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Diophanengeometrie: Eine Einführung von Marc Hindry: Neu-
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eBay-Artikelnr.:284428124461
Artikelmerkmale
- Artikelzustand
- Book Title
- Diophantine Geometry: An Introduction
- Publication Date
- 2000-03-23
- Pages
- 561
- ISBN
- 9780387989754
Über dieses Produkt
Product Identifiers
Publisher
Springer New York
ISBN-10
0387989757
ISBN-13
9780387989754
eBay Product ID (ePID)
1804964
Product Key Features
Number of Pages
Xiii, 561 Pages
Language
English
Publication Name
Diophantine Geometry : an Introduction
Publication Year
2000
Subject
Number Theory, Geometry / Algebraic
Type
Textbook
Subject Area
Mathematics
Series
Graduate Texts in Mathematics Ser.
Format
Hardcover
Dimensions
Item Weight
76.5 Oz
Item Length
9.3 in
Item Width
6.1 in
Additional Product Features
Intended Audience
Scholarly & Professional
LCCN
99-057467
Dewey Edition
21
Reviews
"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." (Jan-Hendrik Evertse, zbMATH 0948.11023, 2021) "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 readby 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), "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) , "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) , "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." (Jan-Hendrik Evertse, zbMATH 0948.11023, 2021) "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)
Series Volume Number
201
Number of Volumes
1 vol.
Illustrated
Yes
Dewey Decimal
512/.7
Table Of Content
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.2 Discreteness of Algebraic Points.- F.3 Height Bounds and Height Conjectures.- F.4 The Search for Effectivity.- F.5 Geometry Governs Arithmetic.- Exercises.- References.- List of Notation.
Synopsis
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.
LC Classification Number
QA564-609
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- m***m (2298)- Bewertung vom Käufer.Letzte 6 MonateBestätigter KaufI’m thrilled with my recent purchase . The website was user-friendly, and the product descriptions were accurate. Customer service was prompt and helpful, answering all my questions. My order arrived quickly, well-packaged, and the product exceeded my expectations in quality. I’m impressed with the attention to detail and the overall experience. I’ll definitely shop here again and highly recommend from this seller to others. Thank you for a fantastic experience!
- a***n (45)- Bewertung vom Käufer.Letzte 6 MonateBestätigter KaufMistakenly ordered a paperback that I thought was a hardcover, not sellers fault; it was described properly on the listing. Seller still processed a refund the day I went to return the item and let me keep the item anyway. A+++ service. Book arrived quickly in great condition and for a great price. Thank you so much! Amazing seller!
- n***c (94)- Bewertung vom Käufer.Letzte 6 MonateBestätigter Kaufseller was communicative about my shipment, media mail took a while and tracking wasn't updated frequently, but seller communicated to me very quickly on status. the item came new and wrapped as described, though the packaging in it was packed wasn't sturdy and falling apart when it got to me.