So I've read (for instance in the introduction to R.S de Jong's thesis) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection theory. Most notably we lack a "good" canonical class inequality, for instance Bost, Mestre and Moret-Bailly showed in this paper that the analogue of Bogomolov-Miyao is false.

I was wondering if someone could explain the "proof" of Mordell which would rely on this inequality? I might well be explained in the Bost,Mestre and Moret-Bailly paper, but my french is not really up to the task....

So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection theory. Most notably we lack a "good" canonical class inequality, for instance Bost, Mestre and Moret-Bailly showed in this paper that the analogue of Bogomolov-Miyao is false.

I was wondering if someone could explain the "proof" of Mordell which would rely on this inequality? I might well be explained in the Bost,Mestre and Moret-Bailly paper, but my french is not really up to the task....