An integral model of the perfectoid modular curve

TL;DR

This paper constructs an integral model of the perfectoid modular curve, analyzes its coherent cohomology, and describes the dual of completed cohomology in terms of integral cusp forms, advancing understanding of perfectoid modular curves.

Contribution

It introduces an integral model of the perfectoid modular curve and computes duals of its coherent cohomology using local duality at finite levels.

Findings

Vanishing results for coherent cohomology at perfectoid level

Explicit description of the dual of completed cohomology as inverse limit of cusp forms

Application of local duality theorem to compute duals

Abstract

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.