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

N. M. Glazunov

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 112-130

Résumé

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. Ünver 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 Ünver, 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.

Détail

Keywords: $p$-adic interpolation, ($p$-adic) $L$-function, Eisenstein Series, comparison isomorphism, crystalline Frobenius morphism, de Rham fundamental group, ($p$-adic) multiple zeta value, Iwasawa theory, Shimura variety, arithmetic cycles.