p-adic analysis

Abstract: 1 p-adic Numbers.- 2 Finite Extensions of the Field of p-adic Numbers.- 3 Construction of Universal p-adic Fields.- 4 Continuous Functions on Zp.- 5 Differentiation.- 6 Analytic Functions and Elements.- 7 Special Functions, Congruences.- Specific References for the Text.- Tables.- Basic Principles of Ultrametric Analysis.- Conventions, Notation, Terminology.

Abstract: Introduction. I. First Steps to Non-Archimedean Fields. II. The Gauss, Lebesgue and Feynman Distributions over Non-Archimedean Fields. III. The Gauss and Feynman Distributions on Infinite-Dimensional Spaces over Non-Archimedean Fields. IV. Quantum Mechanics for Non-Archimedean Wave Functions. V. Functional Integrals and the Quantization of Non-Archimedean Models with an Infinite Number of Degrees of Freedom. VI. The p-Adic-Valued Probability Measures. VII. Statistical Stabilization with Respect to p-Adic and Real Metrics. VIII. The p-Adic Valued Probability Distributions (Generalized Functions). IX. p-Adic Superanalysis. Bibliographical Remarks. Open Problems. Appendix: 1. Expansion of Numbers on a Given Scale. 2. An Analogue of Newton's Method. 3. Non-Existence of Differential Maps from Qp to R. Bibliography. Index.

Abstract: This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research.