Algebraic Geometry Seminar

Inside the category of Q_p-local systems on a smooth algebraic variety X over a p-adic field K there is a subcategory of de Rham local systems. All local systems arising from relative etale cohomology of families of algebraic varieties over X are de Rham and to any de Rham local system one can canonically associate a filtered vector bundle on X with a flat connection satisfying Griffiths transversality. One might think of de Rham local systems as being analogous to complex variations of Hodge structures.

It turns out that if X is proper and L is any Q_p-local system such that the base change of L to X_{\overline{K}} is absolutely irreducible, then L becomes de Rham after a twist by a power of the cyclotomic character (assuming also that K contains roots of unity of a large enough degree). The proof uses a p-adic Riemann-Hilbert correspondence introduced by Liu and Zhu, its decompleted version and the action of Sen operator. Speaker(s): Aleksandr Petrov (Harvard)