Diophantine Equations
This will be a short introduction to what I know about diophantine equations. Since Fermat this topic has fascinated the best minds in mathematics, as well as many amateurs. Progress resulted not just from bettering one’s predecessors in what they did, bu
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1. Introduction This will be a short introduction to what I know about diophantine equations. Since Fermat this topic has fascinated the best minds in mathematics, as well as many amateurs. Progress resulted not just from bettering one's predecessors in what they did, but also from the introduction of new methods and insights. Kummer's introduction of ideal numbers, Weil's invention of abstract algebraic varieties, and the recent introduction of modular elliptic curves are examples of this. This makes it difficult to predict the future, but since this is the main topic of this volume, I will still try to make some remarks about it. Anyway the difficulty of predicting has not deterred experts from trying and sometimes failing. Hilbert's Paris problems and his estimate about the difficulty of the Riemann hypothesis, respectively proving the transcendence of 2-Jl, are famous examples of success and failure. We leave it to future readers to locate our effort between these two benchmarks.
2. What Has Been Done? In diophantine geometry one looks for solutions of polynomial equations which lie either in the integers, or in the rationals, or in their analogues for number fields. Such polynomial equations {F; (T1 , ... , Tn) = 0} define a subscheme of affine space An over the integers which can have points in an arbitrary commutative ring R. Such points are just the common zeros of the polynomials F;. By gluing affine schemes, one defines general schemes, just as a manifold is obtained by gluing small open sets in .!Rn. For example, projective space lP'n is glued from n + I affine pieces An, and homogeneous polynomials in n + 1 variables define projective subschemes. Our basic problem can be reformulated as the search for integral or rational points on such schemes X. For projective or more generally proper schemes X one need not distinguish between integral and rational points. Of course this formulation of the problem is far too general to be useful, except from the point of view of mathematical logic where one can show that there exists no general algorithm to solve it. For more positive results one has to restrict the choice of X. As a first approach we restrict the dimension, or better, the relative dimension over the integers. In dimension zero one has to check whether a polynomial F(T) has a rational zero. As one can easily bound the possible denominator of such a zero, this amounts to checking a finite list and poses no great theoretical challenges. B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
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G.FALTINGS
Thus the first interesting case occurs for smooth curves. The most important invariant of such a curve is its genus, or the topological genus of the set of complex points of its compactification. For example, a nonsingular plane curve of degree d (defined by one homogeneous polynomial of degree d, in three variables) has genus (d- 1) · (d- 2)/2. In the case of singularities the genus drops. Traditionally we distinguish between the elliptic, para
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