# ($p$-adic) $L$-functions and ($p$-adic) (multiple) zeta values

## TL;DR

This paper explores the theory of ($p$-adic) $L$-functions and ($p$-adic) (multiple) zeta values, discussing their historical development, connections to Galois groups, motives, and modular varieties, with numerical examples included.

## Contribution

It provides a comprehensive overview of ($p$-adic) $L$-functions and zeta values, integrating recent research, historical context, and new interpretations in the framework of motives and Galois theory.

## Key findings

- Connections between $p$-adic zeta values and Galois groups elucidated.
- Unipotent motivic fundamental groups characterized.
- Numerical examples illustrating theoretical concepts included.

## Abstract

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 (plan). Numerical examples are included.