Higher coleman theory and completed cohomology

We explain two connections between Boxer-Pilloni’s higher Coleman theory for modular curves and the completed cohomology of the towers of modular curves. In the first, we use the completed cohomology classes attached to overconvergent modular forms to give a new proof of the slope classicality theorem in higher Coleman theory. In the second, we explain how cup products with the groups in higher Coleman theory furnish a finite level description of (a large chunk of) Pan’s Hodge-Tate decomposition for highest weight vectors in completed cohomology.