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.

'This lucid compact book provides a short and direct access to Vojta-Bombieri's proof of Faltings's celebrated theorem. The text itself is mostly self-contained, with all needed results on diophantine geometry presented without unnecessary abstraction, in as concrete a manner as possible. Without doubt, this excellent course will become a standard for anyone wishing to be introduced to the topic of rational points on curves over the rational numbers, and to one of the crowning achievements of the mathematics of our time.'

Vincent Maillot - Centre National de la Recherche Scientifique (CNRS), Paris

'In less than 200 pages, the authors have given a complete treatment to the two most important results in diophantine geometry in the last 100 years: the Mordell–Weil theorem and Faltings’s theorem. This will be a wonderful reference for everybody interested in diophantine geometry with minimal background in number theory and algebraic geometry.'

Shou-Wu Zhang - Princeton University

'This book is a comprehensive introduction, with plenty of motivations, to Mordell conjecture - a deep theorem of Faltings that has far-reaching influences in modern diophantine geometry. Knowledge of algebraic number theory and height theory is considerately refreshed, and the proof of the Mordell conjecture is meticulously structured with all details, which are most helpful for beginners. More experienced readers will appreciate the insights of the authors into the problem and into the domain of diophantine geometry.'

Huayi Chen - University of Paris, Mathematics Institute of Jussieu–Paris Rive Gauche

‘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 Source: zbMATH