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Moscow Mathematical Journal, 2003, Volume 3, Number 1, Pages 205–247
DOI: https://doi.org/10.17323/1609-4514-2003-3-1-205-247
(Mi mmj83)
 

This article is cited in 41 scientific papers (total in 41 papers)

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

V. Vologodsky

Institut des Hautes Études Scientifiques
Full-text PDF Citations (41)
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Abstract: We study the unipotent completion $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ 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 $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ carries a certain additional structure. That is a $\mathbb Q^{\rm ur}_p$-space $\Pi_{\rm un}(x_0,x_1,X_K)$ 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 $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)\otimes_K \overline K\simeq\Pi_{\rm un}(x_0,x_1,X_K)\otimes_{\mathbb Q_{\rm p}}^{\rm ur}\overline K$. We prove that an analogue of the monodromy conjecture holds for $\Pi_{\rm un}(x_0,x_1,X_K)$.
As an application, we show that the vector space $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ 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 $({\rm rk}E=2)$ and Coleman's $p$-adic iterated integrals ($X_K$ is a curve with good reduction).
In the second part, we prove that, 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 $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)\otimes B_{\rm dR}\simeq\Pi_{r}^{\rm et}(x_0,x_1,X_{\overline K_0})\otimes B_{\rm dR}$ compatible with the action of the Galois group ($\Pi^{\rm dR}_{\rm r}(x_0,x_1,X_{K_0})$ stands for the level $r$ quotient of $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$). 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.
Received: February 21, 2002
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}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-1-205-247}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1996809}
\zmath{https://zbmath.org/?q=an:1050.14013}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000208594100013}
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  • This publication is cited in the following 41 articles:
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