5 - The Proof of Faltings’s Theorem

Summary

Chapter 5 is devoted to giving a detailed proof of Faltings’s theorem (Theorem 5.1), asserting that "any algebraic curve of genus at least two over a number field has only finitely many rational points." We begin by recalling the proof of Liouville’s inequality in classical Diophantine approximation, and compare it with the proof of Faltings’s theorem, that is, Bombieri's version of Vojta's proof. We then explain how to reduce Faltings’s theorem to Vojta’s inequality (Theorem 5.2) and to Theorems 5.4, 5.5, and 5.6, which respectively assert the existence of sections of Vojta divisors having small heights, an upper bound for the indices of the small sections, and a lower bound for the indices of the small sections. We give proofs of Theorems 5.4, 5.5, and 5.6, which completes the proof of Faltings’s theorem. Finally, we give an easy application of Faltings’s theorem to Fermat curves.