p-adic L-functions and p-adic multiple zeta values

The article is dedicated to the memory of George Voronoi. It  is concerned with (p-adic) L-functions (in partially  (p-adic) zeta functions)  and cyclotomic  (p-adic) (multiple) zeta values. The beginning of the article contains a short  summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. "Unver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple zeta values is given by H.  Furusho. Short history and connections among Galois groups, fundamental groups, motives and arithmetic functions are presented in the talk by Y. Ihara. Results  on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov.Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov. The framework of (p-adic) L-functions and (p-adic) (multiple) zeta values is based on Kubota-Leopoldt p-adic L-functions and arithmetic p-adic L-functions by Iwasawa. Motives and  (p-adic) (multiple) zeta values by Glanois and by "Unver, improper intersections of Kudla-Rapoport divisors and Eisenstein series by Sankaran are reviewed. More fully the content of the article can be found at the following table of contents: Introduction. 1. Voronoi-type congruences for  Bernoulli numbers. 2. Riemann zeta values. 3. On class groups of rings with divisor theory.  Imaginary quadratic and cyclotomic fields. 4. Eisenstein Series. 5. Class group, class fields and zeta functions. 6. Multiple zeta values. 7. Elements of non-Archimedean local fields and $ p-$adic analysis. 8. Iterated integrals and (multiple) zeta values. 9. Formal groups and p-divisible groups. 10. Motives and (p-adic) (multiple) zeta values. 11. On the Eisenstein series associated with Shimura varieties. Sections 1-9 and subsection 11.1 (On some Shimura varieties and Siegel modular forms) can be considered as an elementary introduction to the results of section 10 and subsection 11.2 (On improper intersections of Kudla-Rapoport divisors and Eisenstein series). Numerical examples are included.

p-адическая интерполяция, (p-адическая) L-функция, ряд Эйзенштейна, изоморфизм сравнения, кристаллический морфизм Фробениуса, фундаментальная группа де Рама, (p-адическое) кратное дзета-значение, теория Ивасавы, многообразие Шимуры, арифметические циклы.

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