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

This article is cited in 26 scientific papers (total in 26 papers)

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.

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MSC: Primary 14D10, 11G25; Secondary 14D07
Received: February 21, 2002
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
\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

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    This publication is cited in the following articles:
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    2. Faltings G., “A $p$-adic Simpson correspondence”, Adv. Math., 198:2 (2005), 847–862  crossref  mathscinet  zmath  isi
    3. Deninger Ch., Werner A., “Vector bundles on $p$-adic curves and parallel transport”, Ann. Sci. École Norm. Sup. (4), 38:4 (2005), 553–597  crossref  mathscinet  zmath  isi
    4. Kim M., “The motivic fundamental group of $\mathbf P^1\setminus{0,1,\infty}$ and the theorem of Siegel”, Invent. Math., 161:3 (2005), 629–656  crossref  mathscinet  zmath  adsnasa  isi
    5. Besser A., “$p$-adic Arakelov theory”, J. Number Theory, 111:2 (2005), 318–371  crossref  mathscinet  zmath  isi
    6. Besser A., Furusho H., “The double shuffle relations for p-adic multiple zeta values”, Primes and Knots, Contemporary Mathematics Series, 416, 2006, 9–29  crossref  mathscinet  zmath  isi
    7. Furusho H., “p-adic multiple zeta values: a precis”, p-ADIC Mathematical Physics, AIP Conference Proceedings, 826, 2006, 222–236  crossref  mathscinet  zmath  adsnasa  isi
    8. Furusho H., Jafari A., “Regularization and generalized double shuffle relations for $p$-adic multiple zeta values”, Compos. Math., 143:5 (2007), 1089–1107  crossref  mathscinet  zmath  isi
    9. Furusho H., “$p$-adic multiple zeta values. II. Tannakian interpretations”, Amer. J. Math., 129:4 (2007), 1105–1144  crossref  mathscinet  zmath  isi
    10. Olsson M.C., “$F$-isocrystals and homotopy types”, J. Pure Appl. Algebra, 210:3 (2007), 591–638  crossref  mathscinet  zmath  isi
    11. Faltings G., “Mathematics around Kim's new proof of Siegel's theorem”, Diophantine Geometry, Proceedings, CRM Series, 4, 2007, 173–188  mathscinet  zmath  isi
    12. Kim M., “The unipotent Albanese map and Selmer varieties for curves”, Publ. Res. Inst. Math. Sci., 45:1 (2009), 89–133  crossref  mathscinet  zmath  isi
    13. Olsson M.C., “On Faltings' method of almost etale extensions”, Proceedings of Symposia in Pure Mathematics: Algebraic Geometry Seattle 2005, Proceedings of Symposia in Pure Mathematics, 80, no. 1- 2, 2009, 811–936  crossref  mathscinet  zmath  isi
    14. Pridham J.P., “Galois actions on homotopy groups of algebraic varieties”, Geometry & Topology, 15:1 (2011), 501–607  crossref  mathscinet  zmath  isi
    15. Olsson M.C., “Towards Non-Abelian p-adic Hodge Theory in the Good Reduction Case”, Mem Amer Math Soc, 210:990 (2011), 1  crossref  mathscinet  isi
    16. Hadian M., “Motivic Fundamental Groups and Integral Points”, Duke Math J, 160:3 (2011), 503–565  crossref  mathscinet  zmath  isi
    17. Besser A., “On the Syntomic Regulator for K-1 of a Surface”, Isr. J. Math., 190:1 (2012), 29–66  crossref  mathscinet  zmath  isi
    18. Unver S., “Drinfel'D-Ihara Relations for P-Adic Multi-Zeta Values”, J. Number Theory, 133:5 (2013), 1435–1483  crossref  mathscinet  zmath  isi
    19. Andreatta F., Iovita A., Kim M., “a P-Adic Nonabelian Criterion For Good Reduction of Curves”, Duke Math. J., 164:13 (2015), 2597–2642  crossref  mathscinet  zmath  isi
    20. Dan-Cohen I., Wewers S., “Explicit Chabauty-Kim Theory For the Thrice Punctured Line in Depth 2”, Proc. London Math. Soc., 110:1 (2015), 133–171  crossref  mathscinet  zmath  isi
    21. Balakrishnan J.S., Besser A., “Coleman-Gross Height Pairings and the P-Adic SIGMA Function”, J. Reine Angew. Math., 698 (2015), 89–104  crossref  mathscinet  zmath  isi
    22. Katz E., Rabinoff J., Zureick-Brown D., “Uniform Bounds For the Number of Rational Points on Curves of Small Mordell-Weil Rank”, Duke Math. J., 165:16 (2016), 3189–3240  crossref  mathscinet  zmath  isi
    23. Balakrishnan J.S., Besser A., Mueller J.S., “Quadratic Chabauty: P-Adic Heights and Integral Points on Hyperelliptic Curves”, J. Reine Angew. Math., 720 (2016), 51–79  crossref  mathscinet  zmath  isi
    24. Besser A., de Shalit E., “l-Invariants of P-Adically Uniformized Varieties”, Ann. Math. Que., 40:1 (2016), 29–54  crossref  mathscinet  zmath  isi
    25. Dan-Cohen I., Wewers S., “Mixed Tate Motives and the Unit Equation”, Int. Math. Res. Notices, 2016, no. 17, 5291–5354  crossref  mathscinet  isi
    26. Chatzistamatiou A., “on Integrality of P-Adic Iterated Integrals”, J. Algebra, 474 (2017), 240–270  crossref  mathscinet  zmath  isi
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