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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....

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You can find a good explanation of the proof of the (effective) Mordell conjecture based on such an inequality in:

Parshin, A. N.: Application of ramified coverings in the theory of Diophantine equations. Math. USSR Sbornik 66 (1990), no. 1, 249–264.

The corresponding proof for function fields has been published by Parshin before in:

Parshin, A. N.: Algebraic curves over function fields. Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219.

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