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Nevanlinna-Cartan theory over function fields and a diophantine equation. (English) Zbl 0880.11049
J. Reine Angew. Math. 487, 61-83 (1997); correction ibid. 457, 235 (1998).
The primary purpose of this paper is to develop Nevanlinna theory for rational maps from a complex projective variety to projective space, relative to a divisor consisting of hyperplanes in general position. Following E. I. Nochka [Sov. Math., Dokl. 27, 377-381 (1983); translation from Dokl. Akad. Nauk SSSR 269, 547-552 (1983; Zbl 0552.32024)], the image of the rational map may be contained in a proper linear subspace. As an application the paper gives some evidence for a conjecture of S. Lang: if $$X$$ is a projective variety defined over a number field, and if the corresponding complex manifold is hyperbolic, then $$X$$ has only finitely many rational points over the number field. This paper shows that the examples of K. Masuda and J. Noguchi [Math. Ann. 304, 339-362 (1996; Zbl 0844.32018)] have only finitely many rational points over function fields of any dimension. It also shows that such varieties over a number field have only finitely many points whose coordinates are all $$S$$-units; the reason for the weaker conclusion in this case is related to the fact that certain generalizations of the $$abc$$ conjecture are known over function fields but not over number fields.
Reviewer: P.Vojta (Berkeley)

##### MSC:
 11G35 Varieties over global fields 14G05 Rational points 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables 14G25 Global ground fields in algebraic geometry
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