  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 in arXiv:2008.07099,
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.
