The Mordell Conjecture
A Complete Proof from Diophantine Geometry

This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry.

Hideaki Ikoma (Author), Shu Kawaguchi (Author), Atsushi Moriwaki (Author)

9781108845953, Cambridge University Press

Hardback, published 3 February 2022

150 pages
23.5 x 15.7 x 1.4 cm, 0.38 kg

'This concisely written book is a splendid achievement and an indispensable contribution to the mathematical literature on Faltings' theorem and the Bombieri-Vojta approach. It has most definitely succeeded in its intent of giving a compact and complete exposition of its subject matter, namely the elementary proof of one the fundamental results of twentieth-century mathematics.' Jeroen Sijsling, zbMATH

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.

1. What is the Mordell conjecture?
2. Some basics of algebraic number theory
3. Theory of heights
4. Preliminaries
5. The proof of Falthing's theorem.

Subject Areas: Geometry [PBM], Number theory [PBH], Algebra [PBF]