Tomokazu Kashio

Periods and the multiple gamma function in the p-adic case

The Gross-Koblitz formula expresses Gauss sums by division values of Morita's p-adic gamma function. This formula can be regarded as a p-adic analogue of the Chowla-Selberg formula. In this talk, we will define the p-adic logarithmic multiple gamma function and give a conjectural generalization of the Gross-Koblitz formula. Unlike Yoshida's conjecture in the archimedean case, this conjectural formula concerns the leading term of the Taylor expansion of a p-adic L-function. In particular, it gives a refinement of the Gross conjecture, which is a p-adic analogue of the Stark conjecture. In order to emphasize the similarity between the archimedean case and the p-adic case, we will generalize our conjecture to the case of ``the non-leading term'' and explain the relation to the p-adic CM-period. (Joint work with H. Yoshida.)