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...