An Invitation to Arithmetic Geometry (Graduate Studies in Mathematics, Vol 9) GSM/9

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... 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 The interaction between number theory and algebraic geometry has been especially fruitful. 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 one-dimensional 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. "Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... 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 The interaction between number theory and algebraic geometry has been especially fruitful. 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 one-dimensional 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."-- Résumé de l'éditeur This Work Brings Out The Deep Analogies Between Number Theory, Commutative Algebra And Algebraic Geometry. It Has Been Designed To Emphasize The Interconnection Between The Two Fields Of Mathematics, And Presents In A Unified Manner The Basic Tools And Concepts Of Number Theory And Algebraic Geometry. Containing Worked-out Examples And Suggested Exercises, The Work Was Written As A Textbook For A Year-long Introduction To Arithmetic Geometry. A text for a year-long introduction to arithmetic geometry, leading the reader in a logical progression from the local to the global properties of rings and curves through the study of class groups and of the Riemann-Roch theorem. Emphasizes the connection between algebraic number and algebraic curve theory, with worked examples and exercises. Includes reference appendices and a glossary. For graduate students and research mathematicians Presents some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry. This book presents the deep analogies between them. It also stresses on the geometric viewpoint.