Specialization of the p-adic polylogarithm sheaf to p-th power roots of unity.

K. Bannai

Abstract

In this talk, I will explain the method of specializing the p-adic polylogarithm sheaf, which is a filtered oveconvergent F-isocrystal on $\mathbb{P}^1 \setminus \{ 0, 1, \infty \}$, to p-th power roots of unity. There is slight technical difficulty due to the fact that p is ramified in the field obtained by adjoining p-th power roots of unity to $\mathbb{Q}_p$.

The main result is a result compatible with the results by M. Somekawa and also Besser-de Jeu on the calculation of the syntomic regulator of cyclotomic fields. In other words, we show that the specialization of the p-adic polylogarithm is again related to Coleman's p-adic polylogarithm function.