The Mordell Conjecture

A Complete Proof from Diophantine Geometry

Hideaki Ikoma, Shu Kawaguchi, Atsushi Moriwaki

• Publisher: Cambridge University Press
• Year: 2022
• Print ISBN: 9781108845953
• eBook ISBN: 9781108998192
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Contents
• Preface
• 1 What Is the Mordell Conjecture (Faltings's Theorem)?
• 2 Some Basics of Algebraic Number Theory
• 3 Theory of Heights
• 4 Preliminaries for the Proof of Faltings's Theorem
• 5 The Proof of Faltings's Theorem
• References
• Notation
• Index of Symbols
• Index