Regularization of $p$-Adic Distributions Associated to Functions on $p$-Adic Fields With Moderate Variation

Year 2021, Volume: 9 Issue: 2, 74 - 80, 01.06.2021

Altan Erdoğan

https://doi.org/10.36753/mathenot.794789

https://izlik.org/JA36LU98HD

Abstract

The $p$-adic distributions attached to ordinary functions defined on $p$-adic fields with moderate variation are studied. We first give a sufficient growth condition on ordinary functions to construct $p$-adic distributions. Then a moderate variation condition on functions for regularization of these $p$-adic distributions is imposed which provides a general method to construct $p$-adic measures. The $p$-adic integrals against these measures are also explicitly transformed to integrals against Bernoulli measures.

Keywords

p-adic analysis , p-adic integration , Bernoulli measures , p-adic Gamma measures

References

• [1] Kubota, T., Leopold H.W.; Eine p-adische Theorie der Zetawerte, I. Journ. Reine Angew. Math., 214/215, 328-339, (1964).
• [2] Coleman, R.F.: Dilogarithms, Regulators and p-adic L-functions, Invent. Math.,69, 171-208, (1982).
• [3] Washington, L.C.: Introduction to Cyclotomic Fields, Springer-Verlag, New York, (1982).
• [4] Diamond, J.: The p-adic Gamma Measures, Proc. Amer. Math. Soc., 75(2), 211-217, (1979).
• [5] Diamond, J.: The p-adic Log Gamma Function and p-adic Euler Constants, Trans. Amer. Math. Soc., 233, 321-337, (1977).
• [6] Lang, S.: Introduction to Modular Forms, Springer-Verlag, Berlin Heidelberg, (1976).
• [7] Katz, N.: p-Adic Properties of Modular Schemes and Modular Forms, In: Kuijk W., Serre JP. (eds) Modular Functions of One Variable III, Lecture Notes in Mathematics, vol 350. Springer, Berlin, Heidelberg, (1973).
• [8] Katz, N.: p-adic L-functions via Moduli of Elliptic Curves, Proc. 1974 AMS Arcata Summer Institute in Algebraic Geometry, PSPM 29, AMS, Providence (1975).
• [9] Koblitz, N.: p-adic Numbers, p-adic Analysis and Zeta-Functions, Springer-Verlag, New York, (1984).
• [10] Koblitz, N.: p-adic Analysis: a Short Course on RecentWork, London Mathematical Society Lecture Note 46, Cambridge University Press, Cambridge, New York, (1980).