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 |