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Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over characteristic zero.has anyone used Manin ideas to prove theorem for positive characteristic ?

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Implicitly, my proof (Inv. Math. 104 (1991) 643-646) is an extension of Manin's approach to char p. I first extended Manin's ideas to char p for elliptic curves (Comp. Math. 74 (1990) 247-258) and realized the connection with p-descent so, for the Mordell conjecture, I just used the p-descent directly. See also my papers with Buium (Crelle 460 (1995) 117-126 and Comp. Math. 103 (1996) 1-6).

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  • $\begingroup$ thanks.this papers are about ordinary curves are these methods works in general case? $\endgroup$
    – ali
    Commented Jul 16, 2018 at 17:55
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    $\begingroup$ @ali They work in general but that requires extra effort, see my paper with Abramovich IMRN 5 (1992) 103-115 $\endgroup$
    – Felipe Voloch
    Commented Jul 16, 2018 at 19:54

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