Geodesic
A~function-field analog of the Mordell conjecture: a~noncompact version
Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 61-81.

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The author proves a geometric analog of Manin's theorem on curves over function fields in the noncompact complex analytic situation. The proof relies on a “theorem on domination” in which the construction of the semistable reduction of a family of noncompact curves is carried out in terms of hyperbolic complex analysis. Bibliography: 23 titles. 
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M. G. Zaidenberg. A~function-field analog of the Mordell conjecture: a~noncompact version. Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 61-81. http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a2/

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