Koblitz N. p-adic Analysis: A Short Course on Recent Work

Cambridge University Press, 1980. — 167 p. — (London Mathematical Society Lectures Notes, 46). — ISBN 0-521-28060-5, 978-0-511-52610-7.This book is a revised and expanded version of a series of talks given in Hanoi at the Vien Toan hoc (Mathematical Institute) in July, 1978. The purpose of the book is the same as the purpose of the talks: to make certain recent applications of p-adic analysis to number theory accessible to graduate students and researchers in related fields. The emphasis is on new results and conjectures, or new interpretations of earlier results, which have come to light in the past couple of years and which indicate intriguing and as yet imperfectly understood new connections between algebraic number theory, algebraic geometry, and p-adic analysis.
I occasionally state without proof or assume some familiarity with facts or techniques of other fields: algebraic geometry (Chapter III), algebraic number theory (Chapter IV), analysis (the Appendix). But I include down-to-earth examples and words of motivation whenever possible, so that even a reader with little background in these areas should be able to see what's going on.Basics
History (very brief)
Basic concepts
Power series
Newton polygonsp-adic ζ-functions, L-functions and Г-functions
Dirichlet L-series
p-adic measures
p-adic interpolation
p-adic Dirichlet L-functions
Leopoldt's formula for L_p(1, x)
The p-adic gamma function
The p-adic log gamma function
A formula for L'_p(0, x)Gauss sums and the p-adic Gamma function
Gauss and Jacobi sums
Fermat curves
L-series for algebraic varieties
Cohomology
p-adic cohomology
p-adic formula for Gauss sums
Stickleberger's theoremp-adic regulators
Regulators and L-functions
Leopoldt's p-adic regulator
Gross's p-adic regulator
Gross's conjecture in the abelian over Q caseAppendix
A theorem of Amice-Fresnel
The classical Stieltjes transform
The Shnirelman integral and the p-adic Stieltjes transform
p-adic spectral theoremBibliography
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