罗森编著的《函数域中的数论》内容介绍:基本数论和整数环的算术性质有关,在早期数论的发展过程中,学者已经注意到整数环和有限域上的多项式环之间的很多共同性质,例如,fermat和euler定理、wilson定理、二次(更高)互反性、素数定理以及算术级数中素数上的dirichlet定理,他们都存在着极大的相似性。本书在介绍完函数域上的基本资料以后,接下来深入剖析全局函数域和代数数域之间的相似性。内容丰富,包括abc-猜想、素数原根的artin猜想、brumer-stark猜想,drinfeld模型,类数公式和平均值定理。本书的前几章高年级本科生也可以理解,后面的章节更适合于研究生和数学专业以及相关专业的专家学者,增加了许多研究代数数域和代数函数域之间的关系的内容,本书也可以作为深入学习的基础教程。
Preface
1 Polynomials over Finite Fields
Exercises
2 Primes, Arithmetic Functions, and the Zeta Function
Exercises
3 The Reciprocity Law
Exercises
4 Dirichlet L-Series and Primes in an Arithmetic Progression
Exercises
5 Algebraic Function Fields and Global Function Fields
Exercises
6 Weil Differentials and the Canonical Class
Exercises
7 Extensions of Function Fields, Riemann-Hurwitz,and the ABC Theorem
Exercises
8 Constant Field Extensions
Exercises
9 Galois Extensions - Hecke and Artin L-Series
Exercises
10 Artin's Primitive Root Conjecture
Exercises
11 The Behavior of the Class Group in Constant Field Extensions
Exercises
12 Cyclotomic Function Fields
Exercises
13 Drinfeld Modules: An Introduction
Exercises
14 S-Units, S-Class Group, and the Corresponding L-Functions
Exercises
15 The Brumer-Stark Conjecture
Exercises
16 The Class Number Formulas in Quadratic
and Cyclotomic Function Fields
Exercises
17 Average Value Theorems in Function Fields
Exercises
Appendix: A Proof of the Function Field Riemann Hypothesis
Bibliography
Author Index
Subject Index