Book:Neal Koblitz/p-adic Numbers, p-adic Analysis, and Zeta-Functions

Subject Matter

 * Analysis
 * $p$-adic Numbers

Contents
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 of $\Q_p$

$\:\:\:$ Exercises

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

$1.$ A formula for $\map \zeta {2k}$

$2.$ $p$-adic interpolation of the function $\map f s = a^s$

$\:\:\:$ Exercises

$3.$ $p$-adic distributions

$\:\:\:$ Exercises

$4.$ Bernoulli distributions

$5.$ Measures and integrations

$\:\:\:$ Exercises

$6.$ The $p$-adic $\zeta$-function as a Mellin-Mazurka transform

$7.$ A brief survey (no proofs)

$\:\:\:$ Exercises

Chapter III Building up $\Omega$

$1.$ Finite fields

$\:\:\:$ Exercises

$2.$ Extensions of norms

$\:\:\:$ Exercises

$3.$ The algebraic closure of $\Q_p$

$4.$ $\Omega$

$\:\:\:$ 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