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 Mosc. Math. J., 2003, Volume 3, Number 1, Pages 205–247 (Mi mmj83)

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

V. Vologodsky

Institut des Hautes Études Scientifiques

Abstract: We study the unipotent completion $\Pi^dR_un(x_0,x_1,X_K)$ of the de Rham fundamental groupoid of a smooth algebraic variety over a local non-Archimedean field $K$ of characteristic 0. We show that the vector space $\Pi^dR_un(x_0,x_1,X_K)$ carries a certain additional structure. That is a $\mathbb Q^ur_p$-space $\Pi_un(x_0,x_1,X_K)$ equipped with a $\sigma$-semi-linear operator $\phi$, a linear operator $N$ satisfying the relation $N\phi=p\phi N$, and a weight filtration $W_\cdot$ together with a canonical isomorphism $\Pi^dR_un(x_0,x_1,X_K)\otimes_K \overline K\simeq\Pi_un(x_0,x_1,X_K)\otimes_{\mathbb Q_p}^ur\overline K$. We prove that an analogue of the monodromy conjecture holds for $\Pi_un(x_0,x_1,X_K)$.
As an application, we show that the vector space $\Pi^dR_un(x_0,x_1,X_K)$ possesses a distinguished element. In other words, given a vector bundle $E$ on $X_K$ together with a unipotent integrable connection, we have a canonical isomorphism $E_{x_ 0}\simeq E_{x_1}$ between the fibres. This construction is a generalisation of Colmez's p-adic integration $(rkE=2)$ and Coleman's $p$-adic iterated integrals ($X_K$ is a curve with good reduction).
In the second part, we prove that, for a smooth variety $X_{K_0}$ over an unramified extension of $\mathbb Q_p$ with good reduction and $r\leq\frac{p-1}{2}$, there is a canonical isomorphism $\Pi^dR_un(x_0,x_1,X_K)\otimes B_dR\simeq\Pi_{r}^et(x_0,x_1,X_{\overline K_0})\otimes B_dR$ compatible with the action of the Galois group ($\Pi^dR_r(x_0,x_1,X_{K_0})$ stands for the level $r$ quotient of $\Pi^dR_un(x_0,x_1,X_K)$). In particular, this implies the crystalline conjecture for the fundamental group (for $r\leq\frac{p-1}{2}$).

Key words and phrases: Crystalline cohomology, Hodge structure, $p$-adic integration.

Full text: http://www.ams.org/.../abst3-1-2003.html
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Bibliographic databases:

MSC: Primary 14D10, 11G25; Secondary 14D07
Language: English

Citation: V. Vologodsky, “Hodge structure on the fundamental group and its application to $p$-adic integration”, Mosc. Math. J., 3:1 (2003), 205–247

Citation in format AMSBIB
\Bibitem{Vol03} \by V.~Vologodsky \paper Hodge structure on the fundamental group and its application to $p$-adic integration \jour Mosc. Math.~J. \yr 2003 \vol 3 \issue 1 \pages 205--247 \mathnet{http://mi.mathnet.ru/mmj83} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1996809} \zmath{https://zbmath.org/?q=an:1050.14013} 

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