Slope classicality via completed cohomology

2021/11/30 by Sean Howe, Howe, Sean
Mathematics · #11F33 #11F77 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F33 #msc:11F77

paper · pdf · doi:10.48550/arxiv.2111.15576

5 pages + references. Comments welcome!

arxiv created 2021/11/30 · arxiv updated 2021/12/01

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 GL₂(ℚ_p). The U_p operator becomes a double-coset, and unitarity yields the slope vanishing.

Related

• p-adic Cohomology and classicality of overconvergent Hilbert modular forms
• Higher Coleman Theory
• A Gluing Lemma And Overconvergent Modular Forms
• Slopes of modular forms and the Ghost conjecture
• Critical Lambda-adic modular forms and bi-ordinary complexes