On p-adic absolute CM-periods I

The second author defined the absolute CM-period symbol using the multiple gamma function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic multiple gamma function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.