On certain uniformity properties of curves over function fileds

Details On certain uniformity properties of curves over function fileds On certain uniformity properties of curves over function fileds

1999-06-23

arxiv.org/abs/math/9906156v2

Mathematics

Algebraic Geometry

Improved exposition, added references, to appear on Comp. Math

Scientific paper

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain a uniform version of Manin theorem (the Mordell conjecture on rational points for curves of genus g over function fields). By a "function field" is meant the function field of a complex variety V of any dimension. If V is a curve, the uniform bounds will depend on g, on the genus of V and on the cardinality of the degeneracy locus.

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