Slope classicality via completed cohomology

TL;DR

This paper presents a new proof of the slope classicality theorem for modular curves using completed cohomology, linking overconvergent modular forms to unitary representations and simplifying the understanding of slope conditions.

Contribution

It introduces a novel proof method for slope classicality leveraging completed cohomology and unitary representations, applicable to arbitrary levels in classical and higher Coleman theory.

Findings

Completed cohomology classes embed obstruction spaces into unitary representations.

The $U_p$ operator is interpreted as a double-coset, facilitating slope analysis.

Unitarity implies slope vanishing, confirming classicality conditions.

Abstract

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at 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 $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.