Non-Archimedean analytic functions, measures and distributions

Abstract

In this chapter we give an exposition of some standard facts from the theory of continuous and analytic functions over a non-Archimedean local field. We start by recalling the definitions and notations concerning p-adic and S-adic numbers. Then we discuss the theory of continuous p-adic functions and their p-adic interpolation, and also the basic properties of p-adic analytic functions. In ¡ì3 we introduce distributions and measures and give a general criterion for the existence of a non-Archimedean measure with given values of integrals of functions belonging to certain dense family (“generalized Kummer congruences”). The next ¡ì4 is devoted to a description of the algebra of bounded measures in terms of their non-Archimedean Mellin transforms (Iwasawa isomorphism). The chapter is completed with an exposition of a general construction of measures, attached to rather arbitrary Euler products.This construction provides a generalization of measures first introduced by Yu.I.Manin [Man4], B.Mazur and H.P.F.SwinnertonDyer [Maz-SD]. Our construction [Pa5], [Pa9] was already successfully used in several problems concerning the p-adic analytic interpolation of special values of Dirichlet series [Ar], [Co-Schm], [Co-Schn], [Sch].