Slope classicality in higher Coleman theory via highest weight vectors in completed cohomology

Abstract

We give a proof of the slope classicality theorem in classical and higher Coleman theory for modular curves of arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of GL2(ℚp). The Up operator becomes a double coset, and unitarity yields slope vanishing.