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

Abstract

We consider the partial zeta function ζ(s,c) associated with a narrow ray class c of a totally real field F. We have a canonical decomposition ζ′(0,c)=∑ιX(c,ι) where ι runs over all real embeddings of F. The symbol X(c,ι) denotes Hiroyuki Yoshida's class invariant defined as a finite sum of log of Barnes' multiple gamma functions and some correction terms. Yoshida studied the relation between the values of exp⁡(X(c,ι)), Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the p-adic analogue Xp(c,ι), and obtained an explicit relation between the “ratios” [exp⁡(X(c,ι)):expp⁡(Xp(c,ι))] and Gross–Stark units. In a previous paper, the author proved the algebraicity of some products of exp⁡(X(c,ι))'s. In this paper, we prove its p-adic analogue. As an application, we obtain an explicit relation between the same “ratios” and Stark units. As a result, we can discuss the rank 1 abelian Stark conjecture with respect to real places and Gross' p-adic analogue uniformly.