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