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An Invitation to Arithmetic Geometry
Dino Lorenzini, University of Georgia, Athens, GA

Graduate Studies in Mathematics
1996; 397 pp; hardcover
Volume: 9
Reprint/Revision History:
reprinted with corrections 1997
ISBN-10: 0-8218-0267-4
ISBN-13: 978-0-8218-0267-0
List Price: US$75
Member Price: US$60
Order Code: GSM/9
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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 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.


Graduate students and research mathematicians interested in number theory.


"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. 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 thought-out."

-- 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)
  • Zeta-functions (Chapter VIII)
  • The Riemann-Roch Theorem (Chapter IX)
  • Frobenius morphisms and the Riemann hypothesis (Chapter X)
  • Further topics (Chapter XI)
  • Appendix (Chapter XII)
  • Glossary of notation
  • Index
  • Bibliography
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