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Non-Archimedean analytic functions, measures and distributions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1471)

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].

Keywords

  • Power Series
  • Zeta Function
  • Dirichlet Series
  • Projective Limit
  • Bernoulli Number

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1991 Springer-Verlag Berlin Heidelberg

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Panchishkin, A.A. (1991). Non-Archimedean analytic functions, measures and distributions. In: Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Lecture Notes in Mathematics, vol 1471. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21541-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-21541-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54137-0

  • Online ISBN: 978-3-662-21541-8

  • eBook Packages: Springer Book Archive