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

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) = ι X(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.

p-Adic measures associated with zeta values and p-adic log multiple gamma functions

“…Roughly speaking, the above conjecture states a relation between the ratios [ p-adic multiple gamma functions : multiple gamma functions ] and Stark units associated with the finite place p ι . We also studied a relation between the same ratios and Stark units associated with real places in [Ka3]. We found a more significant relation between the ratios [ p-adic gamma function : gamma function ] and cyclotomic units in [Ka2].…”

“…We recall the definition and some properties of the symbol Y p defined in [KY1], [Ka3]. We denote by R + the set of all positive real numbers.…”

“…We need such fundamental domains in order to construct both of the invariants Y p , u T . In §3, we recall the definition and some properties of Y p , which is essentially defined in [KY1] and slightly modified in [Ka3]. The classical or p-adic log multiple gamma function is defined as the derivative values at a = 0 of the classical or p-adic Barnes' multiple zeta function, respectively.…”