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

Vadim Vologodsky

Abstract

We study the unipotent completion Π_{un}^{DR}(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 Π_{un}^{DR}(x_{0}, x_{1}, X_{K}) carries a certain additional structure. That is a ℚ_{p}^{ur}-space Π_{un}(x_{0}, x_{1}, X_{K}) equipped with a σ-semi-linear operator ϕ, a linear operator N satisfying the relation Nϕ = pϕN and a weight filtration W_{•} together with a canonical isomorphism Π_{un}^{DR}(x_{0}, x_{1}, X_{K}) ⊗_{K} \overline{K} ≃ Π_{un}(x_{0}, x_{1}, X_{K}) ⊗_{ℚ_{p}^{ur}} \overline{K}. We prove that an analog of the Monodromy Conjecture holds for Π_{un}(x_{0}, x_{1}, X_{K}).

As an application, we show that the vector space Π_{un}^{DR}(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 a canonical isomorphism E_{x_{0}} ≃ E_{x_{1}} between the fibres. The latter construction is a generalisation 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 case, when X_{K} is proper and has a smooth model over the ring of integers R ⊂ K, a similar construction has been independently found by Besser