On Vojta's proof of the Mordell conjecture

TL;DR

This paper revisits Vojta's proof of the Mordell conjecture, reformulating it through Arakelov geometry and introducing Yuan's arithmetic Siu inequality as a key new element.

Contribution

It provides a re-organization of Vojta's proof using Arakelov geometry and replaces Gillet--Soule's theorem with Yuan's inequality, offering new insights.

Findings

Reformulation of Vojta's proof in Arakelov geometry

Introduction of Yuan's arithmetic Siu inequality into the proof

Enhanced understanding of the proof structure

Abstract

This paper re-organizes Vojta's proof of the Mordell conjecture (i.e. Faltings' theorem) in terms of Arakelov geometry. A new ingredient is to replace an application of Gillet--Soule's arithmetic Riemannn--Roch theorem by that of Yuan's arithmetic Siu inequality.