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Softcover ISBN:  9781470467258 
Product Code:  GSM/9.S 
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Book DetailsGraduate Studies in MathematicsVolume: 9; 1996; 397 ppMSC: Primary 11; 14; Secondary 12; 13;
In this volume the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the onedimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes which will aid the reader who goes to the next level of this rich subject.
ReadershipGraduate students and research mathematicians interested in number theory.

Table of Contents

Chapters

Description of the chapters

Chapter I. Integral closure

Chapter II. Plane curves

Chapter III. Factorization of ideals

Chapter IV. The discriminants

Chapter V. The ideal class group

Chapter VI. Projective curves

Chapter VII. Nonsingular complete curves

Chapter VIII. Zetafunctions

Chapter IX. The RiemannRoch Theorem

Chapter X. Frobenius morphisms and the Riemann hypothesis

Chapter XI. Further topics

Chapter XII. Appendix


Additional Material

Reviews

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebrogeometric aspects of number theory, on the other hand. Detailed discussions, full proofs, much effort at thorough motivations, a wealth of illustrating examples, numerous related exercises and problems, hints for further reading, and a rich bibliography characterize this text as an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature.
Zentralblatt MATH 
In order to come straight to the point: this book represents an excellent introduction to Algebraic Number Theory and to Algebraic Curves as well by viewing both theories as part of Commutative Algebra ... all proof are given in full detail and its concept as well thoughtout.
Monatshefte für Mathematik


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In this volume the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the onedimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes which will aid the reader who goes to the next level of this rich subject.
Graduate students and research mathematicians interested in number theory.

Chapters

Description of the chapters

Chapter I. Integral closure

Chapter II. Plane curves

Chapter III. Factorization of ideals

Chapter IV. The discriminants

Chapter V. The ideal class group

Chapter VI. Projective curves

Chapter VII. Nonsingular complete curves

Chapter VIII. Zetafunctions

Chapter IX. The RiemannRoch Theorem

Chapter X. Frobenius morphisms and the Riemann hypothesis

Chapter XI. Further topics

Chapter XII. Appendix

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebrogeometric aspects of number theory, on the other hand. Detailed discussions, full proofs, much effort at thorough motivations, a wealth of illustrating examples, numerous related exercises and problems, hints for further reading, and a rich bibliography characterize this text as an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature.
Zentralblatt MATH 
In order to come straight to the point: this book represents an excellent introduction to Algebraic Number Theory and to Algebraic Curves as well by viewing both theories as part of Commutative Algebra ... all proof are given in full detail and its concept as well thoughtout.
Monatshefte für Mathematik