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) 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).