p进数，p进分析和ζ函数

Chapter Ⅰ p-adic numbers

1. Basic concepts

2. Metrics on the rational numbers

Exercises

3. Review of building up the complex numbers

4. The field of p-adic numbers

5. Arithmetic in Qp

Exercises

Chapter Ⅱ p-adic interpolation of the Riemann zeta-function

1. A formula for .（2k）

2. p-adic interpolation of the functionf（s） = as

Exercises

3. p-adic distributions

Exercises

4. Bernoulli distributions

5. Measures and integration

Exercises

6. The p-adic -function as a Mellin-Mazur transform

7. A brief survey （no proofs）

Exercises

Chapter Ⅲ Building up Ω

1. Finite fields

Exercises

2. Extension of norms

Exercises

3. The algebraic closure of Qp

4. Ω

Exercises

Chapter Ⅳ p-adic power series

1. Elementary functions

Exercises

2. The logarithm, gamma and Artin-Hasse exponential functions

Exercises

3. Newton polygons for polynomials

4. Newton polygons for power series

Exercises

Chapter Ⅴ Rationality of the zeta-function of a set of equations over a finite field

1. Hypersurfaces and their zeta-functions

Exercises

2. Characters and their lifting

3. A linear map on the vector space of power series

4. p-adic analytic expression for the zeta-function

Exercises

5. The end of the proof

Bibliography

Answers and Hints for the Exercises

Index

在数论，表示理论等许多现代数学研究领域中，p进分析占据着非常重要的地位。《P进数，P进分析和ζ函数（第2版）（英文版）》是p进分析的入门教材。主要分两部分内容，首先论述p进分析理论的基本思想，其次介绍p进理论的两个重要应用，即在黎曼ζ函数值为负整数时的p进内插和在一个有限域内方程组的δ函数有理性的证明。可供数学系数论专业的研究生和研究人员参考。