Arithmetic Geometry

We describe the obstruction to lift Frobenius on certain Shimura varieties and apply this to the theory of companion forms for classical modular forms (Gross, Coleman-Voloch, Faltings-Jordan) and also in higher dimension.

Schubert varieties are known to be Frobenius split in positive characteristic by the work of Mehta and Ramanathan. More recently, Bhatt showed that the full flag variety for $GL_n$ is a splinter by entirely different methods. In this talk, we will...