INTRODUCTION

This book demonstrates the applications of a fundamental inequality to the resolution of various general families of equations over function fields. The families concerned are the analogues of certain classical equations over number fields, and therefore we shall commence by discussing the researches of Thue, Mordell, Siegel and Baker on the theory of Diophantine equations. This discussion will be followed by an exposition of the problems surrounding the function field analogues, and an analysis of the contribution made towards this latter subject, principally by Osgood and Schmidt. The introduction will conclude with a summary of the results to be established herein, as consequences of the fundamental inequality.

The techniques of Diophantine approximation have long been applied to the study of general families of Diophantine equations. In his celebrated paper [42] of 1909, the Norwegian mathematician Thue employed approximation methods to prove that the equation f(x,y) = 1 has only finitely many solutions in integers x and y; here f denotes any irreducible binary form with rational coefficients and with degree at least three. The connexion between Thue equations, Diophantine approximation and transcendence will be discussed further in Chapter VI. The Johnian Mordell, in his Smith's prize essay of 1913 [25], demonstrated that for certain values of the non-zero integer m, the equation y2 = x3 + m has only finitely many integer solutions. Actually Mordell then conjectured that for certain other values of m there exist infinitely many integer solutions, but this was confounded by his own result [27] that the elliptic equation y2 = g(x) has only finitely many solutions in integers x, y, where g denotes a cubic polynomial with integer coefficients which contains no squared linear factor.

At the International Congress of Mathematicians held in Paris in 1900, David Hilbert proposed a list of 23 problems intended to engage the attention of researchers in mathematics. The tenth problem requested a universal algorithm for deciding whether any Diophantine equation is soluble in integers. Although as proved by Matijasevic in 1970, no such general algorithm actually exists, this century has generated a substantial body of knowledge on the more limited, but tractable, problems associated with equations in two variables only. In particular, the works of Thue, Siegel and Baker, which are discussed within in some detail, have been particularly fruitful, and mention should also be made on a most recent outstanding result, namely Faltings' establishment of the Mordell conjecture.

The aim of this book is to provide a connected and self-contained account of researches on similar problems appertaining to equations over function fields. The purpose throughout is the production of complete algorithms for the effective resolution of the families of equations under consideration. The algorithms obtained turn out to be much simpler and more efficient than those provided by the analysis for number fields. This is a reflection of the strength of the fundamental inequality which forms the crux of the approach, and which may also be regarded as a loose analogue of Baker's celebrated lower bounds for linear forms in logarithms. The particular families of equations treated are those of the Thue, hyperelliptic, genus zero and genus one types. Several worked examples are given illustrating the general methods.

PRELIMINARIES

In this chapter we shall expand the results obtained in Chapter IV on the complete resolution of equations of genera 0 and 1 by determining explicit bounds on the heights of all their integral solutions, as expressed in Theorems 9 and 12. It is to be remarked that these bounds are linearly dependent on the height of the equation concerned, in contrast with the classical case when the bounds established by Baker and Coates [8] are of multiply exponential growth. Our method of proof consists of a detailed analysis of the construction of the algorithms derived in Chapter IV, coupled with an estimation of the various parameters involved at each stage thereof. Central to the constructions are Puiseux's theorem (see Chapter I) and the Puiseux expansions; in this section we shall establish the requisite bounds on the coefficients in any Puiseux expansion. First, however, we shall require a bound on the genus of any finite extension of k (z). Throughout this chapter we shall denote by L a sufficiently large finite extension of K, and, unless otherwise stated, for f in L H(f) will denote the sum − Σ min(0,v(f)) taken over all the valuations v on L. If K' is any field lying between K and L then we denote by GK, the integer [L:K'] (gK,−1), where gK, is the genus of K'/k and [L:K'] is the degree of L over K'; we also recall that the height in K' of any element f is given by HK'(f) = H(f)/[L:K'].

INTRODUCTION

This chapter will be devoted to the construction of algorithms whereby may be determined the complete set of integral solutions of all equations of genera 0 and 1 over an arbitrary algebraic function field K. As usual [40] we shall assume that in the former case the curve associated with the equation possesses at least three infinite valuations. As for the Thue and hyperelliptic equations already solved, in each case we shall establish a simple criterion for the equation to possess an infinity of solutions in 0. In the next chapter a constructive examination of the algorithms will be used to establish explicit bounds on the heights of the solutions. The chief ingredient in those proofs is a direct recursive technique for determining the coefficients in a Puiseux series (see Lemma 5 and [10]). The bounds obtained are linear functions of the height of the original equation and thus are not exponential, as are the bounds established by Baker and Coates [8] in the case of algebraic numbers. This provides further evidence of the strength and power of our fundamental inequality which again plays the crucial role in the analysis. These results on the heights of solutions may be viewed in another way, as a complement to the celebrated theorem of Manin and Grauert, which had proved the analogue for function fields of Faltings' recent result. We recall that, as a consequence of this theorem, the heights of all solutions in K, not just those in 0, of any equation of genus 2 or more, are bounded.

Hitherto this book has effectively resolved the problem of determining the complete set of integral solutions to certain general families of equations. The fundamental inequality has formed the crux of the argument in each analysis, and it has led to the solution in Chapter II of the Thue equation, in Chapter III of the hyperelliptic equation, and in Chapter IV of equations of genera 0 and 1. In Chapter VII we succeeded in dealing with the Thue and hyperelliptic equations over fields of positive characteristic, and it was the appropriate extension of the fundamental inequality to such fields which again provided the crucial step. In the case of positive characteristic it is possible for the heights of the integral solutions to be unbounded, but this cannot occur over fields of characteristic 0, and for that case we determined explicit bounds for each of the various families solved. The fundamental inequality contributed the essence to each of those proofs also, and it is the purpose of this concluding chapter to illustrate a further range of applications for the inequality by studying briefly the superelliptic equation. Here the inequality is employed in a rather different fashion from previously, and this new approach will in fact lead to explicit bounds on the heights of all the solutions, not just those integral. Explicit bounds for non-integral solutions have only been obtained by Schmidt [36] in the case of certain Thue equations, and stronger bounds may be deduced from our methods as below.