Hodge structure on the fundamental group and its application to p-adic integration

Abstract

We study the unipotent completion ${\Pi }_{un}^{DR}(x_0,x_1,X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space ${\Pi }_{un}^{DR}(x_0,x_1,X_K)$ possesses a distinguished element. In the other words, given a vector bundle E on $X_K$ together with a unipotent integrable connection, we have {\sf a canonical} isomorphism $E_{x_0}\simeq E_{x_1}$ between the fibers. The latter construction is a generalization of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals ($X_K$ is a curve with good reduction). In the second part we prove that, if $X_{K_0}$ is a smooth variety over an unramified extension of ${\mathbb{Q}}_p$ with good reduction and $r\le \frac{p-1}{2}$ then there is a canonical isomorphism ${\Pi }_r^{DR}(x_0,x_1,X_{K_0})\otimes B_{DR}\simeq {\Pi }_r^{et}(x_0,x_1,X_{{\overline{K}}_0})\otimes B_{DR}$ compatible with the action of Galois group (Here ${\Pi }_r^{DR}(x_0,x_1,X_{K_0})$ is the level r quotient of ${\Pi }_{un}^{DR}(x_0,x_1,X_K)$). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for $r\le \frac{p-1}{2}$) .

44 pages