# $p$-adic measures associated with zeta values and $p$-adic $\log$ multiple gamma functions

## TL;DR

This paper explores the relationship between two $p$-adic symbols related to Gross-Stark units, connecting $p$-adic gamma functions and multiplicative integration to deepen understanding of $p$-adic $L$-values and units.

## Contribution

It establishes an explicit relation between Yoshida's $Y_p( au)$ symbol and Dasgupta's $u_T( au)$ symbol, linking two approaches to Gross-Stark units.

## Key findings

- Derived an explicit formula relating $Y_p( au)$ and $u_T( au)$
- Connected $p$-adic gamma functions with multiplicative integration methods
- Enhanced understanding of $p$-adic units in abelian extensions

## Abstract

We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $\tau \in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(\tau)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(\tau)$. Dasgupta constructed the symbol $u_T(\tau)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by $u_T(\tau)$. In this paper, we give an explicit relation between $Y_p(\tau)$ and $u_T(\tau)$.