Graduate Studies in Mathematics 1996; 397 pp; hardcover Volume: 9 Reprint/Revision History: reprinted with corrections 1997 ISBN10: 0821802674 ISBN13: 9780821802670 List Price: US$75 Member Price: US$60 Order Code: GSM/9
 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. Readership Graduate students and research mathematicians interested in number theory. 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 Table of Contents  An invitation to arithmetic geometry (entire volume)
 Description of the chapters
 Integral closure (Chapter I)
 Plane curves (Chapter II)
 Factorization of ideals (Chapter III)
 The discriminants (Chapter IV)
 The ideal class group (Chapter V)
 Projective curves (Chapter VI)
 Nonsingular complete curves (Chapter VII)
 Zetafunctions (Chapter VIII)
 The RiemannRoch Theorem (Chapter IX)
 Frobenius morphisms and the Riemann hypothesis (Chapter X)
 Further topics (Chapter XI)
 Appendix (Chapter XII)
 Glossary of notation
 Index
 Bibliography
