Diophantine Geometry: An Introduction / Edition 1

Paperback (Print)
Buy New
Buy New from BN.com
$61.07
Used and New from Other Sellers
Used and New from Other Sellers
from $51.09
Usually ships in 1-2 business days
(Save 21%)
Other sellers (Paperback)
  • All (11) from $51.09   
  • New (7) from $51.09   
  • Used (4) from $80.74   

Overview

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

Read More Show Less

Editorial Reviews

From the Publisher
"In this excellent 500-page 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 Mordell-Weil 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 C—E form the core of the book and can be read by any reader already acquainted with algebraic number theory, classical (i.e., not scheme-theoretical) 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 algebraic-geometric 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 C—E, 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 higher-dimensional 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)

Read More Show Less

Product Details

  • ISBN-13: 9780387989815
  • Publisher: Springer New York
  • Publication date: 3/1/2000
  • Series: Graduate Texts in Mathematics Series , #201
  • Edition description: 2000
  • Edition number: 1
  • Pages: 561
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.18 (d)

Table of Contents

A The Geometry of Curves and Abelian Varieties.- A.1 Algebraic Varieties.- A.2 Divisors.- A.3 Linear Systems.- A.4 Algebraic Curves.- A.5 Abelian Varieties over C.- A.6 Jacobians over C.- A.7 Abelian Varieties over Arbitrary Fields.- A.8 Jacobians over Arbitrary Fields.- A.9 Schemes.- B Height Functions.- B.1 Absolute Values.- B.2 Heights on Projective Space.- B.3 Heights on Varieties.- B.4 Canonical Height Functions.- B.5 Canonical Heights on Abelian Varieties.- B.6 Counting Rational Points on Varieties.- B.7 Heights and Polynomials.- B.8 Local Height Functions.- B.9 Canonical Local Heights on Abelian Varieties.- B.10 Introduction to Arakelov Theory.- Exercises.- C Rational Points on Abelian Varieties.- C.1 The Weak Mordell—Weil Theorem.- C.2 The Kernel of Reduction Modulo p.- C.3 Appendix: Finiteness Theorems in Algebraic Number Theory.- C.4 Appendix: The Selmer and Tate—Shafarevich Groups.- C.5 Appendix: Galois Cohomology and Homogeneous Spaces.- Exercises.- D Diophantine Approximation and Integral Points on Curves.- D.1 Two Elementary Results on Diophantine Approximation.- D.2 Roth’s Theorem.- D.3 Preliminary Results.- D.4 Construction of the Auxiliary Polynomial.- D.5 The Index Is Large.- D.6 The Index Is Small (Roth’s Lemma).- D.7 Completion of the Proof of Roth’s Theorem.- D.8 Application: The Unit Equation U + V = 1.- D.9 Application: Integer Points on Curves.- Exercises.- E Rational Points on Curves of Genus at Least 2.- E.I Vojta’s Geometric Inequality and Faltings’ Theorem.- E.2 Pinning Down Some Height Functions.- E.3 An Outline of the Proof of Vojta’s Inequality.- E.4 An Upper Bound for h?(z, w).- E.5 A Lower Bound for h?(z,w) for Nonvanishing Sections.- E.6 Constructing Sections of Small Height I: Applying Riemann—Roch.- E.7 Constructing Sections of Small Height II: Applying Siegel’s Lemma.- E.8 Lower Bound for h?(z,w) at Admissible Version I.- E.9 Eisenstein’s Estimate for the Derivatives of an Algebraic Function.- E.10 Lower Bound for h?(z,w) at Admissible: Version II.- E.11 A Nonvanishing Derivative of Small Order.- E.12 Completion of the Proof of Vojta’s Inequality.- Exercises.- F Further Results and Open Problems.- F.1 Curves and Abelian Varieties.- F.1.1 Rational Points on Subvarieties of Abelian Varieties.- F.1.2 Application to Points of Bounded Degree on Curves.- F.2 Discreteness of Algebraic Points.- F.2.1 Bogomolov’s Conjecture.- F.2.2 The Height of a Variety.- F.3 Height Bounds and Height Conjectures.- F.4 The Search for Effectivity.- F.4.1 Effective Computation of the Mordell—Weil Group A(k).- F.4.2 Effective Computation of Rational Points on Curves.- F.4.3 Quantitative Bounds for Rational Points.- F.5 Geometry Governs Arithmetic.- F.5.1 Kodaira Dimension.- F.5.2 The Bombieri-Lang Conjecture.- F.5.3 Vojta’s Conjecture.- F.5.4 Varieties Whose Rational Points Are Dense.- Exercises.- References.- List of Notation.

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)