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Summary

Chapter 4 is devoted to several fundamental results of Diophantine geometry such as Siegel's lemma (Lemma 4.1 and Proposition 4.3) and Roth's lemma (Theorem 4.20). Besides them, we also introduce Guass’s lemma, the Mahler measure, the height of a polynomial, Gelfond’s inequality, the index with respect to a weight, the Wronskian, the norm of an invertible sheaf, the height of a norm and the local Eisenstein theorem. We will use them in Chapter 5. Because our purpose is to give a proof of Faltings's theorem in not too many pages, we touch on only the essential results of Diophantine geometry.

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.

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4 - Preliminaries for the Proof of Faltings’s Theorem

Summary

The Mordell Conjecture

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2 results

4 - Preliminaries for the Proof of Faltings’s Theorem

Summary

The Mordell Conjecture