f we obtain from Lemma 1.1 of Ch.III:

(o e -6U

j=, Lp,-~.'~J .~=1 Lq)J

({

=

1,2,...

, S ;

j = 1,2,

9

9

.>r)

,

provided that for given 6, (0 < 6 < v ~ P " ! ) - : ) , the inequality U > (c96-1(R + hlnp)2s4)4("+~+:)(ln IAI + l n H f + R + h s l n P )

(1.21)

90

V. The Thue-Mahler Equation

holds, where c9 = cg(n); the same goes for c10,..., c18 below. To estimate the exponents uk exceeding (1.19), take the prime ideal ~3 in IG dividing p(2) and note that from the equality (1.15) it follows that

[ ,,~(i) ~(2)\ (1) (~1 --'1 )Po

fl

ttJ1 - u1 )Po

j=1

[pJi)] uj

[ ?.]Ji)] vj

fl

--

/"k

(1.22)

j=l

where ek is the ramification index of ~3k in pk. Comparing this with (1.18) we see that each number in square brackets is a 9~k-adic unit. To obtain a lower bound for the left-hand side of (1.22), we apply Lemma I.I of Ch.III arguing as in obtaining the estimate (1.20), whilst recalling that

T=Tk=N(~

21k k )(1

1

N( k)

)~

p2~!

.

The left-hand side of (1.22) is bounded from below by the value p ~ U if U > (ClOS-ls4(R + hln P)2)4(s+~+l)x • p16(~+s+l)~!+l (In ]A I + In H / + R + hs in P ) .

(1.23)

Having assumed this inequality, which is stronger than (1.21), we find from (1.22) that uk ~ k

[1- (2) ~ 1>

e_ U , ( j = 3 , 4 , . . . , n ) ,

yield for (j = 2 , 3 , . . . , n ) : i=~viln -{i~

<cugU(R+hlnP)+cl2(ln[AJ+lnHf+R+hslnP).

It follows from this system of inequalities that max [vj[ < claSU(R + h l n P ) + c14(ln]A[ + l n H f + R + h s l n P ) s . l ~_j~_r Taking ~ = min((2n!cla (R + h In p ) ) - l , (x/~p,~!)-l), we find that U < 2c14(In[A[ + l n H f + R + h s l n P ) s . This inequality is inconsistent with (1.23), so we have to suppose that U < (c15(R + h in p)3s4)4(s+n+l) x x P2~

(ln ]A I + l n H / + R + hs in P)

while the left-hand side of (1.22), estimated from below by Lemma 8.3 Ch.III, gives

uk _ 3 and S some finite set of distinct prime ideals p of the field Q(0). Then for rational integers x , y with (x, y) = 1 we have the inequality

Ix - Oyl H Ix - Oylr~p > c24X -'~+1+1/c~,

x = max(Ixl, lyl),

(2,1)

pES

where p runs through all the ideals in S and the np denote the local degrees [Qp(0) : Qp]. Moreover c2 is a positive quantity effectively determined in terms of n, the regulator R and the class-number of the field Q(0), and the number and the maximal norm of the ideals p C S; c24 > 0 is determined by those same parameters and the height of O. Proof. Let K = Q(0). We apply the 'product formula' in the following form: foraC K,a#0, H Ic~l;~ = I N m ( a ) l - l " (2.2) P

Here p runs over all (finite) prime ideals of the ring I~. If a is an integer and Nm(a) = uv with rational coprime integers u, v then one has

Hl

l; ~ = fur 1,

IIl

plu

t; ~ = IvJ-1.

(2.3)

r; o0

(2.a)

p]v

Indeed, each of the products

IIl

l; o0,

[Ij

plu

ply

is a natural number since we have ]a]p

=

p-(1/%)ordpa

~

T/,p ~-- e p f p ,

where p and fp are defined by the equality N(p) = pfp and % is the ramification index of p in p. The numbers (2.4) are coprime, the first containing only prime divisors of u, and the second only prime divisors of v. By (2.2) their product is luvl; hence (2.3) holds. Let P l , . . . , P s be those prime numbers which are divided by at least one of the prime ideals in S. First, we shall suppose that 0 is an integer, and we

2. Rational Approximation to Algebraic Numbers in Several Metrics

93

consider Nm(x - By) for rational integers x, y with (x, y) = 1. Separating powers of the prime numbers P l , . . . , P s from Nm(x - By), we obtain N m ( x - By) = A p ~ ' . . . p : ' ,

(A, p l . . . p s )

= 1.

(2.5)

We now apply the above remarks by setting ~ = x - By, u = A , and v = p~l .. 9ps=o. T h e n we have

l-I Ix - eyl; ~ = IA1-1 PlA Since none of the prime ideals from S divides A, it follows t h a t

l-[ Ix- eyl; ~ _< 1-I Ix - eyl; ~ = IA1-1 PlA

pCS

Now applying the product formula (2.2) and using (2.6), we find that

1 = I N m ( x - 8y)l l"I Ix - 8yl'~ ~ 1--[ Ix - 8yl'~ ~ (in X) 1/~(20n!+21). The estimate (1.26) gives slightly sharper inequalities of type (3.2), (3.3) for all forms of degree n >_ 3. Indeed, it shows t h a t in in X < 200s ins + (hi + 1) in P + c29s(ln In P' in Hf),

3. The Greatest Prime Factor of a Binary Form

95

where c29 depends only on n. As s < 7r(P) ..~ P~ in P, it follows t h a t in the denominator of (3.2) one can put 200 + 5 with any 5 > 0, provided t h a t X > X2(n, H f, 5). Accordingly, (3.3) becomes a little stronger. It is not necessary to suppose irreducibility of the form f for the t r u t h of an inequality similar to (3.2), but it is a necessary condition t h a t the form f have at least three distinct roots. We shall prove that an inequality of type (3.2) holds under this assumption ([211], [190]). T h e o r e m 3.1 Let f be an integral binary form with at least three distinct roots. Then for any rational integers x, y with (x, y) = 1, and f ( x , y) ~ 0 with

X = max(lxl, ly[) -> x3(n, H I ) , we have the inequality P [ f ( x , y)] > c30 l n l n X .

(3.4)

Here X 3 ( n , H / ) is effectively expressed in terms of n and H f , and c30 > 0 in terms of n. Proof. By Gauss' lemma, the form f may be expressed as the product of irreducible integral forms f = f l .-. fk; and it is obvious t h a t P [ f ( x , y ) ] -- maxP[f~(x,y)] (~)

(i--1,2,...,k).

So if at least one of the forms f~ is of degree not less than 3, inequality (3.4) follows from (3.2). Thus we must consider the case in which all the forms fi are linear or quadratic. Since the form f has at least three different roots, it is enough to obtain (3.4) in the following special cases: 1) f is the product of three (non-proportional) linear forms; 2) f is the product of one linear and one irreducible quadratic form; 3) f is the product of two different irreducible quadratic forms. In the first case we have ( a l x - bly)(a2x - b2y)(a3x - b3y) = p ~ . . . p ~ ,

and we can assume that all bi # 0 and that all the quotients ai/bi are distinct (i = 1, 2, 3), whilst P l , . . . ,p~ are distinct prime numbers. Since (x,y) = 1, the common factors of the numbers a i x - biy, a j x - bjy, (i # j), divide aibj - ajbi # O, so t h a t a i x - b~y = dip~ ' l . . . p ~ ' ,

(i = 1,2,3),

(3.5)

where the di are determined by the numbers ai, bi, (1 __ i , j 3 with integer coefficients from the field K and irreducible over K. Let a, i l l , . . . ,fi~ and a be nonzero integers in K such t h a t the (flj) are powers of distinct prime ideals of K, (j = 1, 2 , . . . , s). Then the equation f(x,y)

= ~

...f12 ~ ,

(x,y)la,

(4.1)

in which x, y are unknown integers in K, and zl > 0 , . . . , z~ _> 0 are unknown rational integers, is a natural generalisation of (1.2). However, it has a peculiarity arising from the extension of the domain of solutions x, y from rational integers to algebraic integers in K. It is natural to relate the analysis of (4.1) with that of the equation

f(z,y)

= o fl;

Zs

(x,y)la,

(4.2)

where ~ is a new unknown number, which is to be some unit of K. If the field K contains a unit e of infinite order, then for every solution x, V, Zl,. 9 9 zs, ~ the new set of numbers r ey, Z l , . . . , zs, ~e ", also satisfies the equation, so t h a t the number of solutions is infinite. However, it turns out t h a t if one identifies solutions in which the pairs z, V differ only by a factor which is a unit in K, then the number of 'solutions' will be finite, and all of t h e m may be effectively found ([115], [116]). In particular, t h a t means that the exponents Z l , . . . , zs are determined by a, the coefficients of the form f and the parameters of the

4. The Generalised Thue-Mahler Equation

99

field K. Since (4.2) implies (4.1), we obtain a bound for these exponents in (4.1) as well. It therefore reduces to a finite number of the generalised Thue equations (see w Ch.IV). Our first objective is the following theorem. Theorem

4.1 For every solution of (4.2) there exists a unit rl E N such that max([-~'], [-~--~,p~,... ,psz,) < c43(Nm(~a)Hy)

TM

,

(4.3)

where the pj are rational prime numbers divisible by the corresponding prime ideals of the/~j, (j = 1, 2 , . . . , s), and H I is the height of the f o r m f , whilst c43 and c44 are effectively determined in terms of the degree, the regulator and the ideal class number of the decomposition field of the form f , and by s and P = max(j) pj, (1 _< j _< s). Proof. Let f ( x , y ) = ao(x - O~y)... ( x - O~y) be a decomposition of the form f ( x , y) in the field G = N(0~... 0~), and set [G: Q] = g, [G: IN] = d. By considering the decomposition into prime ideals in IG of the left and right-hand sides of (4.2), we obtain - 0 y') =

(i = 1, 2,...,

where x ~ = aox, y~ = y, Oi = aoO~, and the pj are prime ideals contained in the numbers ~ 1 , . . . ,~8; ai is an integral ideal contained in a ~ - l a , and t < sd. Setting Uji = u j i h + r j i , with (0 < rji < h), where h = ha is the class-number of G, we find that x' - Oiy I = 7r,p ,~1, " " "Pt~" ei/

(i = 1, 2 , . . . , n),

(4.4)

where the (pj) = ph and (Trl) = aip~ 1'... p~" are principal ideals, pj, 7ri c IG, and the e~ are units of G. Lemma 2.2 of Ch.lI allows one to take numbers pj, rri so that _< e~45R(N(ph)) 1/z _< eC45RP h

(j = 1, 2 , . . . , t),

~ ec45R(X(aip~ll... ~)~tl))l/g < c45R INm(~-lc~)F1/gpth < < e ~ 4 S n H ] - l [ N m ( ~ ) l l / g P ~ah

(i -- 1 , 2 , . . . , n ) ,

where c45 depends only on g, and R = RG is the regulator of the field G. Therefore we may assume that in (4.4) the integers pj and ~ri are such that 5, or n = 4 but f is not an exceptional form over K (see w Ch.III). We first consider how to estimate some value uki, say, u n .

4. The Generalised Thue-Mahler Equation

101

It is easy to see by considering the common divisor of the numbers X - 0 I Y ,

X - OiY t h a t if h(Ull - 1) _> [c46(lnHf+lnlNm(c~)I+lnINm(a)l)] = U0,

(4.12)

each of the numbers within square brackets in the equality t

(ex - e2)~,

L (el

e-~--~,~'l

r

I [I

1-I[,7~] ~'~-~,,

k=2

l----1

(4.13)

is a Pl-adic unit, and we have the congruence (01 - 0i)#2 _ 0

1

(01 -- 02)]ti

--

(mod u h~ll-U~ ri

(4.14) '

The multiplicative structure of the right-hand side of (4.13), and Lemma 7.1 of Ch.III, allows us to estimate from above the order of the left-hand side of (4.14) with respect to the prime ideal Pl. In view of (4.6) the logarithm of the height of the number (01 - 0~)cr2/(01 - 02)a~ is bounded by

c ~ ( a + in HI + In INm(~)l + ~h in P), and the logarithms of the heights of the numbers pk and ~z are respectively bounded by c48(R § h ln P) and c49R. Moreover, T = Nm(p~)(1 1 ) Nm~pl) satisfies T < p2g. Let q be a prime number in the range P < q < 2P, and denote by I ]q the q-adic valuation of the field G induced by a prime ideal q contained in q. To apply Lemma 7.1 of Ch.III we distinguish two cases. Either (01 - O 2 ) ~ i q (01 0i) Er2 # 1,

(4.15)

f o r some i, (3 < i < n), or this norm is equal to 1 for every i = 3, 4 , . . . , n. If (4.15) holds, then under the condition

H > (c505-184(R + h In p)2)4(sd+9+l) pX6g(sd+g)+189x • (ln I Nm(aa)] + in Hf + R + shin P)

(4.16)

we obtain

hull - U0 < ~ e H ,

e = ordmpl,

(4.17)

where 5 is arbitrary in the range 0 < ~ _< (v~P9) -1. If (4.15) does not hold, then in accordance with Lemma 7.1 Ch.III we must establish the inequality

log ( (Ol - ~ Oi)a2 ) s

-

~(~

-

u~,)log(;~) ~-

k=l

-

~ ( v z 2 - vii)log(~ s) q > q - - ~ - ' ( " + ~ ) - ' H , l=l

(4.18)

102

V. The Thue-Mah]er Equation

where ~ is bounded above by a value of the form S = Nm(q2e')(1

C515-184(R -}- h in p ) 2 ,

1 )' Nm(q)

e'

ordqq.

Assuming that for all i = 3, 4 , . . . , d the opposite to the inequalities (4.18) holds, and arguing as in w Ch.lV, we obtain the relations 01 - Ok 02 - 0.~ 0 2 - 0~01 - 0j

1 - (~ 1 - 4j'

(4.19)

where ~i,(j # 1 are distinct S-th roots of 1, (i # j; i , j = 3 , 4 , . . . , n ) . But from the results of w Ch.IV we know that if [0 : E] _> 5 the equalities (4.19) cannot hold, otherwise the form is exceptional. Hence (4.18) is true at any rate for some i, and again we obtain (4.17). Therefore, under condition (4.16) we have (k=l,2,...,t;

U = maxuk~ < 5 9 H + Uo + 1 (k,0

i=1,2,...,n).

(4.20)

To bound V ----m a x (z,0

Ivz l

(z

= 1, 2 , . . . , r;

i ---- 1, 2 , . . . , n)

(4.21)

we apply L e m m a 1.1 of Ch.III in the same way to obtain archimedean values of the left-hand side of the equality

which follows from (4.11); here r denotes an isomorphism of the field G into the field of complex numbers. The multiplicative structure of the equality (4.13) plays the determining role again. As before, we assume that H satisfies (4.16) and distinguish two cases: For at least one i, (3 < i < n) we have

(0 I- 0~) a[

# 1,

or equality holds for all i = 3, 4 , . . . , n. In the latter case we check whether an inequality of t y p e (4.18) holds when 01,02,0~,pk, 77t are replaced by their images under the isomorphism r , and see that (4.18) holds for at least one i (3 < i < n) if 0 is non-exceptional. For that i we have

(e:- (el" e:-)#;

> e_6.%

and then for all r we find that

I,i/,

l >

(4.22)

4. The Generalised Thue-Mahler Equation

103

Since #1/#2 = (x' - 01y') / (x' - 02y'), where x', y' E K, and 01 and 02 are conjugates over K, the absolute norm of the number #1/#2 is equal to 1, and the product of the left-hand sides of (4.22) over all T is 1. Hence, we find that max Iln IP'~/#;I[ < g6H + e53 i n H I . (-)

(4.23)

Turning to (4.10) and taking into account the estimates (4.5), (4.6), (4.20), we obtain for all T : ~--~(Vjl -- Vj2 ) in

Iv2l

lnH/+ In INm(a)l+

< cs4(R +

j=l

+ s h l n P ) + 2(@H + Uo + 1)(c45R+ h l n P ) . This shows that

max Ivjl - vj2l < 2@H(c45R + h l n P ) + c55(lnINm(aa)l+ (J) +lnHf+s)(e4sR+hlnP).

(4.24)

Obviously, we can find inequalities of type (4.22) for p ~ / p [ with distinct k, l = 1, 2 , . . . , n in the same way. Hence,

V'= maxlvjk-vjkl (j,k,l)

(j=l,2,...,r;

k,l=l,2,...,n;

kr

is bounded above by a value of the form (4.24). To be specific, set V = Ivul, where V is determined by (4.21). Then we see that the previous arguments give an upper bound for I ( n - 1 ) v u - (v12 + . ' - + v l . ) l o f the form (4.24) multiplied by n. Then (4.9) shows that V is bounded above by a value of the form (4.24). Consequently, we have an upper bound of the same type for H. In view of the supposed inequality (4.16) we must have 1 < (c45R + h l n P ) ( 2 @ + (C46~))4(sd+g+l) . If we take = m i n ( ( v ~ P g ) -1 , (49) -1 (c45R + h in p ) - I

(2c46)-1) ,

then the last inequality is impossible and we conclude that with such 6 inequality (4.16) does not hold. Hence, it follows that H _< Ho = (c5784(R -t- h in p)3)4(sd+9+l)p2Og(sd+g+1)+229 • • (in I Nm(cxa) I + in H f + R + sh in P ) . We now obtain from (4.10) [ -~ 5 and for non-exceptional forms of degree n = 4. We have explicit representations for the values c43 and c44 in terms of the regulator and class-number of the field as well as 9, s and P. Similar arguments, relying on direct estimates of the linear forms in logarithms of algebraic numbers give a proof for the theorem also in the remaining cases (n = 3, 4). Turning to equation (4.1), we observe that from the bound (4.3) a similar bound follows for I/3~~.-./32" l, and then by Theorem 6.1 of Ch.IV we obtain T h e o r e m 4.2 All solutions of (4.1) satisfy m a x ( H , [ ~ , p ~ , . . . ,pZ~) < (F57Hf I Nm(a)l)c~,

where c63, c64 are effectively determined by the degree, regulator and ideal class number of the decomposition field of the form f, and by s, P and the maximal

size of Zf~,...,Z2. Inequality (4.26) yields a corollary which supplements the results of the previous section. T h e o r e m 4.3 Let x, y E I~, with (x, y) = 1, and let P be the maximal prime

number with (P, f ( x , y)) 7~ 1. Then P _> ~

1

lnlnN,

N = max(i Nm(x)l , I Nm(y)l) > No,

(4.27)

where No is effectively determined by f and the parameters of the decomposition field of the form f, and 9 is the degree of that field. Indeed, one applies the bound (4.26) in the case c~ = a = 1 and observes that

I Nm(x) l = I Nm(wx) I < [-621[~:QI ; and similarly for Nm(y). A simple computation then gives (4.27). It is easy to see t h a t in the analysis described above of equations (4.1) and (4.2) one can avoid supposing the irreducibility of the form f , if one applies

5. Approximations by Algebraic Numbers of a Fixed Field

105

direct estimates of the linear forms in the logarithms of algebraic numbers. It is enough to assume t h a t the form f has three distinct roots. In particular, application of Lemmas 8.2 and 8.3 of Ch.III gives the inequality of the form (4.27) for all such forms.

5. A p p r o x i m a t i o n s to Algebraic N u m b e r s by Algebraic N u m b e r s o f a F i x e d Field Theorem 4.1 allows one to obtain the deepest and strongest generalisation of the Liouville inequality on approximation to algebraic numbers by rational numbers [116]. Before we proceed with this topic directly, we simplify the assumptions of Theorem 4.1 concerning the greatest common divisor of the numbers x and y and the choice of the number a, satisfying (4.2). Let the numbers i l l , . . . , fl~ be powers of the prime ideals p l , . . . , P~ of the ring Ix: respectively, (fli) = ph,, (i = 1, 2 , . . . , s) with hilh, where h = h~ is t~

t/

the class-number of the field N. We set (x, y) = apl 1 ... p~s, where the ideal a is relatively prime to the prime ideals Pl,. 9 9 P~ and t~ = hiti+ri, 0

_

c69(h(~))

r//

~d+

1 +2

r H

~ .

(5.10)

yES

It is clear that inequality (5.8) may be considered as the most general form of Liouville's inequality on rational approximation to algebraic numbers. We shall now prove that Theorem 5.1 implies a strengthening of the inequalities (5.3) and (5.10). T h e o r e m 5.2 Let K be an algebraic number field of degree [K : Q] = m, and let 0 be an algebraic number of degree [K(0) : K] = d > 4 over K. Denote by S an arbitrary finite set of valuations on L = K(0). Then for all ~ E K we have I I min(1, IlO - ell. ) > CTo(h(e)) -rod+c71 ,

(5.11)

yES

while for all e E I s we have

II iio - ellv > c72(h(e)) -md+r+c71 , yES

(5.12)

108

V. The Thue-Mahler Equation

where r is the number of archimedean valuations in S (complex values are counted twice), and c7o, c71, c72 are positive numbers effectively determined by the parameters of L and S. Proof. Let, as before, t and k' be the leading coefficients of the minimal polynomials of 0 and a respectively, and set k = tk'. Write ka = l, and (Nm~/K(I - Ok)) = n p ~ - . . p ~ ' ,

(n,p~...p~) = 1

where P i , . . . , Ps are those prime ideals of the ring IK which are divisible by prime (finite) ideals of the set S, ph~ = (/3~), (i = 1, 2 , . . . , s). Then i

t

N m L / ~ ( I - Ok)) = c~/~1~.../32"~,

(5.13)

where ~ is some unknown unit in IK, and c~,/~1,... , ? , E I~. Equation (5.13) has the form (4.2), where f ( z , y) = N m ; / ~ ( x - Oy), but without the condition (x,y) I a. Therefore we may apply Theorem 5.1, which gives

h(I/k) = h(n) < c731Nm(c~)l ~ 9

(5.14)

From (5.13) we find that I~l; 1.

pfp~p,

ptp~p~

p

(5.15)

plp~"'p,

Since ordpc~ < h• it follows that Ic~Ip> p - h ~ where P is the maximal rational prime divisible by a prime ideal from the set Pl, 99 9 Ps. Hence,

I-[ I~1;-1 < p s ~ = c74 PIPl""Ps

(5.16)

Moreover Ic~lp = IIc~tll/'~",where np = [IKp: Qp] _< rn. Hence, H Ic~lp < (l-I Ilc~llp)l/re' P

(5.17)

P

By the product formula we have

I gm(c~)l H II~llp = 1, p so that by (5.17) and (5.16), H

[Nmr~/~(l

-- k0) lp
_ 4 over K, let A be a rational integer, and let Pl , 9 9 9 Ps be rational primes. T h e n all solutions x l , . . . , x m of the diophantine equation Nm(Olxl + . . . + Omxm + O) = A p ~ . . . pZ" satisfy the inequality

X =

max

l 0 is arbitrary ([136], [137]).

VI.

Elliptic

and

Hyperelliptic

Equations

The equations considered in this chapter are in essence different both from the Thue equation and from its direct generalisations. It is possible to prove the existence of an effective bound for solutions of these equations by purely arithmetic methods, by reducing them to the Thue equation or to its generalisations over relative t~elds. However the bounds obtained in this way are not quite satisfactory in the general case, and we again turn to exponential equations to obtain better results. We give an analysis of the hyperelliptic equation and of integer and S-integer points on elliptic curves.

1. T h e S i m p l e s t E l l i p t i c E q u a t i o n s In this chapter we mainly consider diophantine equations of the form

f(x) = y2

(i.i)

where f ( x ) is a polynomial of degree n > 3. At first we suppose t h a t the coefficients of f ( x ) , and the unknowns x, y, are rational integers, but subsequently we shall also consider (1.1) over a finite extension of the ring of rational integers. The famous problem on the difference between cubes and squares of natural numbers leads to an equation of the form of (1.1):

z 3 - y~ = k,

k # O.

(1.2)

This problem has been systematically studied since Bachet (1621) and has its own deep history [144]. Nevertheless, many natural questions concerning (1.2) remain unclear. In particular, it is not known for which numbers k the equation is solvable. Mordell [141] was the first to prove that (1.2) has only finitely many solutions; he did so by directly connecting solutions of the equation with the solutions of a finite number of Thue equations. Mordell observes that all the solutions of the diophantine equation X 2 + kY 2 = Z s

are given by the formulae

(X,Z)

= 1

(1.3)

112

VI. Elliptic and Hyperelliptic Equations 1

X = ~a(u,v),

Y = f(u,v),

Z = H(u,v),

(1.4)

where u, v are integer parameters, and f(u, v) is a certain integral binary cubic form of discriminant 4k; G(u, v) and H(u, v) are the cubic and quadratic covariants of this form. Consequently, if we consider solutions x, y of (1.2) with (x, y) = 1, then (1.3) and (1.4) imply f(u, v) = 1 for some binary cubic form of discriminant 4k. However, it is known that there exist only a finite number of classes of integral binary cubic forms with given discriminant, where a class consists of forms obtained one from another by means of unimodular integral transformations of the variables. Since equivalent forms (forms of the same class) represent the same numbers, we may consider the form (1.4) to be chosen from a fixed finite set of representatives of the classes of forms with discriminant 4k. Thus we obtain a finite number of Thue equations f(u, v) = 1. From these we first determine all possible values for u, v and then, by the formulae x = H(u, v) and y = G(u, v)/2, we find all x, y satisfying (1.2) and the condition (x, y) = 1. Similar arguments allow one to remove the condition (x, y) = 1. Let f(u, v) be a binary cubic form f(u,v)

=

u 3 -

3zuv

2 -

(1.5)

2yv a .

It follows from (1.2) that the discriminant of this form is 108k. By the reduction theory of binary cubic forms it is not difficult to show t h a t f(u, v) is equivalent to a form

g (u', v') = f(c~u' +/%', 3"u' § 6v'), of height at most

(1081k1)1/2 (cf.

c~6 -/3",/= + 1,

[14]). On noting that

(1.6)

9(eu - / 3 v , c~v - 3"u) = i f ( u , v)

and comparing the coefficients of u 3 on the two sides of this equality we obtain the Thue equation to determine 6 and 3":

9(6,-3') = J : l .

(1.7)

Recalling t h a t the form f(u, v) does not contain a term in u2v, we now obtain c~ and /3 from (1.6) and the equality c~6 - 137 = +1. Finally, we see from (1.5) and (1.6) that x and y are represented by polynomials in c~,/3, 7 and 6 and the coefficients of the form 9(u', v'). Therefore, for fixed k # 0, evidently max(lz], lY[) is bounded. It is obvious from the foregoing arguments that to obtain the explicit bound for solutions of the equation (1.2) it suffices to have a bound for the solutions of the Thue equation (1.7). Thus Baker [14] obtained the inequality

max([xl, lYl) < exp{(10101k[)l~ improved by Stark [227] to the inequality

,

1. The Simplest Elliptic Equations

max(Ixl, lYl) < exp{(cllkl)l+*},

113

(1.8)

where c > 0 is arbitrary, and cl = cl (c) is effectively determined by ~. It is clear that bounding the solutions of the equation (1.7) in the light of inequality (4.4) of Chapter IV yields

max(l*l, I'TI) < exp((c21kl)(ln(Ikl + 1))6},

(1.9)

where c2 is a computable absolute constant. To derive this estimate, we recall that the field K, is generated by a root of the form f ( u , v), and therefore the discriminant of K divides the discriminant 108k of a root of the polynomial. Hence the IDI appearing in (4.4) of Chapter IV, is bounded above by a quantity of order Ikl, implying (1.9). If the form g(u', v') is reducible, the bound for the solutions of (1.7) is essentially stronger than (1.9). In that way we obtain the following: T h e o r e m 1.1 All solutions of the diophantine equation (1.2) satisfy max(]xl, lYl) < exp{(c3[kl)(ln(lkl + 1))6}, where c3 is a computable absolute constant. A similar idea underlies Mordell's proof [142] of the finiteness of the number of solutions of (1.1) in the case of an integral polynomial f ( x ) without multiple roots. Let f ( x ) = ax 3 + bx 2 --F cx + d, where all the coefficients are rational integers, and a # 0. On multiplying (1.1) by (27a) 2 we obtain 1

S3 - - - - a 2 S - 4 ~

1

"Q3

=

(1.10)

t2

4 ~

where s = 9ax+3b,

t = 27ay,

g2 = 108(b2-3ac),

g3 = 108(9abc-2b3-27a2d),

and both the discriminant of the polynomial on the left of (1.10), and the discriminant of f ( x ) are non-zero: A = g23- 27g32 # 0. Mordell proved that all the solutions of the diophantine equation X

3 -

g 2 X Y 2 - g3Y 3 = Z 2 ,

(X, Y) = 1,

(1.11)

are given by the formulae X = H(u,v),

Y = f(u,v),

(1.12)

where f ( u , v) is a certain integral binary biquadratic form with invariants g2,g3, and H ( u , v) is the Hessian of this form. Since there exist only a finite number of classes of integral biquadratic forms with given invariants, it is sufficient to examine the finite number of forms f ( u , v) in the formulae (1.12), after taking representatives of the classes. Equation (1.10) is obtained from

114

VI. Elliptic and Hyperelliptic Equations

(1.11) on setting Y = 1, which leads by (1.12) to a finite number of Thue equations f ( u , v) = 1. On determining u, v from these equations we find s and t, and then x and y. Thus we obtain the finiteness of the number of solutions of (1.1) with f(x) a cubic polynomial without multiple roots. There exists another direct method to analyse (1.10), relying on the theory of binary biquadratic forms [15]. One associates a solution s, t of the equation with a binary form f(u,

V ) = U 4 --

6SU2V2 fi- 8tuv 3 q- (g2 -- 3S2)V4 ,

with invariants g2, ga. The form is equivalent to a form g(u', v'), one of the finite number of representatives of all the classes of forms with invariants g2, g3. Consequently, similarly to (1.6), (1.7), one obtains a Thue equation g ( ~ , - 7 ) = 1,

(1.13)

from which to determine 6 and 3'. Since the coefficients of f(u, v) are polynomials in a, fl,7, 6 and the coefficients of the form g(u', v'), one can first find a, fl, and t h e n s and t. Baker [15] was the first to obtain a bound for solutions of (1.1) with a cubic polynomial f ( x ) without multiple roots. He proved that (1.1) implies max(]xl, lyl) < exp{(10~H)l~ where H is the height of the polynomial f ( x ) (the maximum of the absolute values of its coefficients). To derive this estimate, Baker used Mordell's arguments above, supplementing them with his own estimate for the value of solutions of the equation (1.13) and with a lemma t h a t any form f ( u , v ) with invariants g2,g3 is equivalent to a form g(u~,v ~) of height at most

10 4 max(Ig21 s, Ig3 ?/3). An application of inequality (4.4) of Chapter IV allows one to obtain the following result. T h e o r e m 1.2 Suppose that in (1.10) g2,g3 are arbitrary integers with A = g~ - 27g~ r 0. Then all solutions of this equation satisfy max(ls[, It]) < exp{cn[lAll/2(]A] 1/1 + lnG)]l+~},

(1.14)

where G = max([g21, Ig3l); e > 0 is arbitrary, and c4 is effectively determined by e. Therefore, all solutions of (1.1) with a cubic polynomial f ( x ) without multiple roots, satisfy the inequality max(lx], [Y[) < exp{cs[laIIDil/2(lallDI1/l + lnH)]l+r

9

(1.15)

Here D is the discriminant of the polynomial f ( x ) and H is its height; c > 0 is arbitrary, and cs effectively determined by c. We leave the detailed proof of the inequalities (1.14) and (1.15) to the reader. We remark only that to prove (1.14) on the basis of the estimate (4.4)

1. The Simplest Elliptic Equations

115

of Chapter IV, one must recall that in the most important case, t h a t of irreducibility of the form g(u ~, v~), a root of this form generates a field coinciding with that generated by a root of the form f ( x , y). Hence the discriminant of this field is at most IA[. To prove (1.15) one utilises (1.14) and the identity 27Da 2 = 4(b 2 - 3ac) 3 - (9abe - 2b3 - 27a2d) 2 . Thus all the solutions of the diophantine equation (1.1) with a cubic.polynomial without multiple roots, satisfy an inequality of the form max(Ix], [Yl) < exp{c6H6+~},

(1.16)

with an arbitrary a > 0, and c6 effectively determined by E. The ideas described above allow one to analyse S-integer solutions of the corresponding equations as well. Indeed, consider the basic equation (1.10). Obviously, if (s, t) is a rational point on the curve (1.10), t h e n if s = p/q, with (p, q) = 1, we find t h a t q is a square of a rational integer: q = r 2. On using the formulae (1.11), (1.12) we come to equations of the form

f ( u , v) = r 2 ,

(1.17)

where a binary form may be regarded as a representative of one of the finite number of form classes with invariants g2 and g3 - - in particular, we may consider its height to be bounded by Baker's lemma. If we consider S-integer points on the curve (1.10), then (1.17) is a Mahler equation of t y p e (1.1) of Chapter V. For in this case the collection of prime divisors of the n u m b e r r must be fixed, and relative primality of u and v follows from (1.12) and the homogeneity of the polynomials H(u, v) and f ( u , v). On employing Theorem 1.2, Chapter V to estimate solutions of (1.17), we obtain the following assertion: Theorem

1.3 Let (s, t) be a rational point on the curve (1.10), with s = p/q

and (p, q) = 1, and with P > 2 and I > 1 denoting the greatest prime divisor and the number of different prime divisors of q respectively. Finally, set h( s) = max(Ip[ , [q[). Then h(s) < exp{(c7p)2~

G = max([g3[ 2, ]g2]3/2) ,

(1.18)

where c7 is a computable absolute constant. In particular, we see from inequality (1.18) that as the height h(s) grows the magnitude of P increases as the iterated logarithm of h(s). Similarly we can use the formulae (1.3) and (1.4), and T h e o r e m 1.2 of Ch.V to bound the solutions of the equation

z3 _ y2 = kp

l.., kp;',

(x, y) = 1,

(1.19)

where Pl, 9.- ,Pl are fixed prime numbers, and x, y, z 1 ~ 0 , . . . Zl ~_ 0 are unknown integers (cf. also [48]). As a consequence we find t h a t the greatest

116

VI. Elliptic and Hyperelliptic Equations

prime divisor of the difference x 3 - y2, (x, y) = 1, increases with growing X = max(Ixl, lYl) as the square root of the iterated logarithm of X. It is interesting to note, t h a t these results are completely consistent with those on the greatest prime divisor of binary forms (cf. w Ch.V). The effective analysis of equation (1.19) turned out to be useful for the explicit determination of elliptic curves with given conductor [48]. We have only sketched these results because the arguments just now described are very specific and admit no generalisations to polynomials f(x) of degree greater t h a n 3 nor to polynomials f(x) of third degree with algebraic coefficients. So in the general case we have to analyse equation (1.1) in an essentially different way. Below we shall describe in detail a method of analysis leading to inequalities of type (1.15), (1.16), (1.18) in the general case.

2. The General Hyperelliptic Equation The first step in analysis of the general equation (1.1) was made by Siegel [192] who proved the finiteness of the number of it's solutions under the assumption that the polynomial f(x) has at least three simple roots. We notice that in the general case, for (1.1) to have just a finite number of solutions it is necessary to suppose that f(x) has at least three different roots of odd degree. After that one has no trouble in turning to the case of three simple roots. Let a l , a2, c~a be three simple roots of the polynomial f(x) and let K denote the algebraic number field Q(c~l,a2,a3). On factoring x - aj as a product of prime ideals in K, (1.1) yields x

_

ai =

/~

i~i2

(i = 1,2,3),

(2.1)

where the lambdai and ~i lie in the field K, and the ,~i belong to a fixed set of numbers depending on the parameters of K. The (i are unknown integers (cf. below, w Eliminating x from the equalities (2.1) we obtain

aj - ai = Ai~ - A j ~

(1 _< i,j 0 is arbitrary and c22 = c22 (n, ~). Setting #1 -- ~A1, #2 = ~2A2, we find from (4.10) that ~v/-~ - ~ v / ~ - = #~53,

~ v / ~ - ~ 2 v ~ = #~e3

(4.13)

4. Main Theorem

123

and (4.11), (4.12) imply max([#i-], [-fi2]) < exp{c23Rl+~(R + l n H ) l + e } , where c23 = c23(n, r

Eliminating r

r

(4.14)

from the relations (4.13), we obtain

-

.lV

.s)v '

By the estimate (4.8), similar bounds for [~] and ~ we find t h a t

.2v

(4.15)

.lye

and the bound (4.14),

max(h(cr), h(~-)) < exp{cs4Rl+~(R + l n H ) 1+~ + c25(ln A + R*)},

(4.16)

where R* = max(RK, RK1, R~2), c24 = c24(rt,r and c25 = css(n). Now taking two of the equations (4.9) we have O:1 - - C~ ~- .,/r

__ . y 1 r

O~2 - - (~ = .),~2 __ " ) ' 1 ( ~ r -[- 7- r

2

,

where 6 = aX//~/71 and ~ = ~-v/~/71. Eliminating r from this system, we see that ~ satisfies (1r _}_ r 1 6 2 .}_ r =- 0 (4.17) where (1 = 4~ST2"yI"Y -- (7 -- ~S3' -- a271)s,

@ = 4~s~s'yl(a - a l ) + 2(7 - f2-r - ~271)(fs(a - a l ) - a + a 2 ) , r

= "r2( Ol - - C~1) - - O: + OLs .

At least one of the numbers 43, (s is non-zero, since otherwise a = 0, which would imply a l = as. Consequently, (4.17) is non-trivial. Using (4.8) to find similar bounds for [7-i], [7-2], and (4.16), we obtain from (4.17) a bound for [~[s in a form similar to the right hand side of (4.16), though with other constants. In view of x - a = 7r 2 and the bound (4.8) we have for Ix I that

[x I < exp{c26Rl+~(R + in H) 1+~ + cs7(ln [A] + R*)},

(4.18)

where C26 = css(n, r and c27 = c 2 7 ( n ) . To complete the proof of the theorem it remains to bound R and R* from above. Since the fields K, K1 and Ks are contained in L, by Lemma 2.3 Ch.II we have R* < cssR, where c28 = css(n). To bound R we employ L e m m a 2.4 Ch.II: R < cs9[DL[1/2(ln [DL[) z-1 . (4.19) Now it remains to b o u n d the discriminant DL. We view L as a field obtained from Q by means of successive adjunction of the integers a, ax, a2, gx/~, gl v/-~, and g2 V ~ " Because of the multiplicative property of relative fields differents, we see t h a t the different of the field L is a divisor of the product of

124

VI. Elliptic and Hyperelliptic Equations

the differents of the successively adjoined numbers. Therefore the discriminant DL, being the norm of the different, divides the absolute norm of the number

A = f~(~)f;1 (~l)ft~2 ((~2)2gx/~2glx/~2g2x/~ 9 But we have that Nm(A) is given by

Dla/mDla/ml Dla/~n2(8gglg2)l(Nm(7) )I/me(Nm(71) )I/role1 (Nm(72) ) I/m2e2 , where m = [K : Q], e = [K(x/~ ) : K] and m l , m2, el, e2 are defined analogously. Because of 9 < [DR[, noting t h a t [Nm(7)] coincides with [Nm(7')l and is bounded by the inequality (4.6), and because of similar inequalities for gl, g2, I Nm(71)[, and ] Nm(72)[ , we find t h a t ] Nm(A)[
0 arbitrary. In view of (6.14) a similar estimate is true for maxlln with v E S, and because of the symmetry of x and y in (6.7), also for max Iln lyl~l with v c S. Thus we obtain the following main assertion on the equation (6.7).

Izlvl

L e r n m a 6.2 Let oq fl, ~/ be non-zero elements of the field L with heights at most T > e. Then all solutions of equation (6.7) in S-units x , y of the group EL(S) satisfy max fin [x]v] + max [ln [Y]v] < yES

yES

< c54(es2(s + l))2~176

h l n P ) 8 ( R + s h l n P ) R l n r ] 1+~ ,

where s is the number of non-archimedean valuations in S, prime with IPi, < 1 for v e S, and l, R and h are the and class number of the field L respectively. The constant determined by l, and the constant c54 by I and e, where e >

(6.19)

P is the greatest degree, regulator c52 is effectively 0 is arbitrary.

130

VI. Elliptic and Hyperelliptic Equations

For s = 0 this lemma yields a formally worse estimate t h a n Lemma 3.2 (it has R 2 instead of R) but, actually in this case one should replace the factors pZ and R + s h l n P by 1, since those terms appear in the argument only if

sr Prom (6.19) it is easy to turn to estimates for the heights of the numbers x, y in any variant of definition of height. Since we shall proceed to work with the ordinary notion of height, we employ inequality (1.5) of Ch.II to estimate the heights of the numbers x, y. For that, it is necessary to have upper bounds for the sizes F~'I, ~ and for the minimal natural numbers a0, b0 (the 'denominators') such that aox, boy are integers. Because S includes all the archimedean valuations, we obtain the bounds for ~ and ~ from (6.19). Further we observe that for non-archimedean v E S and corresponding prime ideals p iXlv = ixlp = p-~ordp x , e = e p = ordpp, where p is the prime divided by p. Since 1 _< ep _< l, then the product over all p for which ordp x < 0 a0 = 1-I Izl~

(6.20)

P

is a natural number and aox is an integer. Analogously, the natural number b0 -- lip lyl~ is such that boy is an integer. Therefore, applying the inequality (6.19) we obtain: L e m m a 6.3 Under the hypotheses of Lemma 6.2 the heights of the numbers x, y satisfy the inequalities in max(h(x), h(y)) < < e55(s + l)(c52(s + l))2~176

+ h l n P ) S ( R + s h l n P ) R l n T ] 1+~ ,

where c55 is effectively determined in terms of I and e, with c > 0 arbitrary. We make use of this lemma to obtain further results on the bounds for solutions of diophantine equations in S-integers. We also notice that it allows one to obtain an effective analysis of the general ternary exponential equation [222], t h a t is of the equation c~a~l... ~ , + ~ m

... ~ym + ~/3,;~... 3~. = 0,

(6.21)

where c~, ~, ~,, and the c~i, ~j, ~/k are non-zero algebraic numbers, and the xi, yj, zk are non-negative integer unknowns. If this equation has a finite number of solutions, an effective upper bound for them is determined in terms of the parameters of the equation. If there are infinitely many solutions, then the bound for the generating elements of the of the solution family is determined. It is the problem of exceptional interest and importance to achieve a similar result for equations of the type (6.21) with a greater number of terms on its left-hand side.

7. Solutions in S-integers

131

7. Solutions in S-integers We continue with the notation of the previous paragraph. Let IL(S) be the set of all numbers A E L with

I Jv < 1

for all v r S .

(7.1)

It is clear t h a t I~(S) is a ring containing I~. Its elements are called the Sintegers of the field L. We shall deal with solutions of equation (4.1) under the assumption t h a t the coefficients of the polynomial f ( x ) are integers of L, that A is also an integer of this field, and that the unknowns x, y lie in I~(S). Again, as in w we suppose t h a t the polynomial f ( x ) is monic of degree n has three simple roots ~, O~1, O~2. Suppose t h a t x and y are in I~(S) and satisfy (4.1). From the condition (7.1) it follows t h a t the factorisations of the principal ideals (x), (y) as products of prime ideals of L, have only powers of those prime ideals P l , . . 9 P~ determining non-archimedean valuations from S in their denominators. Let be an integral ideal of L from the ideal class inverse to t h a t of the denominator of (x) and with norm at most IDLI 1/2. T h e n we have a representation z = X/z,

x e IL,

z e

( x , z) =

(7.2)

where the ideal (z) contains only the prime ideals P l , . . . P8 and divisors of ~. On defining a binary form f ( X , z) by I ( X , z) -- z'~f(X/z), we find from t h a t (4.1) becomes f ( X , z) = Ay2z n . (7.3) Let ]K = L(c~), where c~ is a root of f(x). From (7.3) we see that every prime ideal of the field ]K which is not contained in OAf~(o~) divides X - c~z with even exponent. Hence, the factorisation of X - c~z as a product of prime ideals of ]K is of the shape ( X - c~z) = ap 2 ,

a I ~Af'(c~).

On arguing further as in w we find a representation X - c~z = 3,~2, where 7, ~ E E, and g, V, ~ are integers. We obtain bounds for g and ['7-] of the same kind as in w Setting ]K1 = L(c~l), ]K2 = L(c~2), we obtain similar equalities X - c~iz = Vi~ (i = 1, 2) in these fields, whence (c~j - o~i)z = 7 ~

- ~,j~2

(0 - 3 and ordu(g) > 0 for all the other valuations U of ~ , one may construct a basis of the form

x'hwj

(1 < j _< 1,0 < h < •j)

(8.3)

with certain integers l, A1,..., Al, where x I = (x - b) -1, b is an integer in the interval 0 < b < n 3, and where W l , . . . , wl are elements o f ~ with the following properties: wy has a Puiseux expansion at Qi of the form

wj = (x') - ' / e ' ~

w,ykx 'k/e'

(1 < i < r, 1 < j < l),

(8.4)

k=0

where vi = 3 for i = 1, vi = 0 for i # 1. All the coefficients wiyk belong to the field K = Q(0) with an algebraic integer 0 of degree at most 8 "6 and of size

8. S-integer Points on Elliptic Curves

135

< (2H) t' where tt = 8'~8. There exists a natural number A such that all the numbers A k + l w q k are algebraic integers of K and max( Ak+l, Ak+l [w~q) < (2H) t*(k+l)

(k = 0, 1,...).

(8.5)

In addition, each basis element (8.3) may be represented as a rational function of x, y over K. By the Riemann-Roch Theorem the dimension of the space 9)2 is equal to the degree of the divisor determined by 9)l; that is, it is 3. This shows that 1 = 3 and that one may take Aj = 0, (0 < j < 3). Among the linear combinations of the three functions wl, w2, w3 one can choose two, X1 and X2, say, such that ordQl(X1 ) = -2, ordQl(X2 ) = - 3 , and ordu(X1) > 0, ordv(X2) > 0 for all the other valuations U of the field 9~. These functions will have an expansions of the type (8.4) and, roughly speaking, will satisfy the inequalities max(A k+l, A k+l [ - ~ )

< (2H) "(k+l)

(k = 0, 1,...).

(8.5)

OO

Xj =

(x') -~'/e' x2_-, - ~ u ijk z 'k;~ ' k=0

(j = 1,2),

(8.6)

where uqk C K, Ak+luijk E I~:, and max(A k+l, A k+l [K~]) < (3H) 2~'(k+l)

(k = 0, 1,...).

(8.7)

On these grounds, one establishes by direct computation that X1 satisfies an equation X F q- Pl(x)Xr~ -1 + . . . q-- Pro(x) = 0, (8.8) where the Pj (x) are quadratic polynomials Pj(x) = qjox 2 + qjlz + qj2 ,

A'~Sqjk E IK.

(8.9)

It follows from (8.7) that max(A'~5, A'~5 [ ~ ] ) < (2H)"5

(l<j<m;k=l,2).

We now consider the seven functions

1,xl,x ,x ,x ,x ,xlx2. The inequalities (8.6), (8.7) imply that at Q1 these functions have expansions of the type OO (x,)-6/~ v-, ,~,k/z, 2__. vjk(j = 1, 2 , . . . , 7) k=0 respectively, where A6(k+X)vjk E IK, and

max(z~6(k+l), z~6(k+l) [ ~ )

~ (4H) 12"(k+1) .

As they have no other poles, except for the one at Q1 of order at most 6, they are linearly dependent by the Riemann-Roch Theorem. The coefficients

136

VI. Elliptic and Hyperelliptic Equations

z j ( j = 1, 2 , . . . , 7) of their linear dependence may be chosen in I~; to satisfy

the inequalities < {7(4H)S4t'} 6 < (2H) "2 .

(8.10)

Thus X l -]- x 2 X l

-'}- x 3 2 2

+ x 4 x 2 -1- 3;5 X 2 -1- x6231 -}- x T X l X 2

:

0,

(8.11)

where it is not difficult to observe that x5 r 0. Now on assuming that X l = X1 ,

yI = 2x522

a' = --4XsX6 , e t = 2 x 3 x 7 -- 4 x 2 x 5 ,

-~ x T X 1 -]- x3 ,

b' = x~ - 4x4x5 , d ~ : x 2 -- 4 X l Z 5 ,

we obtain the equation (8.12)

y a = a~Xt3 + b~X~2 + c~X ~ + d ~"

We can convince ourselves that the polynomial on the right of (8.12) has no multiple roots. Indeed, if the equation is reduced to (z'/(x'

-

: a'(x'

-/9)

with certain c~,/9, then, since X ' - / 9 has only a pole at Q1, Y ' / ( X ' - o~) also has a pole only at Q1- But x5 # 0 and ordQ1 (X1) = -2, ord c61 lnmax(h(x), h(v)) ~176 . Here r is the rank of the group of rational points on the curve and c61 depends only on the coefficients and degree of the equation determining the curve (the value of c61 is ineffective). This deep inequality is the first and as yet unique case in which an essentially stronger estimate than an iterated logarithm of the height of the argument has been obtained in problems on the arithmetic nature of numerical values of polynomials and rational points on algebraic curves.

VII. Equations of Hyperelliptic Type

We apply the methods and considerations described in the previous chapter to more general situations so as to obtain new facts generalising our former results. Then we proceed to a new type of equations in which at least one of the unknowns is a power of an unknown integer. Our aim is to bound the unknown exponent so as to reduce these new equations to those of the kind considered before. In this way we determine, for example, that any integral polynomial having at/east two simple roots represents only a finite number of powers of integers with exponents greater than 2. We also give an analysis of S-integer solutions of the Catalan equation.

1. E q u a t i o n s

with

Fixed

Exponent

The subject of this chapter is the equation of the form f ( x ) = A y '~,

Ar

m>3,

(1.1)

in which f ( x ) is a polynomial with at least two simple roots. We shall suppose that the coefficients of the polynomial and the unknowns x, y are rational integers, but this supposition is not essential, and just as in the discussion in the previous chapter we shall be able to consider (1.1) and its solutions in finite extensions of the field of rational numbers. In this paragraph we assume that the exponent m is fixed. This makes the equation close to (4.1) of Ch. VI; to analyse it we apply the arguments similar to those given in w Ch. VI. Those methods for analysing (1.1) were based on reducing it to a generalised Thue equation (over a finite extension of the field of rational numbers). By this route Baker [16] found the first effective bound for its solutions. He proved that all solutions satisfy the inequality max(Ixl, lY]) < exp exp{(5m)l~176

'~2},

where n and H are respectively the degree and the height of the polynomial f ( x ) . The next theorem gives an essential improvement to this bound and is similar to the Theorem 4.1 Ch.VI (cf. [219]). T h e o r e m 1.1 Suppose m >_ 3 is a natural number and that f ( x ) is an integral polynomial of degree n >_ 3, with height at most H > e, with leading coefficient

1. Equations with Fixed Exponent

1, and having two simple roots a, al. Let fc`, fc`l be the minimal and Dc`, D~ 1 the discriminants of a and al respectively. Denote the resultants of the polynomials f l and fc` and of f f and fc`l Let K, K1, and 1F denote the fields of algebraic numbers Q(a), Q(a, a l ) respectively, and set d=[IF:K],

dl=[IF:K1],

s=[K:Q],

139

polynomials by Rc`, R~I respectively. Q ( a l ) , and

sl=[Kl:Q].

Then all integer solutions of the equation (1.1) satisfy max(Ix], lYl) < exp{cllAl"~(m-1)~(m)(d+d~)+r162

(1.2)

wh ere B = [Dc`Dc`~sslrn3(rn--1)q~

]?d Rdl m(rn-1)2~o(rn) DdD dl [rn(m--1)w(m) --c`--c`l c`

ot 1

Here cl is effectively expressible in terms of n, m and e; ~(m) is the Euler function, and e > 0 is arbitrary. Proof. We assume that A does not contain prime divisors with exponents greater than m - 1, as we may without loss of generality by separating any ruth power from A and joining it to ym, noting that this does not weaken inequality (1.2). Suppose the rational integers x, y satisfy (1.1). In the field K the ideal (x - a) may be represented in the form (x - c~) = ab "~ where a and b axe integral ideals, and a is free of ruth powers. Since f ( x ) ----( x - a ) ( f t ( a ) + ~1 f II ( a ) ( x - a ) + . . . ) , then if p is a prime ideal of the field IK and (if(a), p) = 1, we have ordpf(x) = ordp (x - a). Therefore ordoA=ordpa

(modm)

and, consequently, ordp a < ord, A. Ifp ] ff(a),then at any rate ordp a < m - 1 . Hence

a I A(f'(a)) "~-1

(1.3)

and since f ' ( a ) divides Rc`, then a I ARC` ,~-I . For the conjugate ideals a (i) we shall also have a (~) I A R ~ -1" Consequently the least common multiple [a(1), . . . , a (s)] divides ARC`'~-I. Further, (a (0, a (i)) divides (x - a(i),x - a(d)), which in its turn divides a(i) _ a(d), and then

II (a('),

l ~]. Without changing notation, we also assume that [~I = ~], since we may pass in (2.9) to the conjugate equalities. Now from (2.9) we obtain:

= 1 - fll07?~-" fl0

-'~ ~)11

"'" 7]k

. . . -7rs ~,

711

~,~ 9. 7l11

1"lklT~l~d'l

"''7]1kl

~ , ~fl- ~ , -fll- m

"'" .

(2.10)

We apply Lemma 8.2 Ch.III to estimate the right-hand side of this equality from below. Recalling the inequalities (2.4), (2.6), (2.8) and similar inequalities for the ul~, ill0, and the vl~ we find that the right-hand side of (2.10) is not less t h a n exp {-c24 In ~] inm[in(IAl + 1)]1+~} , (2.11) where c24 is determined in terms of ~, the coefficients and the degree of the polynomial f(x); here c > 0 is arbitrary. Since the left-hand side of (2.10) is bounded from above by the value

IAl~ cp6~-'~,

(2.12)

2. Equations with indefinite exponent

147

where c25 and c26 are determined in terms of the coefficients and the degree of f(x), we obtain from the comparison of (2.11) and (2.12) t h a t

[~,~
eC25c26.

(2.15)

Suppose first t h a t Then we find from (2.14) t h a t m

ln'---m < 2c24[ln([A[ + 1)]1+% If (2.15) does not hold, then we consider (2.9) as a relation of linear dependence of three S-units in the field L, containing the fields K and K1, and for the set of those prime ideals of the ring IL occurring in r~l,..., ~rs,/3 and the (j. We take the numbers 30,/310 and a l - a as the coefficients of this linear relation. Lemma 6.3 Ch.VI shows that the logarithm of the height of the number P = ~]'~ - - . r/k ~ 1

.-. 7r

-ml

is bounded from above by the value c2r [ln(IAl+ 1)] 1+~, where C2r is determined by f(x) and s, with r > 0 arbitrary. If we have In h(p) > m~ i n t o , t h e n we obtain (2.1). If the opposite inequality holds, then we find from the equality x - a =/3oP that

Ixl < c2s(lAlem/lnm) c=", and the condition lYl > 1 shows now that m _< ~0(ln ]AI + 1). By this the proof of Theorem 2.1 is completed. On studying the proof of Theorem 1.1 it is easy to ascertain t h a t the magnitude of Cl as a function of m does not grow more rapidly t h a n m 4'~a. Then (1.2) shows t h a t for large [x[ we have 881

5

lnln Ixl < - ~ - m in ID,~D,~ I + c31m4( 1 + In ]AI) , where c31 depends only on f(x). By Theorem 2.1 we can estimate m in terms of A, which gives lnln Ixl < ca~[ln(lAI + 1)15+e. (2.16) This inequality leads to an interesting arithmetic statement. Let m , N be natural numbers. Denote by Am[N] the m-free part of N, that is, represent N in the form N = N1N~, where all the prime divisors of N1 occur with exponents less t h a n rn, and set Am[N] = N1. Then it follows from Theorem 1.1 t h a t for m _> 3 and an integer x with x ~: O, f(x) ~ 0 we have A,~[f(x)] > c33(ln I3gl)1/m(m-1)~~

148

VII. Equations of Hyperelliptic Type

where c33 is effectively determined in terms of f(x) and m and e > 0. We see that with any fixed m the magnitude of Am[f (x)] grows unboundedly with the growth of Ix], but the estimate of this growth is the worse the bigger m is. Meanwhile, the inequality (2.16) shows t h a t the magnitude

Amin[f(x)] = minAm[f(x)] m_>3

(f(x) 7~ O)

grows unboundedly with Ixl:

Amin[f(x)] > exp{c34(lnln I x l ) 1 / 5 - ~ } , where c34 is determined by f(x) and e. Thus, independently of m we have a guaranteed estimate for the speed of growth of Am[f (x)] (cf. also [236]).

3.

The

Catalan

Equation

In 1844 Catalan [44] conjectured that the only solution of the equation z ~ - y~ = 1;

(3.1)

where all the unknowns x, y, u, v are integers not less t h a n 1, is given by 32 2a = 1. In 1953 Cassels [38], independently, put forward a weaker conjecture that the equation (3.1) has only a finite number of solutions. Over many years the equation (3.1) attracted the attention of mathematicians who considered it for small values of u or v and either with fixed x, y and unknown u, v or with fixed u, v and unknown x, y (see [145] for a review). Recently Tijdeman [232] obtained the principal solution to this problem by proving the existence of a computable bound c35 to the solutions of the equation (3.1): max(x, y, u, v) < c3s. In particular, the t r u t h of Cassels' conjecture follows. Following Van der Poorten [157] we discuss here the solution of (3.1) in S-integers, which is the same as to analyse the following problem: Let S be a fixed set of prime numbers pl,. 9 9 ps. We consider the equation x~ _ yV = (p~,,... p~s)[~,~],

(x, y) = 1,

(3.2)

in which x > 1, y > 1, u > 1, wl > 0 , . . . , w 8 > 0 are unknown integers (excluding u = v -~ 2), and [u, v] is the least common multiple of the numbers u and v. It is obvious that (3.2) implies the Catalan equation (3.1). T h e o r e m 3.1 All the solutions to the equation (3.2) satisfy max(x, y, u, v, w l , . 9 9 ws) < ca6

where c36 is effectively determined by S.

(3.3)

3. The Catalan Equation

149

P r o o f . We shall use L e m m a s 8.2 and 8.3 C h . I I I with the intention of first bounding the exponents u, v in (3.2). We start with the case of u = v where the desired result follows quickly. It is obvious, t h a t if (3.2) has a solution, t h e n not all of the numbers w j a r e equal to zero, and we can assume t h a t all of t h e m are not zero without change in notation. Let p be a prime from the set S. By L e m m a 8.3 C h . I I I we find that ix"-Y'~Ip = ]x '~Y -~' - lip > e -c~'(p)ln~(ln~)2

Turning to (3.2) we find

pES

where cas is the s u m of all the values c37(p) over the p E S. It is apparent, that

x"-y~_>x u - ( x - l ) ~> ( x - l ) ~-1, so we find u < c39(lnu) 2, from which it follows t h a t u < c40 where the m a g n i t u d e of c40 m a y be determined in explicit form in t e r m s of the p r i m e numbers of S. Now we come to the equation of Mahler (see w Ch.V): x

-

=

...

p:',

(x, y)

=

1

from which the bound for x and y is determined. This completes the discussion of the case u = v in (3.2). Yet one more simple case corresponds to v = 2. Here it is enough to consider the equation x ~=y2+z

2,

z=p~...p:s,

(x,y)=l.

In the Gaussian field Q(i) we find t h a t the numbers y + i z and y - i z have 9m a x i m a l c o m m o n divisor dividing 2. Hence, z = c~cr~ + f i t ~,

(a, T) = I

(3.4)

for fixed nonzero numbers a, fi and unknowns a , T, lal > 1, ITI > 1 (all the numbers lie in the Gaussian field). We analyse this equation relying on the ideas which were applied above. In view of the s y m m e t r y of (3.4) we m a y assume t h a t [a] > 171. Applying L e m m a 8.2 Ch.III, we obtain i

ll

l

-

-

>

e -c41 lnlcqlnu,

(3.5)

if one takes into account t h a t the height of an integer in the Gaussian field coincides with its norm. In the case z = 1, from (3.5) it follows t h a t u < c42, which is just w h a t we need. If z > 1 then for a prime divisor p I z which does not occur in r we obtain by L e m m a 8.3 C h . I I I t h a t

P

150

VII. Equations of HypereUiptic Type

If p I z and p I T t h e n p ] ~. Consequently, applying the ' p r o d u c t formula' we obtain Z2 =

Nm(z) < H

t a a u +/~TU]; 2 '~ e c 4 2 1 n l a [ ( l n u ) 2 H

plz

1C~];2"

P

Since the inequality (3.5) yields z > I~llal ~-c,1 ~n~, we again obtain u < c44. Thus, we come to (3.4) with a bounded exponent u i.e. to the generalised equation of Mahler which was discussed in w Ch.V. We m a y suppose from now on t h a t u is odd. Now we proceed with the main case, t h a t is u # v, v >_ 3. We argue as before, b u t now a direct application of L e m m a s 8.2 and 8.3 C h . I I I is not sufficient and first of all we derive a few auxiliary facts on the a r i t h m e t i c structure of the solutions of the equation (3.2). Note t h a t one m a y suppose t h a t the numbers u, v are prime. T h e n we have [u, v] = uv. We set z p~Ol " ' " P sw. " First of all we verify t h a t (Y+Z~'

yv q_ z u" V+Z~ ) =1 orv.

(3.6)

If p is a p r i m e n u m b e r and p~ divides b o t h y + z ~ and (y" + z ~ V ) / ( y + z~), then y-=-z ~

(modp~)

0 =- y,,-1 _ y , - 2 z ~ , + . . . =_ ( _ l ) , v z ~ ( ~ - l )

(rood p~)

Since ( x , y ) = 1 t h e n ( z , p ) = 1, and then p~ divides v, hence a = 1, p = v, which proves (3.6). Now from the equality x~ = (y + z~)Y ~+z~"

y + z '~

we conclude t h a t y + z ~" is, up to a power of v, a u t h power of a natural number. To determine the exponent to which v m a y occur in y + z ", we consider the congruence y~ + z ~

= ( ( y + z ~) - z ~ ) ~ + z ~ -

-

v ( y + z ~ ) z ~'(~-1) - 89

-

- 1)(y + z " ) 2 z ~(~-2)

(mod (y + z~)3).

It follows t h a t if v I (Y + z"), t h e n Y" + z ' ~ - v z ~'('-1) y + z ~'

(mod v2),

i.e. v m a y occur in ( y ' + z ~ " ) / ( y + z ~') with the exponent equal to 1. Therefore v ~-1 divides y + z ~ and we find t h a t y can be represented in the form

3. The Catalan Equation

151 (3.7)

y = v o Y ~' - z ~'

where v0 --- 1 or v -1 and Y is a natural number. Similarly one verifies t h a t (3.8)

x = u o X '~ + z ~',

where u0 or u -1 and X is a natural number. Now the equality (3.2) takes the form ( u o X v + z~') ~' -

(Vo y ~ ' - z~') ~' = z ~''~,

(3.9)

and and it is to it t h a t we apply the analytic ideas indicated above in special cases. From now on we assume t h a t v < u though there is no full s y m m e t r y with respect to u and v in the equality (3.9); nevertheless we shall be able to meet the case u < v with similar arguments. Let p be a prime occurring in z and set w = ordp z. Note that it follows from (3.7),(3.8) that if p coincides with u or v, then the corresponding number u0 or v0 is 1. From the equalities (3.2), (3.7), (3.8) we obtain correspondingly

Ix 'y

-

1Fp = p . . . .

,

(3.10)

(3.11)

[YvoaY -'~ - lip = p-~'~', IxuolX

- v - lip = p - ~ .

(3.12)

Substituting (3.11) and (3.12) into (3.10) we obtain lU~VoV(X/Y)

'~' - 11 ~ m a x ( p - ' ~ ' , p - ~ ' ~ ' , p

....

) =p-'~'

(3.13)

We set A = u ~ v ~ ( X / Y ) '~v - 1 and observe that A r 0. Indeed, otherwise we have ( x - z~') '~ = ( y + z~') ~' s o t h a t ( x - z~') " > y'~ + z ~ ' , which is inconsistent with the main equality x ~ = y" + z ~'. Suppose first t h a t X < Y. Applying Lemma 8.3 Ch.III, we obtain (3.14)

[A[v > e-c4s(p)(ln u) 4 In Y,

where the value e45(p) depends only on p E S. Comparing (3.13) and (3.14), we obtain

pWV < ecas(p)(lnu)4 InY

and taking the product of all such inequalities over all p dividing z, we find

zV < Y c46(In~)4

(3.15)

From this inequality we shall obtain that v < c4~(lnu) 4,

(3.16)

but first we suppose that v > 3c46(lnu) 4. Then it follows from (3.15) that z c4s t h a t y = roY"

- z ~' > y 3 ~ , / 4 _ y , , / 3

> y2,~/3.

(3.17)

152

VII. Equations of Hyperelliptic Type

Now from the equation (3.9) we see that ( u o X " + z") '~ > y 2 , , v / 3 ,

and t h e n we have u o X v > y 2 v / 3 _ z" > y , / 2 .

(3.18)

In view of (3.15) and (3.18) we find from (3.8), (3.7): [x/(

oX

- ii = I z T ( u o X U I


u is dealt with quite similarly. Thus, having proved the boundedness of the values u, v in the equality (3.2), we come to a finite number of equations with fixed values for u, v - - u v, v _> 3. Since, recalling the comments after Theorem 1.2, the greatest prime divisor of x ~ - y" increases unboundedly with the growth of max(x, y), we

154

VII. Equations of Hyperelliptic Type

have to conclude t h a t (3.2) entails the boundedness of x, y. All the constants involved in our estimates are effectively computable, so we obtain the assertion of the theorem. Recently Inkeri and Van der Poorten [102] proved by similar arguments that all the solutions of the equation xp+(x+k)

p = z p,

(x,k)=l

in which natural numbers x, z and the odd prime number p are unknowns, have a bound effectively determined in terms of k. This result is close to the still unproven conjecture by Markoff [139] asserting that the equation x p + y p = ( y + 1) p has no solutions in natural numbers x, y and odd primes p. The solution of this problem would allow one to fill a gap in Abel's arguments on the impossibility of the existence, of a rational point with numerator a power of an (unknown) prime number, on the Fermat curve [1]. In w Ch.IX we give an effective description of all such points on algebraic curves (there are infinitely many of them on some curves).

VIII.

The

Class

Number

Value

Problem

It was ascertained in previous chapters that upper bounds for the solutions of diophantine equations under our consideration depend essentially on the regnlators of certain algebraic number fields related to the equation. N o w we concentrate our attention on this phenomenon and re/ate it to the genera/ problem of the magnitude of ideal class numbers. We show that algebraic number fields with 'small' regulator (hence 'large' class number) occur v e w frequently and in some sense constitute the majority of tields. Bounds for the solutions of the corresponding diophantine equations, e.g., Thue equations, are much better than the genera/bounds.

1. I n f l u e n c e of t h e V a l u e of t h e Class N u m b e r o n t h e Size of t h e S o l u t i o n s The problem of determining the size of the ideal class number of algebraic numberfields, takes its origin in the works of Gauss, Dirichlet and Kummer, and remains one of the central and most difficult problems of contemporary number theory. Though explicit formulae expressing the class number in terms of the main parameters of the field exist for fields of specific types, it is quite difficult to obtain from them satisfactory information either on the value or on the arithmetic structure of the class numbers. Because of that one meets with considerable difficulties in solving problems on the magnitude or the structure of class numbers, and the need to develop special methods becomes evident. The methods used in previous chapters to investigate diophantine equations rely in essence on the theory of algebraic units and lead to the construction of bounds for the solutions of the equations which essentially depend on the magnitude of the regulators of algebraic number fields, and thus in virtue of the Siegel-Brauer formula (see Lemma 2.4 Ch.II and the discussion after it), on the ideal class numbers of these fields. In this chapter we discuss the main aspects of the influence of the ideal class number on the magnitude of the solutions of diophantine equations and see that there is a two-sided influence: the larger the ideal class numbers of the algebraic number fields related to the equation, the better the bounds for the solutions of the equation. That is, the smaller the maximal height of a solution. Vice versa, good bounds for the solutions of specific types of diophantine equations lead to the construction

156

VIII. The Class Number Value Problem

of families of algebraic number fields with quite large class numbers ([218], [219 D. The phenomenon is clearly seen in the simplest non-linear diophantine equations, and we discuss several examples. Suppose B is a square-free natural number and

2 - B y 2=

1

(1.1)

is Pell's equation. All the solutions of the equation are given by powers of the fundamental unit c0 of the field K = Q(v/-B). Denoting h = h~: and R = lnc0 the class number and the regulator of K respectively, we have by Siegel's formula: ln(hR) ~ l l n D (D --* oo) (1.2) 2 where D = B or 4B is the discriminant of IK. Hence, for any 5 > 0 and large B we have e B1/2-Sh-1 < CO < r B1/2+ah-1,

which implies similar inequalities for the components xo, Yo of the fundamental solution co = x0 + Y0v ~ of (1.1). We see t h a t the magnitude of h has essential influence on the magnitude of the fundamental solution and, in particular, we have Co < e B6' with small 6~ > 0 only when h > B 1/2-~'' with small enough 8" > 0. In (1.1) the value of h influences the magnitude of the fundamental solution. If we t u r n to equations of higher degree, we see the same phenomenon in t h a t or other similar form with respect to all solutions. Consider the Delaunay equation x 3 + B y 3 = 1,

(1.3)

where B is a cube-free natural number. If the equation has a solution xo, Yo ~ 0, then it is unique and defines the fundamental unit 7]0 = xo + yo ~ of the ring Z[~/-B] with 0 < r/0 < 1. This unit is not necessarily the fundamental unit r of the field K = Q(~/-B), but in any event, it is a power of the latter. Because of t h a t I in 71ol >- R• and hence

max(lxol, ]yol)> (1 + ~-B)-leR~.

(1.4)

We shall return to this inequality below, but notice now t h a t if B is squarefree and B ~ 1 (mod 9) then the ring 71[~-B] coincides with the ring of all integers of K. Then 7]o = co and the Siegel-Brauer formula takes the form (1.2), where h, R, and D are now the class number, regulator and discriminant of the field K respectively. This produces the inequalities e Bl-~h-1 < 7]0 < e Bl+6h-1,

which imply

e B'-~'h-1 < max(Iz0], ly01) < e B'+~'h-',

1. Influence of the Value of the Class Number on the Size of the Solutions

157

with any 51 > 0 for all sufficiently large B (for which the equation (1.3) is solvable). A similar result holds also for the Nagell equation

x 4 - By 4 = +1, the only solution of which is given by the fundamental unit of the ring Z[C/L--4-B], and also on other equations of the form

x 2~ - B y 2 ~ = l

(n>3),

all solutions of which are determined by the first or second power of the fundamental unit of the ring Z[v/-B] (cf. [145]). Now we t u r n to Thue's equation

f(x,y)-=l,

(1.5)

with an integral irreducible form f ( x , y) of degree n > 3. By Theorem 4.1 Ch.IV all the solutions to this equation satisfy

x = max(Ixl, lYl)
0 is arbitrary and c2 = c2(n, 5) (cf. w Ch.IV). To derive this estimate we used the Landau inequality (Lemma 2.4 Ch.II).

hR < c3D1/2(lnD) '~-1,

(1.8)

and, having no information on the value of class number h, estimated it from below by 1. It is plain t h a t if we could bound h from below by some power of D, then in view of (1.8) and (1.6) this would lead to a strengthening of (1.7): it would be possible to replace the exponent 3(n - 1) + 5 by a smaller quantity. In what follows we shall see t h a t there exist algebraic number fields of every degree with regulators bounded above by c4D ~ with arbitrary c > 0. Such fields even constitute some kind of 'majority' among all fields of bounded

158

VIII. The Class Number Value Problem

degree. Consequently, the inequality (1.7) for the corresponding binary forms can be replaced by the stronger inequality X < exp(c5H~')

(1.9)

with any e t > 0, c5 -~ ch(n, r and in some cases even by an inequality of the type X < exp(c6(ln Hf) '~-1) (1.10) (see below, w167 On looking through the proofs described in the previous chapters giving bounds for other Diophantine equations, it is easy to observe the same influence of the regulators (and class numbers) on the magnitude of the bounds, though such influence does not show itself as directly, as in the case of Thue equation, it is seen in the intermediate stage of the arguments. Indeed, we used Lemma 3.1 Ch.VI in Ch.VI and VII, the assertion of which contains the regulator of the corresponding field as the main parameter; for applications of this lemma to concrete equations the regulator estimates were needed, and we carried out such in the estimates using the inequality (1.8) where h was replaced by 1. It is clear, that if we could replace h by a large enough value, the results would be better in those cases too. The opposite influence, that is the influence of good enough estimates of the solutions of diophantine equations on the values of class-numbers, shows itself in some form or other no less strongly. We describe in the next three paragraphs the families of algebraic number fields with large class number, relying on the results obtained above on the bounds for solutions of hyperelliptic diophantine equations and equations of hyperelliptic type. Our arguments are the inversion of the influence discussed above of class numbers on the bounds for the solutions. More hidden ties exist as well, for which results of Chowla [45] give examples: Let p > 3 be a prime such that the imaginary quadratic field Q ( ~ ) has class number 1. Then the equation x 3 _ p y 2 = -1728

(1.11)

has an integer solution x, y in which x = [e'v~/3], where [z] is the nearest integer to z. If p > 19 with the same 'class number 1' property, then the equation X3 py2 : --8 (1.12) _

_

has an integer solution with x = ~ [e~v~/3]. We see that supposing that the field Q(x/-:-~ has class number 1 leads to the conclusion that there exist 'quite large' solutions to (1.11), (1.12). Though the finiteness of the number of imaginary quadratic fields with class number 1 was not proved by this way, the presence of similar ties shows the deep nature of problems concerning the values of ideal class numbers.

2. Real Quadratic Fields

159

2. Real Quadratic Fields According to the well-known hypothesis called often the hypothesis of Gauss (or of Gauss-Hasse), there exists an infinite number of real quadratic fields K with class number h~ = 1. No approaches to the proof of this hypothesis are known, excluding approaches relying on yet more deep hypotheses. Even the much weaker hypothesis t h a t for some ~ > 0 there exists an infinite number of real quadratic fields K with r)W2-~ h~ < ~ :

(2.1)

remains unproved; here DK is the discriminant of K. There is a variety of results going in the opposite direction (for instance, cf. [92], [147]). These results state t h a t hK --* oo as D~: -~ c~, with K running through a special sequence of real quadratic fields. For example, if m takes integral values such t h a t m 2 + 1 is squarefree, then the fields Q(v/-m-~ + 1) have class numbers h,~ satisfying hm > /)1/2-5 _,~ ,

(2.2)

where Dm is the discriminant of the field, 6 > 0 is any number, and m _> m0(5). It can be proved by an application of the sieve method t h a t in fact there exists an infinitude of squarefree m 2 4- 1. Since for every squarefree integer D > 0 the equation m 2-Dn 2=1

(2.3)

has a solution in integers m, n (n # 0), every quadratic field K = Q(x/D) is contained in the sequence of fields Q(x/-m~ - 1), (m = 2, 3 , . . . ) . But (2.3) has infinitely many solutions for every D, and therefore every given field K occurs in the sequence of fields Q(x/m-~ - 1) with infinite multiplicity. A natural arithmetic way to reduce the multiplicities to a finite number is by making substitutions m ~ g(m) in (2.3) with an integral polynomial g(x) such t h a t the polynomial G(x) = g2(x) - 1 has at least three simple roots. Since the diophantine equation G(x) = D y 2 has only a finite number s of solutions for any D (see Ch.VI), it follows t h a t the multiplicity of each field Q(v/D) in the sequence of fields Q(v/g2(m) - 1) so obtained is finite. In the case of special polynomials g(x) = x k (k k 2) it may be even expected that multiplicities are uniformly bounded in D by a value depending only on k. Each field Q(x/g2(m) - 1) coincides with a field Km = Q(x/-D--~), where Dm is the squarefree kernel of the number g2(m) - 1, because we have g2(m) - D m y ~ = 1,

ym e Z.

(2.4)

Consequently, g(m) + ymv/-D-~ is a unit of the field IKm and is a power of the fundamental unit of the field. Therefore the regulator R,~ of the field K,~ satisfies

160

VIII. The Class Number Value Problem Rm 2) there are grounds for expecting that R,~ is exactly of order ln(m + 1), as this is the same as stating t h a t g(m) + y m ~ / - ~ is a power of the fundamental unit of the field with an exponent bounded by a value depending only on k. In view of Siegel's theorem we have ln(h,~Rm) ~ 89lnDm

(m ---* o~).

(2.6)

Hence, to decide which of the two possibilities (2.1), (2.2) occurs, we have to determine the order of magnitude of in Dm with respect to l n l n ( m + 1) as m -~ 0% or to determine the order of D,~ with respect to powers of ln(m + 1). Thus, we come to the problem of estimating the solutions of diophantine equations (2.4) with fixed Dm and unknown m and ym. Applying Theorem 4.1 Ch.VI we obtain the following result [218]. T h e o r e m 2.1 Suppose that k >_ 2 is a natural number, and that D,~ and hm are respectively the discriminant and the class number of the field Nm = Q ( g r ~- ~ - 1), where m = 1, 2 , . . . . Then we have D,~ > cs(ln m) ~(2k)/12.~(k)-~ , 1

~ h m > C9 D m

(2.7)

12

to~(2~-61(~

~

(2.8)

where T(2k) is the number of divisors of 2k, and $1(k) = 125(k) with

5(k)=

cs = cs(k,r

c9 = c9(k,r

1/6,

if

2,~k,

3/~k,

3/16,

if

21k ,

3~k,

7/24,

if

21k ,

31k ,

5/24,

if

2,~k,

3 I k,

and c > 0 is arbitrary.

We see t h a t if T(2k) > 28, then hm tends to infinity like some power of D,~ (m --* oo). Similar theorems open up vast possibilities for investigating fields with certain properties possessed by their class numbers. For instance, one can prove by the application of sieve methods that the number of genera of the field Q(v/-m~ - 1) m a y be bounded for infinitely many m by a value depending only on k, while the number of classes in each genus increases without bound as m grows. One can prove also the infinitude of the number of real quadratic fields with class number divisible by a given number, and so on. Before we proceed with the proof of Theorem 2.1, we prove the following simple lemma. L e m m a 2.1 Let f l ( x ) , . . . , f s ( x ) be integral polynomials no two of which have a common zero, and set f ( x ) = f l ( x ) " .. f s ( x ) . Then for any integer

2. Real Quadratic Fields

161

x for which f(x) r O, the square-free (positive) kernels A[f(x)], A[A(x)] , ..., Airs(x)] of the numbers f(x), fl(X), ..., f,(x) satisfy the inequality A[f(x)] >_ P-SA[I1 (x)] ... A[f~(x)],

(2.9)

where P = I-Ii P-1A[fi(x)], we obtain (2.9).

Proof of Theorem 2. I. We start with some general considerations and then use the special conditions of Theorem 2.1. Let f(x) be an integral polynomial with at least three simple zeros, and let a, al, c~2 be such a triplet of zeros. Set F = Q(a, al, a2), and let d, dl, d2 denote the respective degrees of that field over Q(a), Q ( a l ) and Q(c~2). Let a ( f ) be the smallest sum d + dl + d2 for all choices of three simple roots of the polynomial f(x). If f(x) does not have three simple zeros we set cr-l(f) = 0. By Theorem 4.1 Ch.VI the square-free kernel A[f(x)] of the number f(x) r 0 with integral x r 0 satisfies A[f(x)] > cl0(ln

Ixl)

(2.10)

where c10 > 0 depends on r and the coefficients and degree of f(x). If f(x) = f l ( x ) . . , fs(x), where the fi(x) are integral pairwise co-prime polynomials, then Lemma 2.1 and inequality (2.10) applied to each polynomial fi(x) yield

A[f(x)] > c11(ln Ix]) 88

(2.11)

where

a-I = ~ c r - l ( f i )

.

(2.12)

i=1

Suppose e(x), a(x), b(x) are integral polynomials related to f(x) by

e(x)a2(x) - f(x)b2(x) = •

(2.13)

We take any b(x) and define e(x) and a(x) as the square-free part and the full square of the polynomial f(x)ba(x) + 1.

162

VIII. The Class Number Value Problem

Suppose t h a t f(x) has a positive leading coefficient and t h a t rn is a natural number with f(m) > O, b(m) 7~ O. Then e(m) > 0 and one can suppose t h a t a(m) > O, b(m) > 0. We observe from the equality (2.13) t h a t an algebraic number

77= a(m) ~

+ b(m) ~

>1

is a unit, and t h a t its square is a positive power of the fundamental unit em of the real quadratic field II/:m = Q(v/e(m)f(rn)). Consequently, lnem _< Cl=ln(m + 1), where c12 is determined in explicit form in terms of the coefficients and degrees of the polynomials f(x) and b(x). It follows from (2.13) t h a t the numbers e(m) and f(rn) are co-prime, so the squarefree kernel of the number e(m)f(m) is no less t h a n t h a t of f(m). Therefore we see from (2.11) t h a t the discriminant Dm of the field Km satisfies D m > c13(ln(rn + 1)) 88

(2.14)

where a is determined by the equality (2.12), and c13 > 0. Hence, r~4a+~ Rrn -- ln~m < ~-14~m

9

Now we obtain from (2.6) that 1/2-4Cr-~ hm > c15 D m

(2.15)

which in the case t h a t a < ~ is a pithy inequality showing together with (2.14) t h a t hm increases without bound as m ~ oe. On setting f(x) = x 2•- 1 we see t h a t (2.13) holds with e(x) = 1, a(x) = x k and b(x) = I. Now decompose f(x) into factors f l ( x ) , . . . , f~ (x). We have x

- 1 = H Pd(x), dl2k

where the Pd(x) are the cyclotomic polynomials. If d ~ 1, 2, 3, 4, 6 the degree of Pal(x), which is ~(d), is no less t h a n 4. In the remaining cases it is 1 or 2. Therefore one should take into account the divisibility of k by 2 and 3. 1) 2/~ k, 3/~ k. As 'main' factors fi(x) we take P~(x) with d ~ 1, 2. Then a(Pd) is equal to 3, and the number of factors is T(2k) - 2 hence, we obtain from (2.12): a -1 = (T(2k) -- 2)/3. (2.16) 2) 2 ] k, 3~ k. For Pal(x) with d 7~ 1, 2, 4 we have a(Pd) ----3, while a(PIP4) = 4. Therefore we have a -1 = 88+ (r(2k) -- 3)/3. (2.17) 3) 2 I k, 3 [ k. For Pal(x) with d ~ 1, 2, 3, 4, 6 we find t h a t a(Pd) = 3 while a(P1P3) = a(P2P4) = 4. Therefore we have o - 1 = 89+

- 5)/3.

(2.1s)

3. Fields of Degree 3,4 and 6

163

4) 2~ k, 3 I k. Similar to the preceding we have a(Pd) = 3 for d # 1, 2, 3, 6, whilst a(P1P3) = a(P2P6) = 4, which gives a - 1 = 1 + (T(2k) - 4)/3.

(2.19)

Now the relations (2.14) - (2.19) give the assertion of the theorem.

3. F i e l d s of D e g r e e 3,4 a n d 6 The next three theorems may be obtained in a quite similar way to Theorem 2.1; see [219]. T h e o r e m 3.1 Let k > 1 be a natural number, and let D,~ and hm denote the discriminant and class number of the field Q( mv/-m-~3 1), where m = 1, 2, . . . . Then IDm I > c16 (ln m) ('r(3k)-1)/24-e , 1

_

24

--E

hm > ClTIDml 2 ~ where c16 arbitrary.

>

,

(3.1) (3.2)

0 and c l v > 0 are expressible in terms of k and e, with ~ > 0

T h e o r e m 3.2 Let k > 1 be a natural number, and let D,,~ and h,n denote the discriminant and class number of the field Q( m~/-m-~ 4 - 1), where m = 1, 2 , . . . . Then the inequalities ]Din] > Cls(ln m) ~r

(3.3)

1

12

hm > Cl9]Dm]-~-'(4~)-6e2(k)-~,

(3.4)

where 6~(k) = 7/12 if3 I }, and *~(k) = 3/8 r 3~ k; and ClS > O, ~19 > 0 are expressible in terms of k and e. A s usual ~ > 0 is arbitrary.

T h e o r e m 3.3 Let k > 1 be a natural number, and let Dm and hm be the diseriminant and the class number of the field Q( mvZm~-fi 8 - 1), where m = 1, 2, . . . . Then we have the inequalities ]D,~I > c20(ln m) (~-(6k)/4-7/8-e, I --

hm > c2,1Dml ~ ~

12

(3.5)

__~

,

(3.6)

where c20 > 0, c21 > 0 may be expressed in terms of k and e, with e > 0 arbitrary.

To prove Theorem 3.1 we need an estimate for the cube-free divisor of q" - 1, which we denote by A3 [q" - 1]. On denoting the eyclotomic polynomials by Pc(x) we have

164

VIII. The Class Number Value Problem x '~ - 1 = H

Pa (x).

(3.7)

Since Pa(x) is a normal polynomial of degree ~(d), by T h e o r e m 1.1 Ch.VII we have Aa[Pd(q)] > c22(ln q)l/24-e, (3.8) when d > 3, where c22 = c22(d,e) > 0. T h e decomposition (3.7) and an analogue of L e m m a 2.1 for cube-free divisors of polynomials now gives: for odd n Aa[q n - 1] > c23(lnq) (r(n)-l)/24-s, (3.9) while for even n

Aa[q'* - 1] > c24Aa[q 2 - I](lnq) (r%)-2)/24-~, which again leads to (3.9). Now set D = d a - I, where d > 0 is an integer, and write a = 4/-D. T h e n 77 = d - c~ is a unit of the field K = Q(a), and 1 < 1,71 < d + 14/-~1 < 2d. Consequently, the regulator R~< < ln(2d). Setting A3[D] = D 1 D 2, where D1, D2 are squarefree numbers, we have one of two representations for the discriminant D~:: DR = - 3 ( D 1 D 2 ) 2,

if

Aa[D] - 1

(mod 9),

DR = - 2 7 ( D I D 2 ) 2,

if

A3[D] ~ 1

(mod 9).

At any rate DK is divisible by A3 [D], and then IDxl > Aa[D]

(3.10)

holds. If we now take, in accordance with T h e o r e m 3.1, d = m k and n --- 3k, q = m, we obtain t h a t R x < ln(2m k) < 2k In m

(m >_ 2),

IDxl > C2s(ln m) (r(ak)-l)/24-e , the latter in view of (3.9), (3.10). This gives (3.1), and together with the Siegel-Brauer t h e o r e m it gives (3.2) as well. Theorems 3.2 and 3.3 are proved by similar arguments. Setting D = d 4 - 1, ~ = ~ we observe t h a t the numbers 771 = d - c~, and 72 = d 2 - c~2 are units of K = Q(c~). The regulator of this system of units is • in la~ R[r/1 ,r/2]= l n l d + a I lnld e c~21 ,

Id- ~1 In

~211 i

from which we find

3. Fields of Degree 3,4 and 6 R[~I, r/2] '-~ 20(ln d) 2

165

(d --~ oo),

taking into account that a = d(1 - d - 4 ) 1/4 = d(1 - 88 -4 + O(d-S)), a 2 = d2(1 - 89 -4 + O(d-S)). Therefore Rx < 21(ln d) 2 holds for sufficiently large d. It follows from the theory of differents of algebraic number fields (that is, multiplicativity of differents in relative extensions) that the discriminant of the field K is divisible by the square of the discriminant of the field Q ( v / ~ - 1). Hence, we have ]OK[ > A~[d 4 - 1], (3.11) where A2 [d4 - 1] is the square-free part of d 4 - 1. If we now take d = m k, then in view of (2.17) and (2.18) we obtain A2 [m 4k - 1] > cg~(ln m) ~(4k)-[~2(k)-~, where f2(k) is determined as in Theorem 3.2. We find from (3.11) t h a t ]Dx[ > c~6(lnm)[ ~(4k)-~2(k)-2~, which in fact coincides with (3.3). Since we have R~ < 21(k In m) 2 for large enough m, by the Siegel-Brauer theorem we obtain (3.4). Finally, in the case D = d 6 - 1 we set a = ~/-D. Then ~1 = d - a, 7?2 = d 2 - a 2 and ~3 = d 3 - a 3 are units of the field K = Q((~). The regulator R[T]I, T]2, 7]3] of this system of units is equal to •

l n [ d _ ~l lnld+ ~ [ 1 h i d - (~[

lnld 2 _~2[ ln]d 2 _~2[ ln[d 2 - (2a2[

in [d3 - ~3 I ln[d 3 + ~3] , in [d3 - (3~31

where r = e ~i/3. To compute this determinant, we add the second row to the first and subtract it from the last. Then we find R[~1,V2,~3] = =t=ln[d3 + a3[

ln[d 2 - a 2 [ . l n

Jd 2 -- ~20~2 ~--~-~

d-(~ ) 9

-21nld2 - a21 " in ~ Taking into account that

a2 = d2(1 _ 89

+ o(a-12)

we obtain for d ~ 0% R[r]I,V]2,~]3] ~ - + 3 l n d ( - 4 1 n d . l n

~2(2-

24(1nd)2 + 8 1 n d . l n

~-~-~ ) ,

166

VIII. The Class Number Value Problem

which is ~ 72(ln d) 3. Once again from the theory of differents it follows that the discriminant DR is divisible by the cube of the discriminant of the quadratic field Q ( x / ~ - 1). Consequently, [Dx[ _> A3[d6 - 1]. Setting d = m k, we obtain from the equality (2.18) that IDa[ > c27 (ln m) 88r holds, which gives (3.5). As we have R~ < 73(k lnm) 3 for large m, the SiegelBrauer theorem gives (3.6). The inequalities for class numbers indicated in Theorems 2.1, and 3.1-3.3 were obtained on the basis of the 'principal part' of the Siegel-Brauer theorem

hR > c(n,c)lDI 1/2-e,

(3.12)

where h, R, D and n are respectively the class number, the regulator, the discriminant and the degree of the algebraic number field K; r > 0 is arbitrary. The value c(n, ~) in this inequality is ineffective, therefore the inequalities for h,~ obtained in this way are ineffective. To obtain effective inequalities one may use the results of Stark [228]. In particular, if K does not contain a quadratic subfield, then an effective strengthening of (3.12) holds, and in the general case for ~ > 1/n and n _> 4 the quantity c(n, r may be represented in effective form. This makes effective the inequality for hm in Theorem 3.1 and in Theorems 3.2 and 3.3 but gives effective inequalities that are a little weaker. The inequalities (3.1), (3.2) show that h,~ increases without bound together with m when T(3k) > 50. Similarly, an analogous assertion follows from the inequalities (3.3), (3.4) if T(4k) _> 28, when 3 ] k; and if 7-(4k) >_ 27, when 3/~ k. The inequalities (3.5), (3.6) lead to a need for the condition T(6k) _> 28. It is obvious that if we have stronger inequalities for squarefree and cube-free divisors of polynomial values than those obtained from Theorem 4.1 Ch.VI and Theorem 1.1 Ch.VII, then it would be possible to weaken or to exclude all (to replace by trivial ones) the restrictions on the number of divisors arising in Theorems 2.1, 3.1-3.3 and those we discussed above. This shows clearly that a strengthening of the bounds for solutions of diophantine equations allows one to extend one's knowledge of the totality of algebraic number fields with large class numbers. From the other side, as we have seen in w the improvements on the bounds to the solutions depend directly on our knowledge about the magnitude of class numbers of the corresponding fields.

4. Superposition of Polynomials

167

4. Superposition of Polynomials Development of the idea described in two previous paragraphs concerning the substitutions x --* x k in the general case requires investigation of algebraic properties of integral polynomials obtained by repeated substitutions x --* g(x), where g(x) is an integral polynomial. We introduce here the main result we need for further applications (Lemma 4.4) We denote by f * g(x) the superposition f(g(x)) of two polynomials f(x) and g(x). If h(x) is a third polynomial then we have

f * g 9 h(x) = f 9 (g 9 h(x)), and similarly for any number of polynomials. In what follows the polynomials under consideration are different from constants. L e m m a 4.1 Let f(x), g(x) be polynomials with complex coefficients, such that f(x) has no multiple root and deg f(x) >_ 2. Then the squareffee kernel of the

polynomial f 9 g(x) (in the ring C[x]) has degree at least deg f ( x ) d e g g ( x ) - 2 degg(x) + 2.

Proof. Set f * g(x) = u(x)v2(x), where u(x) is squarefree in C[x]. Then we have

d 9 g(x)

= (f' 9 g(x))g'(x)

=

where w(x) = u'(x)v(x) + 2u(x)v'(x). Since (f(x), if(x)) = 1, we have

(f * g ( x ) , f ' *g(x)) = 1 and then v(x) I g'(x). Consequently, deg v(x) 3, containing a. Let 0 be an integral generating element of the field F. Since the numbers 0, 0 2 , . . . , Ok are linearly independent over Q we have

168

VIII. The Class Number Value Problem ct = r i o -k r202 + . . . -k rkO k,

ri ~ Q

(i = 1, 2 , . . . , k).

From this follows a relation for the traces: S(aO j) = rlS(O l+j) + . . . + rkS(O k+j)

(j = 0, 1 , . . . , k -

1).

The determinant (i= 1,2,...,k;j=0,1,...,k-1)

d=det(S(Oi+J))

differs from the discriminant D(O) of the number 0 by an integral rational factor Nm(0). Hence it is a rational integer different from zero. Therefore the ri are integral linear combinations of the numbers S(c~OJ)d -1,

(j = O, 1 , . . . , k -

1).

If a is the leading coefficient of f ( x ) then the numbers adri all are integers. Consequently, the polynomial r ( x ) = r l x + r2x 2 + ... + r k x k has rational coefficients, the denominators of which divide ad; to(X) = r ( a d x ) is an integral polynomial. Let p ( z ) be the minimal polynomial of O. Since we have f(r(O)) = f(c~) = O, the polynomial f * r(x) is divisible by the irreducible polynomial p(x), and the integral polynomial f o ( x ) = f * ro(x) is divisible by pl(x) = p(adx). If p l ( x ) occurs in fo(x) to the first power we set g ( z ) = to(x), otherwise we take g(x) = to(x) • pl(x),

where the sign is chosen so that g(x) has degree k. Then we find that f 9 (r0(x)

+pl(x))

- f * r 0 ( x ) + ( y * r 0 ( x ) ) p l ( x ) --

-- + ( f ' * ro(x))p3(x) ~ 0

(mod p~(x)),

since we have ( f ' * r o ( x ) , p l ( x ) ) =- 1. It is clear that pl(x) is a normal polynomial and that the degrees of Pl (x) and g(x) coincide. Finally, we observe t h a t construction of the polynomials g(x), pl(x) may be done effectively. Indeed, if the degree of a is no less t h a n 3, then the field F is obtained by successive adjunction of the roots of the minimal polynomial of a. On replacing the successive extensions by a single extension, we find a generating element of F. If the degree of a is 1 or 2, the construction of F is trivial. L e m m a 4.3 Suppose f ( x ) is an integral polynomial of degree no less than 2 and without multiple roots; let r > 0 be an arbitrary integer. Then there exist effectively determinable integral polynomials g, gl, . . . , gr-1; Pl,P2, . . . ,Pr; f~, q~, such that p l , p 2 , . . . ,PT are n o r m a l polynomials of degree no less than 3, and so that setting

4. Superposition of Polynomials

169

G=g*gl*...*gr-1,

Pi = Pi * gi * g i + l P~ = p~,

* .-.

*

gr--1

(i = 1 , 2 , . . . , r -

1),

we have f * G =

(Pi, P j ) = I ,

P1P2... p~f~q2

(Pi, f~) = 1

(4.1)

(i,j=l,2,...,r;iT~j).

Proof. By Lemma 4.2 there exist polynomials 9, Pl, f l for which f * g = Plfl, with Pl a normal polynomial of degree k _> 3, (pl, f l ) = 1, and with the degrees of g and Pl coinciding. By Lemma 4.1 the squarefree divisor of f * g(x) is of degree no less t h a n nk - 2k + 2, where n is the degree of f(x). If n _> 2, then the square-free part fl0 of the polynomial f l (x) has degree no less than (n - 2)k + 2 > 2. Applying Lemma 4.2 to fl0(X) we find polynomials gl, P2, f2, for which fl0 * gl -- p2f2 and where P2 is a normal polynomial of degree no less t h a n 3, (p2, ./'2) = 1, and deggl = degp2. Then we have f

*

g

* gl

---- ( P l *

gl)(flo

* f f l ) g 2 ----" ( P l *

g l ) P 2 f 2 q 2.

Since (Pl, f l ) = 1, then (pl, fl0) = 1, and then (Pt * gl, flo * gl) = 1. Consequently, we find (501 "91,P2) = 1,

(Pl * g l , f 2 ) = 1.

Next, we separate from f2(x) its squarefree part f20(x) and apply Lemma 4.2 to f20, and so on. As such the arguments may be continued and we obtain the assertion of the lemma. L e m m a 4.4 Let f(x) be an integral polynomial of degree no less than 2 and

without multiple roots, and take B > 0 arbitrary. Then there exists an integral polynomial G(x) effectively determined by f(x) and B, such that for any integer m ~ 0 with f * G(m) ~ 0 we have A2[f * G(m)] > c2s(ln

Im[)B,

(4.2)

where c2s > 0 is expressible in terms of the coefficients of the polynomials f(x), G(x), and the number B. Pro@ Taking an integer r > 12B, we apply Lemma 4.3. It follows from the equality (4.1) and Lemma 2.1 that

A2[f * a(m)] > c29A2[Pl(m)l... A2[P~(m)], where c29 > 0 is expressed in terms of the resultants of the polynomials Pi and Pj, and Pi and f~, (i, j = 1, 2 , . . . , r; i r j). As we have

170

VIII. The Class Number Value Problem Pi(m)=p~(mi),

mi=gi*...*g,--l(m)

(i= 1,2,...,r-1),

the inequality (4.2) follows from the estimate A2[p(m)] > c30(ln(Iml

+ 1)) 1/12-e,

which is true for any normal polynomial of degree no less than 3 (this is a corollary of Theorem 4.1 Ch.VI; see the discussion at the close of w Ch.VI). Now we can supplement the results of two previous paragraphs by the following assertion. T h e o r e m 4.1 Suppose that t = 2, 3,4 or 6, and that f ( x ) is an integral polynomial of degree at least 2 for which the equation at(x) - f ( x ) b t ( x ) = 1 has a solution in integral polynomials a(x), b(x), with (b(x) not identically zero. Then for any real B > 0 there exists an integral polynomials g(x), effectively determined by the polynomials a(x), b(x), f ( x ) and the number B such that the fields Q(~/-]-~ g(m) ) with m >_ mo have discriminant Dm and class numbers hm satisfying [Dml > c31(lnm) B,

h,~ > c32[Dm[ W2-~/B-~

where e = 1, if t = 2 and t = 3; e = 2, if t = 4; e = 3 if t = 6 ; and c31 > 0 , c32 > 0 are values depending only on the coefficients of the polynomials a(x), b(x), and f ( x ) and the numbers B, e, where e > 0 is arbitrary. Proof. In the case of t # 3 one uses Lemma 4.4 and the arguments described in the two previous paragraphs. In the case t = 3 we take as a basis the arguments of w but Lemmas 4.1, 4.3 and 4.4 should be replaced by the analogous lemmas concerning the cube-free divisors of polynomials, by inserting obvious changes into the arguments described above.

5. T h e A n k e n y - B r a u e r - C h o w l a Fields In 1956 Ankeny, Brauer and Chowla [4] proved the existence of an infinite sequence of algebraic number fields K,~ (rn = 1, 2,...) having any prescribed degree n > 2 and extremely large class numbers: hm >

ID~r1/2-~,

(5.1)

where hm and Dm are respectively the class number and the discriminant of the field Era; r > 0 is arbitrary. For n > 3 it turned out to be possible to find such a sequence in the set of the fields generated by the roots of the polynomials f N ( x ) = (x -- al) . . . (x -- a,~-l)(x -- N) + 1, (5.2)

5. The Ankeny-Brauer-Chowla Fields

171

where a l , . . . , a , - 1 are any fixed distinct integers, and N runs through all natural numbers starting with some appropriate one (it is easy to see that fN(X) is then irreducible). Fields generated by the roots of the polynomials (5.2) are purely real, but if hi, n2 are any natural numbers with nl + 2n2 = n, and the integers al, a 2 , . . . , a , 1+,~2-1 are such that

ai ~ aj

(i ~ j),

a~l+l>0,...,a~l+~2_l>0

,

then the fields generated by the roots of the polynomials gN(X)

:

(X --

al)...(X--an-1)(x

2 -4- a n 1 + 1 ) " " "

9. - ( x 2 +

ana+n2-1)(X

2

+ N) + 1

(5.3)

for sufficiently large N are irreducible and have exactly nl real and n2 complex conjugate fields. In the set of these fields one can also find a sequence with condition (5.1). We prove a few assertions on the sequence of fields generated by the roots of the polynomials (5.2) and satisfying the inequality (5.1). First of all, we prove the possibility of an effective choice from the set of all fields of those which satisfy (5.1) and then we prove that such fields constitute the 'vast majority' of the set. T h e ideas we apply also suffice to deal with, in a similar way, the fields generated by the roots of the polynomials of the form (5.3), as well as more complicated families of the fields generated by the roots of the polynomials (z - al(m))... (x - a n ( m ) ) + 1, where the ai (y) are integral polynomials with max

l(i maxlail, (i = 1 , 2 , . . . , n - 1). Then the regulator of the field generated by the root of the polynomial (5.2) does not exceed c33(lnN) n - l , where c33 depends only on a l , . . . ,a,~-l, and n.

Proof. Denote by ON a root of the polynomial fN(x). Then ON -- al, . . . , ON -- a,~-i are units of the field K = Q(0N). The roots 0~ ) of the polynomial f g (x) may be enumerated in such a way that as N -~ c~ 0~ ) --* ai

(i = 1 , 2 , . . . , n -

1).

Consequently, for j r i we have

]0~) - aj] ---* lai - aj] # 0 and from the equality

(Y ~

oo),

172

VIII. The Class Number Value Problem (0~)

-

al)

. . . (0~)

-

ai)

. . " (0~)

-

a,~-l)(O~

) -

N)

=

-1

w e o b s e r v e that

I0~ )

1

-

-

all ~ -~

H ]ai

-- aj1-1.

j=l

j#i

Thus,

as N

--* ec we find that

lnlai-ayl, iCj,

lnl@-asI ~

i=j.

-lnN,

Hence, it follows t h a t det(ln I @ - a j l ) , j = l , 2 ...... --1 ~ ( - - l n N ) n-1. Since the regulator of E does not exceed the absolute value of the latter determinant, we obtain the assertion of the lemma. L e m m a 5.2 Let c~1,..., ~,~-1 be distinct complez numbers such that Ic~i - c~jl > 2

(ir

i,j=l,2,...,n).

(5.4)

Set f ( x , y ) = (z - c~l)... (x - c ~ _ l ) ( x - y) + 1, and let D(y) denote the discriminant of the polynomial f (x, y) with y fixed. Then D(y) is a polynomial in y of degree 2(n - 1) and without multiple roots. Proof. T h e equation f ( x , y) = 0 defines n functions : ~ l ( y ) , . . . , &,~(y)analytic in a neighborhood of y -- oc for which as y --~ oo we have ~i(y)--*c~i

(i=1,2,...,n-1),

~,~(y)-y~O.

(5.5)

Since D(y) may be represented in the form

D(y) it follows from

=

H (zi(Y) - 2j(y))2, l 0 arbitrary. In the next paragraph we discuss the distribution of fields with large class number among all the fields of fixed degree; consideration of the AnkenyBrauer-Chowla fields will again play an essential role.

6. A Statistical Approach We now discuss the distribution of algebraic number fields IN with class numbers satisfying the inequality

hK ~ ID~[~,

(6.1)

where 5 is a fixed number in the range 0 < 5 < 1/2, from a purely statistical viewpoint. The results of the previous paragraphs suggest that it is reasonable to estimate the number of different fields of fixed degree with a bounded regulator and then to compare that with the number of such fields subjected to the additional condition (6.1) (cf. [212], [213]).

6. A Statistical Approach

177

We start with real quadratic fields. 6.1 For any x > e let E(x) be the number of fundamental units < x of all real quadratic fields, and for a fixed 6 in the range 0 < 6 < 1/2 let E~(x) be the number of such units in the fields satisfying (6.1). Then we have

Theorem

[In ~]

E(x) = 2 E

It(k)xl/k -[- O ( l n x ) ,

(6.2)

k----1

where #(k) is the MSbius function, whilst E6(x) < c4o(lnx) 2/(1-2~'),

(6.3)

where 6' is any number in the interval 6 < 6' < 1/2, and cao depends only on 6' - 6. Proof. Any quadratic unit 77 > 1 and its conjugate satisfies an equation of the form z 2 - m z = +1 with positive integer m = 77+ 7'. T h e number of those ~ which lic in the interval 1 < 77 < x is equal to 2x + O(1) and is also E(x) + E(x 1/2) + . . . + E(x 1/k) + . . . . Consequently, OO

F(x) = E E(xWk) = 2x + O(1).

(6.4)

k=l

Since the MSbius function It(1) satisfies the equations

E --

It(l)

fin

1,

n = 1,

o,

#

i,

we find t h a t

E

It(1)F(xWz) = E #(1) E

/=1

/=1

E(xl/kZ) = E

k=l

E(xl/'~) E

n=l

It(1) = E(x).

l]n

Thus we obtain O4)

=

OO

it(1)F(x 1/')

=

l l n x / l n 2 . It now follows from the right-hand side of (6.4) that [In x / I n 2]

E(x) = 2 which yields (6.2).

E l=l

"(l)xi/l "-~O(lnx),

178

VIII. The Class Number Value Problem

Inequality (6.3) follows directly from Siegel's theorem. Indeed, if c0 is a fundamental unit of the field K, co _< x and (6.1) holds, then we find from (1.2) t h a t for any T > 0 with 6 + ~- < 1/2 we have the following inequalities.

c(T)D 1/2-~ < hln60 _< D ~ lnx, D < c41(lnx) 2/(1-26')

(6' -- 6 + T).

Because each real quadratic field is determined by its discriminant, we find (6.3). Formulae of the type (6.2) also hold for the number of fundamental units c ~ x with the condition Nm(c) = +1, respectively with the condition Nm(c) = - 1 . Denote these numbers by E + (x) and E - ( x ) respectively. Then we easily find similarly to (6.4) t h a t

E(rood 2) E - ( x 1/k) = x + O(1),

=

k--1

from which we obtain

E-(x) =

~

#(k)x Uk + O(inx).

(6.5)

k--I (mod 2)

l_ 3 and t, with 0 < t < n/2, and reals Z > 0 and 5, with 0 _< 6 < 1/2, let N(t)(Z) denote the number of distinct (nonisomorphic) algebraic number fields K of degree n with regulator R~ exp(c44Z1/(n-t-l)), where c42, c43 and c44 > 0 depend only on n, and (0 N~,~(Z) < c45Z '~/(]-26'),

(6.8) (6.9)

(6.10)

where 5~ is any number in the range 6,< 5~ < 1/2, and c45 depends only on n and 5~ - 5. Proof. The proof of the theorem splits into three parts corresponding to the inequalities (6.8), (6.9), and (6.10). We shall prove these successively, relying on the following lemmas.

180

VIII. The Class Number Value Problem

L e m m a 6.1 Suppose that 7? is an algebraic integer of degree m > 3 and 1

[-~ _< i + 30m2 ln(6m) ;

then 77 is a root of unity. Proof. S ~ [31]. L e m m a 6.2 Suppose N is an algebraic number field of degree n >_ 3 and t is

the number of pairs of complex conjugate isomorphisms of N, so 0 c69(ln ]DK[) '~+1 holds, where c69 > 0 depends only on n, and the primes p l , . . . ,p,~. Yamamoto notes that the fields Km = Q(x/-D-~), where Dm = (p2mq + p + 1)2 _ 4p

(m = 1, 2 , . . . )

(7.5)

with p, q distinct prime numbers, satisfy his theorem, so that one obtains an infinite sequence of real quadratic fields K, the fundamental units r of which satisfy eo > eCT~(ln [DK[) 3. The discriminants of the fields Km may be estimated from below as the squarefree divisors of the numbers Din, using the following equality D,~=q2xa+2q(p+l)2+(p-1)

2,

x:pm,

obtained from (7.5), and Theorem 4.1 Ch.VI. It is not known whether the hypotheses of Yamamoto's theorem hold when n>3. It is not too hard to see that for a proof of existence of an infinite sequence of real quadratic fields with relatively small class numbers it suffices to prove the existence of infinitely many not too large intervals of natural numbers (N, N + M) containing natural numbers m with m 2 - 1 having a relatively small square-free kernel. Such an approach leads to the following question. Let MN be the smallest M for which -I p<MM plN(N+I)...(N+M) holds. Is it true that MN < (ln N) c71 holds infinitely often, where c71 > 1 is an absolute constant? If the answer is affirmative, then there exists an infinite number of real quadratic fields K, the fundamental units e0 of which satisfy r ,-~1/3c~1 (ln DK) -1 }, r > explc~2JJ~

hence, the class numbers h~ are less t h a n C73 = DK/2-1/(3c71) in D~;,

7. Conjectures and Perspectives

187

where c72 > 0 and C73 > 0 are absolute constants. Probably, we do not have enough information on the distribution of prime numbers to give the answer to the above question. Many problems concerning the structure of the products of consecutive natural numbers show a deep connection with the problems of the distribution of prime numbers. For instance, there is a conjecture due to Grimm [80]: if the numbers N + 1 , . . . , N + M are composite, then the number of distinct prime divisors of the product, ( N + 1)... ( N + M) is no less than M. It follows from this that the consecutive prime numbers pn satisfy

for sufficiently large n. Recently Ramachandra, Shorey and Tijdeman [162] have made some progress on Grimm's conjecture, while Dobrowolski [57] has obtained an improvement of Lemma 6.1, close to the hypothesised assertion.

IX. Reducibility of Polynomials and Diophantine Equations

In this final chapter, we deal with a 'meta'-topic, one that lies 'over' the theory of diophantine equations, as it were; namely arithmetic specialisation of polynomials. The main result asserts that under such specialisations the multiplicative structure of the numbers obtained goes some considerable way towards determining the multiplicative structure of the original polynomials. This allows one to give effective versions of Hilbert's irreducibility theorem and to describe a11 abelian points on algebraic curves. The methods used are quite independent of the theory of linear forms in the logarithms of algebraic numbers, and rely on the study of the arithmetic structure of sums of algebraic power series in all metrics of the field of rational numbers.

1.

An

Irreducibility

Theorem

of Hilbert's

Type

Let F ( x , y ) be an integral polynomial irreducible in the polynomial ring Q[x, y]. There are two major problems on the arithmetic properties of such polynomials that have attracted attention for a long time. The first problem is to determine or to describe all rational points on the curve

F ( x , y ) = O.

(1.1)

Obviously, this problem is equivalent to the one of determining or describing all rational z0 for which F(zo, y) has a linear factor in Q[y]. However, even for the case of special polynomials F(x, y), unsurmountable difficulties sometimes arise in solving this problem, and it has been considered as one of the most difficult problems ever since the time of Fermat and Euler. The second problem is much younger (some 90 years old), but not any easier. This is the problem of determining all rational x0 for which F(xo, y) is irreducible in Q[y]. Let ~ F and -gF be the sets of rational x0 which constitute the solution of the first and to the second problems respectively. If degy F(x, y) > 2, these sets have no point in common:

~ F A h F =~. At first sight it seems very difficult to predict any relation between the multiplicative structure of a number x0 c Q and the multiplicative structure of a corresponding polynomial f(xo, y) E Q[y]. However, such a relation exists, as

1. An Irreducibility Theorem of Hilbert's Type

189

we shall see below, and that will allow us to characterise DF as an 'extensive' set and will consequently allow us to characterise ~F as well. In 1892 Hilbert [98] proved that DF is infinite for an irreducible polynomial F(x, y). This is the celebrated 'Hilbert Irreducibility Theorem'. Hilbert himself, and later Emmy Noether [148], gave applications of this theorem to the solution of the problem of constructing algebraic equations with given Galois group. In our time Hilbert's theorem remains a corner stone in investigations on the inverse problem of Galois theory. Hilbert's proof of the irreducibility theorem is ineffective and does not make it possible to determine elements of the set ~F. However, Hilbert noticed that his reasoning can be supplemented by appropriate estimates to meet this failing. The theorem has been generalised repeatedly and several new proofs of it have been suggested. In particular, one now knows generalisations to systems of polynomials in many unknowns over a finite extension of the field of rational numbers: these generalisations are of a purely technical nature and the case of polynomials in two unknowns over the field of rational numbers remains the basic one (see [120], [171]). Siegel [193] observed that finiteness theorems for the sets ~G for certain polynomials G E Q[x, y] connected directly with the initial polynomial F allow one to prove Hilbert's theorem. In that way he obtained a proof of Hilbert's theorem from his theorem on the finiteness of integral points on curves of genus greater than zero. This shows that there is a close connection between the two problems mentioned above on the sets ~F and DF. In whatever form, such types of connections constitute the basis for many investigations on Hilbert's theorem. In many papers the set ~g is described as an 'extensive' set from various points of view (for example, ~F contains 'almost all' natural numbers and its elements lie densely in Q and in Qp for all p). Schinzel [169] proved that ~)F contains an arithmetic progression, and Fried [74] recently demonstrated by another method that in any concrete case such a progression may be found by a finite number of computations (the progression, of course, depends on the polynomial). An interesting approach is taken in the work of Cohen [50], who gives a statistical characterisation of the set of those natural numbers x0 for which the polynomial F(x0, y) has the same Galois group over Q as does the polynomial F(x, y) over Q(x). In this chapter we describe a new approach to assertions similar to Hilbert's theorem, which allows us to go further both in respect of obtaining effective results and in permitting a deeper analysis of the set 2)F. More than that, we describe a phenomenon new in principle: the influence of the multiplicative structure of the number x0 on the multiplicative structure of the polynomial F(xo,y). Our arguments rely on the ideas of the theory of diophantine approximation. As a first application of these notions we consider the following theorem [220]. T h e o r e m 1.1 Let F(x,y) be an integral absolutely irreducible polynomial,

with

190

IX. Reducibility of Polynomials and Diophantine Equations

V(O,O)=O,

n=deguF(x,y)>_2,

~yV(O,O)

0.

(1.2)

Suppose that a and b are rational integers, with (a, b) = 1, and let ap denote the maximal power of some prime p in a (the p-component of a). Let n

max(]a[ ]b]) < Ca~ -1

-5

(0 < 5
(1 - 3e)nh, we find [Alp < p-Ordpa.ordW(~) _< p--m(1--3E),h,

m = ordp a .

Comparing this inequality with (2.9), taking (1.3) into account and letting h increase without bound, we obtain

(1_3r

dlnc14 emln---~ + d ( l + e l )

(

n-

n

1

5

)

.

If we suppose t h a t m l n p > nz -2 lnc14 and take e = }~(ncr + 2) -1, then we obtain d > n - 1, and hence d = n. This completes the proof of the theorem f o r p ~ E l . I f p t E f , w e c o n s i d e r V(w) in the disc Iwlp < IEflp (w c Qp) and argue as before.

3. Details and Sharpenings To obtain further results we have to know how the height of the polynomial F(x, y) influences all the auxiliary values involved in the above discussion. In this paragraph we carry out the necessary additional work. As a first corollary we obtain an explicit form of the dependence of cl on HF, the height of F(x, y). Apart from t h a t , our arguments will give the reader a more complete understanding of the m e t h o d described briefly in the previous paragraph.

198

IX. Reducibility of Polynomials and Diophantine Equations

L e m m a 3.1 Under the conditions of Lemma 2.1 one may take c5 = (4Hp) c, where c = k 2~, k = 89 + nl - 1 ) ( n + nl), and n l = deg~ F .

Proof. See [184], p. 193. 3.2 Suppose that F ( x , y ) is integral polynomial, with F ( 0 , 0 ) = 0,

Lemma

and (O/Oy)F(O,O) 7~ O, and that f ( x ) is a power series, defined by (1.6), with E] the Eisenstein number of this series. Then

Es I ( ~ ( 0 , 0 ) ) ~Proof. (cf. [97], 19th lecture). On making the substitution y = au, x = a2v, where a = (O/Oy)F(O, 0) ~r 0, we find t h a t the power series g(v) = a - l f ( a 2 v ) satisfies the equation G(v,g(v)) = 0, where G(v, u) is an integral polynomial of the form G(v, u) = bo(v)u '~ + . . . + b,~_l(v)u + b,~(v), with b,~_ 1(0) = 1. We have the formal expansion oo

,

(b,~_l(V)) - l = ( l § 2 0 0

d~ E 77.

v=0

Setting oo

g(v) = ~ g ~ v ~

(g. = f~a2~-l),

and making the substitution u = 9(v) in the equation G(v, u) = 0 we obtain

b,~(v) b,~-2(V) u 2 + . . . + b,~-l(V---""~+ bn-l(V"""~

-U-

bo(v) u,~, bn-l(V)

oo

-g(~) =b~(v) Z d~v~+

oo

g.~") 2 + . . . .

b~_2(~)(~ d~)(~

v=0

~=0

~=1

Consequently, it follows that oo

g(v) =

~

e.v" + v=0

oo

e~v ( v=0

g.lg.2v "'+"~) it1 ,~2=1 oo

""" + ~--~ e ~ n - - 1 ) V V ( u=0

+...

oo E

g " l ' " " g"r* V " I - t - ' " ~ - " n )"

#1 ,...,/~=I

Here a11 the numbers e(j) are integers. Comparing the coefficients of identical powers of v on the left and right sides of the last equality, we obtain the infinite system of equations (m = 1, 2,...)

3. Details and Sharpenings +..

g m --= e m -t-

(n--l)

+

ev

gm

199

" " " g~,,-1

"

,u,1~1 ..... ~ . _>1

~I ~ I , ~ 2 ~ I

Thus gl = el, g2 = e2 + e~og2, ..., gm is integral polynomial in gl, g2,.. 9 9m-1 and we see t h a t all the gm are integers. Since g m = a2m--lfm, t h e n at any rate the a2mfm (m = 1, 2 , . . . ) are integers and we obtain the assertion of the lemma (supposing t h a t Ef is the least natural E which satisfies the condition t h a t the E~f~ are integers for all u = 1, 2 , . . . ) . L e m m a 3.3

Under the conditions of Lemma 2.3 one may take A

=

LrS(n,~l) 2

C151~F

where c15 is determined by the degree of F(x, y). Proof. T h e power series (1.6) defines a regular function of the complex variable x in the disc [xl < P0, where P0 is the distance from the point x = 0 to the nearest singular point of the algebraic function y(x) defined by the equation F(x, y) = 0. Provided that 0 < p < P0 we find that f~-

27ril -,~-;*1=of(z)z-~-ldz'

Prom the equation either If(z)l _< 1 or

If-[ -
n in Lemma 3.4, and for k = 0, 1 , . . . , n - 1 by the equalities

{~ a~k =

fori=n-k for i r n - k.

Hence, the product S T has the form which is specified in the lemma, and 2n--2

ur = ~

Uka,~-r,k

(r = 0, 1 , . . . , n - - 1).

k=0

Applying Lemma 3.4 we obtain maxdegu~(x) < maxdeg si(z) + maxdegti(z) + n l ( n - 1), and since

< c~gmin(maxdegs,(x),maxdegt,(x))iI~jaxk ~

~ ,

we obtain the estimate for heights of the polynomials u~ which is specified in the lemma; c19 depends only on n. L e m r n a 3.6 Under the conditions of Lemma 2.3 one can take c7 = 2nnl, and cs = c2oH~'~, where c2o is expressible in terms of n and hi.

Proof. Apply the previous lemma and the equalities

202

IX. Reducibility of Polynomials and Diophantine Equations

A -- a~(x) + . . . + a i ( x ) f n - l ( x ) , B = -an-l(X) ..... oB

_

a"

(n - 1 ) a l ( x ) f n - 2 ( x ) + n f n - l ( x ) ,

.....

cOx

cOB = -a,~-2(x) . . . . . Oy

(n - 1)(n - 2 ) a l ( x ) f ' ~ - 3 ( x ) + n ( n - 1)f'~-2(x),

Pz = Aoz(z) + A l l ( z ) f ( x ) + ' " . cOPz . A~oz(X) +. A ~ z.( z ) f ( x ) + Ox

OPl -

COy

+ A,~-lz(x)fn-l(x), + A',~-lzt[X "~r JJ

{X ~ "~ J,

glz(X) + " " + (n - 1)A,~_lz(x)f'~-~-(x).

Using the notation introduced in the proof of Lemma 2.4, we find

hl+l __INm(b'U('~-UA)lp, and the assertion of the lemma follows. We a r e now in a position to give a sharpened version of Theorem 1.1 relying on the improved lemmas suggested above. Theorem

3.1 The assertion of Theorem 1.1 holds with 1/5 2 rr2n 2 (2n+c)+( 6n)t n21(n2rtl + l )2 /5 2 C1 ~ C22

/-/F

where c22 is defined in terms of C, n and nl, and the quantity c is as indicated in Lemma 3.1. Proof. First we estimate % , the maximal divisor of s! made up of powers of prime numbers bigger than cs, and not occurring in EI. Since E I does not exceed ( n - 1)2H~ (this follows from Lemma 3.2), to estimate % it is enough to exclude from s! the factor

H

9< H / / ( P - ' >

p 1, has only a finite number of solutions in integers u > 0 and rational y, and t h a t all the solutions may be found effectively. In fact the polynomial F(r ~', y) is irreducible in Q[y] for all u > c2s = c2s(r, F).

4. Theorems on Reducibility

207

Theorem 4.1 in its turn admits a strengthening in which the condition [bI < C[a[ is excluded. That leads to a corresponding change in the formulae (4.2). Now let v signify a prime number or the symbol oo. We define the vcomponent (x0)~ of a rational number x0 = a/b, (a, b) = 1, by the equality (x0). =

ap, max(l,

V = p, Ib/al),

v = oo.

Then we have

h(xo) = max(lal, Ibl) = I ' I ( x o ) . , and Theorem 4.1 changes as follows. T h e o r e m 4.2 Under the hypotheses of Theorem 4.1 we have for any rational number Xo with h(xo) > 1 that ln(x0), dj + O E lnh(x0-----~- n e~EF~

(i)

lnHF lnh(xo)

(1 _< j _< r),

(4.5)

where O,~ denotes the sum (which can be seen to be finite) of the series (1.6) at x = xo in each metric corresponding v, and the symbol 0 involves a value which is effectively determined by the degree of F ( x , y). It is apparent that this theorem implies both Theorem i.I and Theorem 4.1; it also leads to several new corollaries. For instance, we see that the polynomial F(xo,y) is irreducible for all xo -= a/b, (a,b) = i, provided that Ial < Ibl1/'~-~, with c > 0 an arbitrary fixed number and Ibl large enough. Conversely, if F(xo, y) has a linear factor, then there exists a v such that (x0)v < c29(h(xo)) I/'~+~. We see now that the equation (4.4) has only a finite number of solutions if 0 ~ Irl < i. Together with the previous result this shows the finiteness of the number of its solutions and the possibility of their determination for all r ~ 0, +i (if r -- 0 or r = 4-1, then all the solutions are determined trivially). An interesting corollary concerns 'Abel points' on the curve F(x, y) = 0. It is well known that Abel tried to prove the impossibility of a rational point with x = p"/b (p prime, (p, b) = 1) on the Fermat curve x '~ + yn = 1, but overlooked a gap in his argument, as was subsequently noticed by Markoff [139]. Abel's conjecture still remains unproved (its almost up to date state is discussed in [101]). We shall call a rational point on the curve with x = p'~/b an 'Abel's point'; here all the numbers p, u > 0 and b are unknowns. On considering such points on the curve F ( x , y) = 0 with F ( x , y) satisfying the conditions of Theorem 4.2 (that means: F ( x , y) is absolutely irreducible, degy F ( x , y ) = n >_ 2, F(O,O) = O, (O/Oy)F(O,O) ~ 0) we find as a direct consequence of this theorem: Given any ~ > 0 there are only ~nitely m a n y Abel's points on the curve with Ix[ >_ ~ and all of them may be found by effective procedure.

208

IX. Reducibility of Polynomials and Diophantine Equations

Indeed, for an Abel point on the curve it must be that p~ < Ibl (if not, the polynomial F(p~'/b, y) is irreducible), and we have to consider two quotients: ln(xo)p _ u l n p in h(xo) in [bl'

ln(xo)~ _ lnmax(1, Ib/p~]) = 1 - uln_____~p in h(xo) in h(xo) In ibl"

Hence, by Theorem 4.2, the only limit points for lnp~'/in IbI (with Abel's point xo = p~'/b on the curve) can be 1/n and 1 - 1/n. In both cases the limit points for x0 are formed only by the point x = 0. It is interesting to note that the curve x - y(y + 1) "-1 = 0 contains infinitely m a n y Abel points with both limit points 1/n and 1 - 1/n for the quotients lnp~'/in Ibl.

5. P r o o f s o f t h e R e d u c i b i l i t y T h e o r e m s We start with Theorem 4.1. Suppose (a, E l ) = 1, ~ > 0, and h > 0 is an integer such t h a t cnh > 1. For a prime p I a we construct the auxiliary power series by the method described in w1673

W(x) = Bo(x) + B1 (x)f(x) + . . . + B,~-l(x)f "-1 (x),

(5.1)

where the Bj (x) are integral polynomials of degrees at most g (1 - 3c)nh, W(xo) ~ 0, where on substituting x = xo into (5.1) the value f(xo) is determined by the sum of the series (1.6) in the p-adic metric (for the chosen p ] a). Setting b g W ( x o ) = Bo + Blt~p + . . . + B,~_l~p

,

B~ = bgB~(xo) c 7/,

we obtain an integral polynomial

L(y) = Bo + B l y + ... + B,~-ly '~-1 for which ( h + l ) / e t,-~l,

,~h(l+em)

B = maxlB~ I < c30 [t~ lal) , cl = 4~n2nl , 0 ~ IL(~p)lp < p-(1-3e)nhordpa ,

(5.3)

where ~p = f(x0)p, and c30 is a value of the type (5.2) with some other numerical value for c25, and C' = max(l, C). Equality (5.1) and the condition ordW(x) > (1 - 3c)nh now show t h a t when we substitute Oq = f(xo)q in L(y) for primes q t a we obtain estimates like those in the right side of (5.3) (it is not guaranteed t h a t L(~q) r 0).

5. Proofs of the Reducibility Theorems

209

Let Op be a root of FI(y). Since L(Op) # O, it follows that Fl(y) and L(y) do not have common zeros, so their resultant R(F1, L) # 0. Denote the roots of Fl(y) in Qp, the algebraic closure of Qp by 01p,..., Odp, where d = deg Fl(y). Then the 8~p axe p-integral, one of them is 0p, and d

IR(FI, L)lp = I] IL(O,p)ip (4.4) -1, where A is defined as in Lemma 2.3, then the assertion of Theorem 4.2 follows from Theorem 4.1. Indeed, in this case Ibl < 4A[a I which corresponds to C = 4A. The inequality (5.5) and analogous inequalities for the other polynomials Fj(y) show that C influences the remainder term in (4.2) as O(ln max(l, C)/in laD, while Lemma 3.3 gives an estimate of O(ln HF/in lal) for this value. One takes account of the archimedean component on the left-hand side of (4.5) only in the case of convergence of the series (1.6) at x = x0. The radius of convergence of the series coincides with the distance from x = 0 of the nearest root of the discriminant D(x), introduced in the proof of Lemma 3.3. Consequently, it is bounded from above by c33H2(n-1)nl F , where c33 = c3a(n, nl). It follows that Ix01 is bounded by the same value. We see that I ln]a/bll = O(lnHF), and then we find

lnHF

lnHF

lnla I -- lnlb ] + O ( l n g F )

lnHF / / l n H F ~ 2) -- lnlb----~+O \ \ lnlb I ] / "

This shows that when Ix01 :> (4A) -1 the contribution of the archimedean component to (4.5) may be moved to the remainder term. The difference between the contributions of the archimedean components in the equalities (4.2) and (4.5) is at most

5. Proofs of the Reducibility Theorems

Elna, pla

lnla ]

lnib ]

211

=O\l-~,].

Thus, indeed, when Ixol _> (4A)-; (4.5) follow from (4.2). Suppose now that Ix01 _< (4A) -1. After taking suitable numbers e and h we have to construct an integral polynomial L(y) which will provide an analogue of (5.3) in the archimedean metrics; 8p is replaced by 8oo and IIp by the ordinary absolute value. In the non-archimedean metrics we are to have inequalities of the type on the right side of (5.3). To attain this aim we rely on the construction of formal power series as described above. Passing to the numbers (substituting x = x0) we do the estimates in archimedean metrics. First of all we estimate the coefficients bl of the power series V(x) (see the proof of Lemma 2.3). We find that

Ibll _h0(n, e)

(5.9)

(we used the estimates for H0, A and 10). Since 10 - s _> (1 - 3e)nh and g < h(1 + el), el = 4n2nle, we finally obtain rr~+~(h+l)~/~ 0 # IL(O~)I < .~ ~ar-:u ~as Is0 [0-a~h)lblh(l+~) '

(5.10)

Note t h a t the height B of the polynomial L(y) is estimated as before, but it should be taken into account t h a t

h(xo) = max(la[, [bl) = [b[,

9"h/~l~lh(l+~) B = m a x l B , I < ~30 ,~, 9

(5.11)

Now let Fl(y) be an irreducible divisor of F(xo, y), with 0~ as a zero. Since L(9~r ~ 0, it follows t h a t F~ (y) and L(y) are coprime and we find from (5.10) and from the estimate (3.4) for the complex roots of FI(y) t h a t

0 r IR(FI,L)] = ]L(O~)I H

]n(~)l
_ n / k (2" = 1, 2 , . . . , r), and so F(a/b, y) factor. If we assume again, as in w that x0 E ~ F and x0 (u, v) = 1 and Ivl < lul then we find that all the p-components inequality up (HF + 1) c4~

where 5 > 0 is arbitrary. It then follows that all the solutions of the Diophanfine equation

F(p', y) = 0 in unknown primes p, integers t > 0, and y satisfy the inequality

max(p', lul) < (HE + 1 p ' , where c41 can be determined explicitly by the degree of F(x, y). A more important corollary of Theorem 6.1 is given by the following theorem concerning binary diophantine equations in which one unknown belongs to a special infinite set of natural numbers [223]. Theorem

6.2 Let A = { a ,~}1o o be a sequence of natural numbers satisfying

the following two conditions: a) For each m the number a,n has a prime divisor p such that the p-component (am)p Of am satisfies the inequality (am)p > Al--x(rn)

6. Further Results and Remarks

215

where x(m) > O, and x(m) ~ 0 as m ~ oo. b) For each prime q all the numbers a,~ with m > mo (q) are divisible by q. Then for any integral polynomial F(x, y), irreducible in Q[x, y], of degree at least 2 with respect to y, and such that F(0, y) has at least one simple root, the diophantine equation F(x,y)=O,

zEA,

yE2r,

(6.2)

has only a finite number of solutions. All the solutions can be determined effectively if the functions x(m) and mo(q) are defined effectively. The proof of this theorem is as follows. We distinguish two cases corresponding to condition a) and b) imposed on the sequence A: the polynomial F(0, y) is reducible or irreducible in Q[y]. If it is reducible, we would obtain the theorem as a corollary of Theorem 6.1 if F(x,y) were absolutely irreducible. By arguments similar to those we described at the beginning of this paragraph we find that if (6.2) has a solution (x0, Y0) with (O/Oy)F(xo, Yo) ~ O, then F(x, y) must be absolutely irreducible. Since F(x,y) is irreducible in Q[x, y] and (O/Oy)F(x, y) has lower degree in y, the equalities

F(xo, yo) = O,

~-~F(xo, Yo) = 0

are independent and only a finitely many points (xo,yo) satisfy them. It is apparent that we may suppose the existence of a solution of (6.2) different from these special points, and hence the absolute irreducibility of F(x, y). If F(O,y) is an irreducible polynomial of degree at least 2, then by Frobenius' theorem there exists an effectively determinable prime number q such that F(0, y) (modq) has no linear factor. Then F(am, y) (modq) for m > too(q) likewise has no linear factors, and the equality F(a,~, y) = 0 for integral y is impossible. Finally, the case of a linear polynomial F(0, y) is covered by Theorem 6.1. Theorem 6.2, being effective, may replace Siegel's theorem on integer points on algebraic curves to give an effective version of Hilbert's irreducibility theorem in its full generality. The corresponding result is as follows [223]. T h e o r e m 6.3 Let F(x,y) be an integral polynomial, irreducible in Q[x,y]. Then for all integers a, except for a finite number of effectively determinable singular numbers a, and for all m > mo(a, F) the polynomials F(a + am, y) are irreducible in Q[y]. Here the am are terms of the sequence A introduced in Theorem 6.2, and m0(a, F) is a quantity which is explicitly determinable from a and the parameters of F. In its turn, Theorem 6.3 supplied with some extra estimates allows one to construct in explicit form the universal subsets of Hilbert sets: those sequences of natural numbers bm (m = 1, 2,...) such that for any irreducible polynomial F ( x , y ) E Q[y] the polynomials F(b,~,y) are irreducible in Q[y], whenever

216

IX. Reducibility of Polynomials and Diophantine Equations

m > too(F). Here mo(F) may be determined in explicit form in terms of the height and the degree of F(x, y). For example, one can take bm = a(m) + m!Pm

(m = 1,2,...),

where a(m) is an unbounded increasing sequence of natural numbers subject to the condition [

lnlnm

a(m)

,

and Pm is a power of a prime number with Pm > (m!) r as m --* ec. Setting, in particular,

a(m) = [exp(lnlnm)l/2],

r

where ~b(m) ---* oo

= lnlnm,

we find for the value too(F) an expression of the form m0 (F) = exp exp max @42, (ln HF) 2),

(6.3)

where c42 is defined by the degree of F(x, y). Hence, bm = [ e x P e l +

m'2 m2

m >9

is an example of a universal subset of Hilbert sets. In 1955 Gilmore and Robinson [79] had proved t h a t universal subsets of Hilbert sets exist, but the sets were thought of as mysterious until Theorem 6.1 became known. Theorem 4.2 admits a generalisation to polynomials F(x, y) from fK[x, y], where K is a field of algebraic numbers of finite degree k over Q. Suppose t h a t S is a full system of nonequivalent canonical valuations (metrics) I Iv on the field K, and Kv is a completion of K in the metric ] Iv. Denote by K" the isomorphic image of K in Kv. For 0 r ,r E K we set (~)v = max(i, I~lSk"),

kv = [Kv: Qv],

H~:(~) = n ( ~ ) v . yES

Let F(x,y) be an irreducible polynomial in I~[x,y], satisfying (1.2) and let f(x) be the power series (1.6) (in which now fv C K). For x0 C K we define the sums f(Xo)v of the series (1.6) in the metric v C S, when the series converges. Suppose t h a t F(x, y) has a decomposition of the form (4.1) in K[y]. Then in K v [y] we have the 'conjugate' decompositions

FV(zo, y) =

F:(y),

and f(xo), = 0v turns out to be a root of one of the polynomials Fy = Fy(y), which we signify as 0v E Fy. Similarly to Theorem 4.2, the next theorem shows the way in which the numbers 0v are distributed among the factors of FV(xo,y) as v changes in S (cf. [224], [225 D.

6. Further Results and Remarks

217

T h e o r e m 6.4 Suppose that H~(xo) > 1 and that dj =

degFi(y)

Then for any j = 1 , 2 , . . . , r

(j = 1 , 2 , . . . , r ) .

we have

l n ( x o ) , / l n H ~ ( x o ) = d-i+On

\lnHK(xo)]

'

(6.4)

O.EFi~

where the sum is taken over all v E S for which Ov is a root of F f ( y ) , where [-F-] is the size of the polynomial F ( x , y ) , and the symbol 0 implies a value which can be effectively determined by the degrees of F and of K. Since b o t h Theorems 4.2 and 6.4 wore proved on the basis of the theory of diophantine approximation it is interesting to notice t h a t in fact these theorems are equivalent to some assertions on diophantine approximation to the values 0, (for all v E S(x0), where S(xo) is t h a t subset of S consisting of those v for which 0v is defined). Thus, for instance, Theorem 6.4 both implies and may be derived from the following theorem (cf. [225]). T h e o r e m 6.5 For v E S(xo) set

nk k,,

ln(xo)v In H~:(Xo)

where e~ = 1 in the case of an archimedean v, and ev = 0 for a nonarchimedean v. Then the system of inequalities IP(O~)I. < i ~ - ~

(v e S(x0))

(6.5)

in nonzero polynomials P(y) E I~[y] of degrees at most n - 1, has only a finite number of solutions whenever H~;(xo) >_ ([-F-]+ 1) ~'3/~2,

(6.6)

for e > O, where c43 is effectively determined by the degrees of F ( x , y) and of K. I f r O, then the system (6.5) has infinitely many solutions in polynomials P(y). Instead of inequalities (6.5) it is reasonable to consider inequalities of a different type. Set n ln(x0), (1 + c), #" = k---~" l n H ~ ( x o ) and for P ( y ) = rr0 + 7fly + . . . + 7rn-ly ~-1 E IN[y] define a height

n=(P) = 1-I/Iv, yES

Hv = max([fro[v,..., [rCn_l[~) k~.

218

IX. Reducibility of Polynomials and Diophantine Equations

Then one can show that the system of inequalities [P(Ov)[~ < H~(P)-~H,~

(v E S(xo))

for ~ > 0 has only a finite number of solutions in polynomials P(y) 6 IK[y] whenever a condition of type (6.6) holds, and has infinitely many solutions if ~