**Graduate Studies in Mathematics**

Volume: 9;
1996;
397 pp;
Hardcover

MSC: Primary 11; 14;
Secondary 12; 13

Print ISBN: 978-0-8218-0267-0

Product Code: GSM/9

List Price: $75.00

Individual Member Price: $60.00

**Electronic ISBN: 978-1-4704-2069-7
Product Code: GSM/9.E**

List Price: $75.00

Individual Member Price: $60.00

# An Invitation to Arithmetic Geometry

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*Dino Lorenzini*

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.

#### Table of Contents

# Table of Contents

## An Invitation to Arithmetic Geometry

- Cover Cover11 free
- Title vi7 free
- Copyright vii8 free
- Contents x11 free
- Preface xiv15 free
- Description of the chapters 119 free
- Chapter I. Integral closure 523
- Chapter II. Plane curves 3553
- 1. Introduction 3553
- 2. Rings of functions 3856
- 3. Points and maximal ideals 4563
- 4. Morphisms of curves 4765
- 5. Singular points 5169
- 6. Localization 5775
- 7. More on dimension 6381
- 8. Local principal ideal domains 6785
- 9. Localization of modules 7189
- 10. Hilbert's Basis Theorem 7694
- 11. More rings of functions 7795

- Chapter III. Factorization of ideals 85103
- Chapter IV. The discriminants 131149
- Chapter V. The ideal class group 157175
- 1. Introduction 157175
- 2. Definition of the ideal class group 158176
- 3. Rings with finite quotients 160178
- 4. The case of number fields 163181
- 5. The case of function fields (I) 167185
- 6. Absolute values and valuations 168186
- 7. Archimedian absolute values and the product formula 172190
- 8. The case of function fields (II) 176194
- 9. Valuations and local principal ideal domains 181199
- 10. Nonsingular complete curves 183201

- Chapter VI. Projective curves 193211
- 1. Introduction 193211
- 2. Projective spaces 194212
- 3. Plane projective curves 199217
- 4. Projective transformations 203221
- 5. Conics 205223
- 6. Projections 206224
- 7. The tangent line at a point of a projective curve 209227
- 8. Functions on a projective curve 213231
- 9. Projective curves and valuations 217235
- 10. The intersection of two projective curves 221239

- Chapter VII. Nonsingular complete curves 225243
- Chapter VIII. Zeta-functions 269287
- 1. Introduction 269287
- 2. The Riemann ζ-function 274292
- 3. ζ-functions and Euler products 276294
- 4. Power series 277295
- 5. The zeta-function of a nonsingular curve 279297
- 6. The rationality of the zeta-function 284302
- 7. The functional equation 288306
- 8. Jacobi sums 292310
- 9. Relations between class numbers 296314

- Chapter IX. The Riemann-Roch Theorem 305323
- Chapter X. Frobenius morphisms and the Riemann hypothesis 339357
- Chapter XII. Appendix 375393
- Glossary of notation 383401
- Bibliography 393411
- Index 387405
- Back Cover Back Cover1416

#### 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 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