Diophantine Approximations and Value Distribution Theory
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1239
Paul Vojta
Diophantine Approximations and Value Distribution Theory
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Paul Vojta Department of Mathematics, Yale University New Haven, CT 06529, USA
Mathematics Subject Classification (1980): 11 J 68, 30 D 35 ISBN 3-540-17551-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17551-2 Springer-Verlag New York Berlin Heidelberg
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Introduction Finding all solutions of a given system of diophantine equations has been shown to be an unsolvable problem, in general. More tractable, although still difficult, is the problem of determining whether the system has a finite number of solutions over every ring of integers of every number number field, possibly localized at a finite number of places. Or, one might ask the same question about k-rational solutions of the system. The answer to both questions is known if the system of equations defines a curve, in the sense of algebraic geometry. Indeed, if the genus of the projective closure of the curve is zero, then there are always an infinite number of k-rational points for sufficiently large k , and similarly for integral points if there are at most two points at infinity. If there are three or more points at infinity, however, then finiteness always holds. If the genus is equal to one, then over a sufficiently large number field the curve is an elliptic curve with an infinite number of rational points, but finiteness always holds for integral points. Finally, if the genus is greater than one, then finiteness holds for rational as well as integral points. In each case the answer depended only on algebraic-geometric invariants of the curve, and in fact, the "function field case" of the above questions has provided much insight. More classically, though, the above invariants are also invariants of the associated Riemann surfaces, and the above answers have close parallels in the theory of holomorphic maps to Riemann surfaces. Indeed, there exists a nonconstant holomorphic map from C to a given compact Riemann surface of genus g if and only if g 1 (Picard's theorem), and in the non-compact case, a Riemann surface of genus g missing 8 > 0 points admits such a map if and only if g 0 and s < 3. Thus a curve has an infinite set of rational (resp. integral) points if and only if the associated compact (resp
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