Nevanlinna Theory and Diophantine Approximation

Abstract. As observed originally by C. Osgood, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. For example, if X is a compact Riemann surface of genus> 1, then there are no non-constant holomorphic maps f: � → X; onthe other hand, if X is a smooth projective curve of genus> 1 over a number field k, then it does not admit an infinite set of k-rational points. Thus non-constant holomorphic maps correspond to infinite sets of k-rational points. This article describes the above analogy, and describes the various extensions and generalizations that have been carried out (or at least conjectured) in recent years. When looked at a certain way, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. The first observation in this direction is due to C. Osgood [1981]; subsequent work has been done by the author, S. Lang, P.-M. Wong, M. Ru, and others. To begin describing this analogy, we consider two questions. On the analytic side, let X be a connected Riemann surface. Then we ask: Question 1. Does there exist a non-constant holomorphic map f: C → X? The answer, as is well known, depends only on the genus g of the compactification X of X, and on the number of points s in X \ X. SeeTable1. On the algebraic side, let k be a number field with ring of integers R, and let X be either an affine or projective curve over k. Let S be a finite set of places of k containing the archimedean places. For such sets S let RS denote the localization of R away from places in S (that is, the subring of k consisting of elements that can be written in such a way that only primes in S occur i