# On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues

## TL;DR

This paper explores the relationship between Barnes' multiple gamma functions and their $p$-adic analogues, establishing algebraicity results and connections to Stark units and Gross-Stark units in the context of totally real fields.

## Contribution

It proves the $p$-adic algebraicity of products of ratios of Barnes' gamma functions and their $p$-adic analogues, extending previous algebraicity results and relating them to Stark units.

## Key findings

- Proved the $p$-adic algebraicity of certain products involving Barnes' gamma functions.
- Established a relation between ratios of exponential functions and Stark units.
- Extended previous algebraicity results to the $p$-adic setting.

## Abstract

Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $\iota$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,\iota)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $\zeta(s,c)$ has a canonical decomposition $\zeta'(0,c)=\sum_{\iota}X(c,\iota)$, where $\iota$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp(X(c,\iota))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,\iota)$: In particular, we discussed the relation between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and Gross-Stark units. In a previous paper, the author proved the algebraicity of some products of $\exp(X(c,\iota))$'s. In this paper, we prove its $p$-adic analogue. Then, by using these algebraicity properties, we discuss the relation between the ratios $[\exp(X(c,\iota)):\exp_p(X_p(c,\iota))]$ and Stark units.