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

2017/03/30 by Kashio, Tomokazu · 1 citation
#11R27 #11R42 #11R80 #11S40 #11S80 #33B15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1703.10411

Abstract

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

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• Supersymmetric zeta functions and determinants

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