An Invitation to Arithmetic Geometry (Google eBook)

Front Cover
American Mathematical Soc., Feb 22, 1996
2 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. ... 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.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Integral closure
5
Plane curves
35
Factorization of ideals
85
The discriminants
131
The ideal class group
157
Projective curves
193
Nonsingular complete curves
225
Zetafunctions
269
The RiemannRoch Theorem
305
Frobenius morphisms and the Riemann hypothesis
339
Further topics
361
Appendix
375
Copyright

Common terms and phrases

Popular passages

Page 25 - Let 0 * M' * M ^-* M" . 0 be an exact sequence of A-modules. Then M is noetherian if and only if M' and M
Page 20 - Recall that an ideal / in a ring A is said to be finitely generated if there exist finitely many elements GI , . . . , cr in / such that / = {aici H i- arcr \ tn A, i = 1, . . . , r).
Page 16 - Let B be the integral closure of A in L. Then B is finitely generated as a K -algebra, hence Noetherian.

References to this book

All Book Search results »

Bibliographic information