Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

Abstract We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms the local cohomology at $0$ of sheaf on $\mathbb {P}^1$ , landing in compactly supported completed {C}_p$ -cohomology curve. The group is highest-weight Verma module, non-trivial vector for any form infinitesimal weight not equal to $1$ . For classical $k\geq 2$ has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$ forms, factors through this quotient, giving geometric description ‘half’ locally vectors cohomology; other half described by with roles $H^1$ $H^0$ reversed curve Under minor assumptions, we deduce conjecture Gouvea Hodge-Tate-Sen weights Galois representations attached forms. Our main results are essentially strict subset those obtained independently Lue Pan, but perspective here different, proofs short use simple tools: Mayer-Vietoris cover, product, boundary map cohomology.