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

We study the unipotent completion ${\mathrm{\Pi }}_{un}^{DR}(x_0,x_1,X_K)$ 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 ${\mathrm{\Pi }}_{un}^{DR}(x_0,x_1,X_K)$ 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 $r \leq \frac{p-1}{2}$thenthereisacanonicalisomorphism$\Pi^{DR}{r}(x_0, x_1, X{K_0})\otimes B_{DR} \simeq \Pi^{et}{r}(x_0, x_1, X{\overline K_0}) \otimes B_{DR}$compatiblewiththeactionofGaloisgroup(Here$\Pi^{DR}{r}(x_0, x_1, X{K_0})$isthelevelrquotientof$\Pi^{DR}_{un}(x_0, x_1, X_K)$). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for $r\le \frac{p-1}{2}$) .