Book:Neal Koblitz/p-adic Numbers, p-adic Analysis, and Zeta-Functions
Neal Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions
Published $\text {1984}$, Springer
- ISBN 978-1-4612-7014-0
Subject Matter
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