Algebraic and analytic parallel transport on \(p\)-adic curves
This paper deals with the Riemann-Hilbert correspondence and its generalizations. In the classical case, given a Riemann surface \(S\), the Riemann-Hilbert correspondence links complex representations of the fundamental group of \(S\) and \(C^{\infty}\)-vector bundles with flat connections on \(S\). This is a reflection of the idea of parallel transport. Both algebraic and topological \(p\)-adic analogues of the Riemann-Hilbert correspondence have been studied. In the algebraic \(p\)-adic case, Deninger and Werner associated \(\mathbb{C}_p\)-representations of the algebraic fundamental group of a smooth projective curve \(X\) over \(\overline{\mathbb{Q}_p}\) to a certain class \(\mathfrak{B}^s_{X_{\mathbb{C}_p}}\) of vector bundles on \(X_{\mathbb{C}_p}\). In the topological \(p\)-adic case, Berkovich associated \(\mathbb{C}_p\)-representations of the fundamental group of a smooth \(\mathbb{C}_p\)-analytic space to locally unipotent vector bundles on the space. In the paper under review, the starting point is an integral \(\mathbb{C}_p\)-representation of the algebraic fundamental group of a smooth curve \(X\) over \(\mathbb{Q}_p\) that factors over the fundamental group of a normal, flat, projective model. The author shows that one can attach a vector bundle with connection to such a representation, and that this vector bundle lies in the class for which Deninger and Werner's work applies. Furthermore, the construction is inverse to theirs. Secondly, if one starts with a \(\mathbb{C}_p\)-representation of the topological fundamental group of the Berkovich analytification of \(X\), this gives a vector bundle with connection on \(X_{\mathbb{C}_p}^{an}\). Also, taking the profinite completion gives a representation of the algebraic fundamental group of \(X\) for which the previous paragraph applies, thus yielding an algebraic vector bundle with connection. The author shows that the analytic vector bundle with connection is the analytification of the algebraic vector bundle with connection. The author expects that his theorems can be extended to higher dimensional spaces.
\(p\)-adic geometry, Riemann-Hilbert correspondence
scientific article; zbMATH DE number 6175552