2020/08/18 by Sean Howe, Howe, Sean
Mathematics · #11F33 #11F77 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2008.08029
openalex publication_date 2020/08/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We construct a ( mathfrakgl2, B(\ℚp)) and Hecke-equivariant cup\nproduct pairing between overconvergent modular forms and the local cohomology\nat 0 of a sheaf on \ℙ1, landing in the compactly supported\ncompleted \ℂp-cohomology of the modular curve. The local cohomology\ngroup is a highest-weight Verma module, and the cup product is non-trivial on a\nhighest weight vector for any overconvergent modular form of infinitesimal\nweight not equal to 1. For classical weight k\≥ 2, the Verma has an\nalgebraic quotient H1(\ℙ1, \O(-k)), and on classical forms\nthe pairing factors through this quotient, giving a geometric description of\n"half" of the locally algebraic vectors in completed cohomology; the other half\nis described by a pairing with the roles of H1 and H0 reversed between\nthe modular curve and \ℙ1. Under minor assumptions, we deduce a\nconjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations\nattached to overconvergent modular forms. Our main results are essentially a\nstrict subset of those obtained independently by Lue Pan in arXiv:2008.07099,\nbut the perspective here is different and the proofs are short and use simple\ntools: a Mayer-Vietoris cover, a cup product, and a boundary map in group\ncohomology.\n