This book is a revised and expanded version of a series of talks given in Hanoi at the Viện Toán học (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.

Chapter I contains the basic information about p-adic numbers and p-adic analysis needed for what follows. Chapter II describes the construction and properties of p-adic Dirichlet L-functions, including Leopoldt's formula for the value at 1, using the approach of p-adic integration. The p-adic gamma function and log gamma function are introduced, their properties are developed and compared with the identities satisfied by the classical gamma function, and two formulas relating them to the p-adic L-functions Lp(s,x) are proved.