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 ?
Implicitly, my proof (Inv. Math. 104 (1991) 643646) 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) 247258) and realized the connection with pdescent so, for the Mordell conjecture, I just used the pdescent directly. See also my papers with Buium (Crelle 460 (1995) 117126 and Comp. Math. 103 (1996) 16).

$\begingroup$ thanks.this papers are about ordinary curves are these methods works in general case? $\endgroup$– aliJul 16 '18 at 17:55

2$\begingroup$ @ali They work in general but that requires extra effort, see my paper with Abramovich IMRN 5 (1992) 103115 $\endgroup$ Jul 16 '18 at 19:54