An Invitation to Arithmetic Geometry

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American Mathematical Soc., Feb 22, 1996 - Mathematics - 397 pages
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.

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Integral closure
Plane curves
Factorization of ideals
The discriminants
The ideal class group
Projective curves
Nonsingular complete curves
The RiemannRoch Theorem
Frobenius morphisms and the Riemann hypothesis
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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.

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