  We construct an integral model of the perfectoid modular curve. Studying this
object, we prove some vanishing results for the coherent cohomology at
perfectoid level. We use a local duality theorem at finite level to compute
duals for the coherent cohomology of the integral perfectoid curve.
Specializing to the structural sheaf, we can describe the dual of the completed
cohomology as the inverse limit of the integral cusp forms of weight $2$ and
trace maps.
