Let \(k\) be a finite field of characteristic \(p\), and let \(K\) be a finite extension of the field \(K_0\) of fractions of the ring \(W(k)\) of Witt vectors of \(k\). Given a pair of points \(x_0\) and \(x_1\) of a smooth algebraic variety \(X_K\) over \(K\), the unipotent completion \(\Pi^{\text{dR}}_{\text{un}} (x_0, x_1, X_K)\) of the associated de Rham fundamental groupoid has the structure of a vector space.
In this paper the author obtains a \(\mathbb Q^{\text{ur}}_p\)-space \(\Pi_{\text{un}} (x_0, x_1, X_K)\) equipped with a \(\sigma\)-semilinear operator \(\phi\), a linear operator \(N\) with \(N \phi = p \phi N\), and a weight filtration \(W_\bullet\) exhibiting a canonical isomorphism \(\Pi^{\text{dR}}_{\text{un}} (x_0, x_1, X_K) \otimes_K \overline{K} \cong \Pi_{\text{un}} (x_0, x_1, X_K) \otimes_{\mathbb Q^{\text{ur}}_p} \overline{K}\). He proves that an analogue of the monodromy conjecture holds for \(\Pi_{\text{un}} (x_0, x_1, X_K)\) and as an application he obtains an isomorphism \(E_{x_0} \cong E_{x_1}\) of fibers of a vector bundle \(E\) over \(X_K\) with a unipotent integrable connection. For a smooth variety \(X_{K_0}\) over an unramified extension of \(\mathbb Q_p\) with good reduction and \(r \leq (p-1)/2\), he also proves that there is a canonical isomorphism \(\Pi^{\text{dR}}_r (x_0, x_1, X_{K_0}) \otimes B_{\text{dR}} \cong \Pi^{\text{et}}_r (x_0, x_1, X_{\overline{K}_0}) \otimes B_{\text{dR}}\) compatible with the action of the Galois group, where \(\Pi^{\text{dR}}_r (x_0, x_1, X_{K_0})\) denotes the level \(r\) quotient of \(\Pi^{\text{dR}}_{\text{un}} (x_0, x_1, X_K)\).
This result implies the crystalline conjecture for the fundamental group of A. Shiho [Proc. Am. Math Soc., Lect. Notes 24, 381–398 (2000; Zbl 0984.14010)].