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Table of contents (7 chapters)
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“Unlike several other textbooks in this field, the prerequisites are quite modest, so the book is very useful for instance for a graduate course on diophantine geometry. Each chapter goes along with many exercises.” (JanHendrik Evertse, zbMATH 0948.11023, 2021)
"In this excellent 500page volume, the authors introduce the reader to four fundamental finiteness theorems in Diophantine geometry. After reviewing algebraic geometry and the theory of heights in Parts A and B, the MordellWeil theorem (the group of rational points on an abelian variety is finitely generated) is presented in Part C, Roth's theorem (an algebraic number has finitely many approximations of order $2 + \varepsilon$) and Siegel's theorem (an affine curve of genus $g \ge 1$ has finitely many integral points) are proved in Part D, and Faltings' theorem (a curve of genus $g \ge 2$ has finitely many rational points) is discussed in Part E.
Together, Parts CE form the core of the book and can be readby any reader already acquainted with algebraic number theory, classical (i.e., not schemetheoretical) algebraic geometry, and the height machine. The authors write clearly and strive to help the reader understand this difficult material. They provide insightful introductions, clear motivations for theorems, and helpful outlines of complicated proofs.
This volume will not only serve as a very useful reference for the advanced reader, but it will also be an invaluable tool for students attempting to study Diophantine geometry. Indeed, such students usually face the difficult task of having to acquire a sufficient grasp of algebraic geometry to be able to use algebraicgeometric tools to study Diophantine applications. Many beginners feel overwhelmed by the geometry before they read any of the beautiful arithmetic results. To help such students, the authors have devoted about a third of the volume, Part A, to a lengthy introduction to algebraic geometry, and suggest that the reader begin by skimming Part A, possibly reading more closely any material that covers gaps in the reader's knowledge. Then Part A should be used as a reference source for geometric facts as they are needed while reading the rest of the book.
The first arithmetic portion of the book is Part B, which deals with the theory of height functions, functions which measure the "size" of a point on an algebraic variety. These objects are a key tool for the Diophantine study in Parts CE, and the authors, in their characteristically clear and insightful style, fully prove in Part B most results on heights later used in the book.
The book concludes in Part F with a survey of further results and open problems, such as the generalization of Mordell's conjecture to higherdimensional subvarieties of abelian varieties and questions of quantitative and effective results on the solutions of Diophantine problems.
This book is a most welcome addition to the literature. It is well written and renders accessible to students of Diophantine geometry some of the most elegant and beautiful arithmetical results of the 20th century." (Dino J. Lorenzini, Mathematical Reviews)
Authors and Affiliations
Bibliographic Information
Book Title: Diophantine Geometry
Book Subtitle: An Introduction
Authors: Marc Hindry, Joseph H. Silverman
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/9781461212102
Publisher: Springer New York, NY

eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media New York 2000
Hardcover ISBN: 9780387989754Published: 23 March 2000
eBook ISBN: 9781461212102Published: 01 December 2013
Series ISSN: 00725285
Series EISSN: 21975612
Edition Number: 1
Number of Pages: XIII, 561
Topics: Algebraic Geometry, Number Theory