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2 results

  • 5 - The Proof of Faltings’s Theorem

  • Hideaki Ikoma, Shu Kawaguchi, Doshisha University, Kyoto, Atsushi Moriwaki, Kyoto University, Japan
  • Book:
    The Mordell Conjecture
    Published online:
    13 January 2022
    Print publication:
    03 February 2022, pp 117-159

  • 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.

The Mordell Conjecture
  • A Complete Proof from Diophantine Geometry
  • Hideaki Ikoma, Shu Kawaguchi, Atsushi Moriwaki
  • Published online:
    13 January 2022
    Print publication:
    03 February 2022
  • The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell–Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.