Nevanlinna Theory over Function Fields

Abstract

We develop here Nevanlinna theory described in Sects. 4.1 and 4.2 for holomorphic curves over algebraic function fields. This is understood as an approximation theory of algebraic functions by algebraic functions. Vojta (Diophantine Approximations and Value Distribution Theory, 1987) formulated Diophantine approximation theory from the viewpoint of Nevanlinna theory, noticing their analogies. From that viewpoint Nevanlinna theory is an approximation theory of complex numbers by transcendental meromorphic functions. This brought a new viewpoint to the both theories and has activated their research. The theory over algebraic function fields is considered to be situated in the middle of them.