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.
- Gives explicit and detailed computations for the proof of the Mordell conjecture, some of which are appearing in the literature for the first time
- Offers a concise introduction to Diophantine geometry, treating important theorems and techniques (the theory of heights, Mordell-Weil's theorem, Siegel's lemma, Roth's lemma, etc.)
- Introduces the algebraic number theory and algebraic geometry needed for the proof of the Mordell conjecture
Read more
Reviews & endorsements
'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, zbMATH
See more reviews
Customer reviews

Not yet reviewed
Be the first to review

Log in to review

Review was not posted due to profanity
×

, create a review

(If you're not , sign out)

Please enter the right captcha value

Please enter a star rating.

Your review must be a minimum of 12 words.

How do you rate this item?

Reviews must contain at least 12 words about the product.

×

Product details
- Date Published: February 2022
- format: Hardback
- isbn: 9781108845953
- length: 150 pages
- copublisher: Saiensu sha
- dimensions: 235 x 157 x 14 mm
- weight: 0.38kg
- availability: Available
Table of Contents
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.
Look Inside
Authors
Hideaki Ikoma, Shitennoji University, Osaka
Hideaki Ikoma is Lecturer in the Faculty of Education at Shitennoji University.
Shu Kawaguchi, Doshisha University, Kyoto
Shu Kawaguchi is Professor in the Department of Mathematical Sciences at Doshisha University. He was awarded the Young Scientists' Prize by the Ministry of Education, Culture, Sports, Science and Technology of Japan in 2010.
Atsushi Moriwaki, Kyoto University, Japan
Atsushi Moriwaki is Professor in the Department of Mathematics at Graduate School of Science, Kyoto University. He is the author of Arakelov Geometry (2014) and co-author of Arakelov Geometry over Adelic Curves (2020), and was awarded the Autumn Prize of the Mathematical society of Japan in 2001.