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

By V. Vologodsky. 2001

80 citations recorded by the source corpus.

We study the unipotent completion $\Pi^{DR}_{un}(x_0, x_1, X_K)$ 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 $\Pi^{DR}_{un}(x_0, x_1, X_K)$ 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 $r \leq \frac{p-1}{2}$ then there is a canonical isomorphism $\Pi^{DR}_{r}(x_0, x_1, X_{K_0})\otimes B_{DR} \simeq \Pi^{et}_{r}(x_0, x_1, X_{\overline K_0}) \otimes B_{DR}$ compatible with the action of Galois group (Here $\Pi^{DR}_{r}(x_0, x_1, X_{K_0})$ is the level r quotient of $\Pi^{DR}_{un}(x_0, x_1, X_K)$). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for $r \leq \frac{p-1}{2}$) .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc6f349ae84b142762deb636f722b4fe5f56b6ae

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Citations in the corpus

Uniform bounds for the number of rational points on curves of small Mordell–Weil rank (2015, methodology): “Other references for this integration theory are [18], [54] and the second part of [30], taken with the understanding that the first part uses the Berkovich–Coleman integration theory.”
Vector bundles on p-adic curves and parallel transport (2004, unknown): “Finally we would like to draw the reader’s attention to possibly related works of Berko [2, §9] on p-adic integration, of Ogus and Vologodsky on non-Abelian Hodge theor characteristicp and of Vologodsky [33] on Hodge structures on fundamental groups.”
p -adic multiple zeta values II. Tannakian interpretations (2005, methodology): “have an action of the geometric Frobenius F robp of p. Since the eigenvalues of the F robp-action on H 1 ´et (X K ,Ql) are positive weight Weil numbers, we obtain a unique Frobenius invariant path e p,∗′ by an inductive construction same to [Bes]§3 (or [ Vol ]§4.3). �”
Periods and mixed motives (2002, background): “The family of the mixed Hodge structures Hn(Xn S,BS) forms a unipotent variation Hn of mixed Hodge structures over XS ×XS (see chapter 4 in [G7]). Combining it with the rigidity theorem of Vologodsky [Vol] one immediately sees that this variation is isomorphic to the one constructed by Hain and Zuck”
Toroidal compactifications of integral models of Shimura varieties of Hodge type (2012, methodology): “tion of these results. A useful ingredient here is the interpretation of the log crystalline realization of a semi-stable abelian variety as a space of ‘nearby unipotent cycles’ in the terminology of [Vol03]; cf. (1.4). This allows us to give a construction of the semi-stable comparison isomorphism ”
The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel (2005, unknown): “the class of the point y lies in H(1)f (ΓT , π1,ét)(Qp) for (p − 1)/2 ≥ n + 1 ([13], 1.”