Abstract：

We will talk about the notion of de Rham F-gauges, which are roughly enhanced coefficient objects for de Rham cohomology of smooth algebraic varieties in characteristic p. More precisely, these involve flat vector bundles together with Hodge and conjugate filtrations, and Frobenius structures between the associated Higgs fields and p-curvatures. We will explain how they are related to some other mod p Hodge structures, such as p-torsion Fontaine-Laffaille modules and F-zips. Finally, we discuss some applications to the mod p geometry and cohomology of Shimura varieties, where these objects arise naturally.