Applications of patching the coherent cohomology of modular curves

In this paper, we apply the Taylor–Wiles–Kisin patching method to the coherent cohomology of modular curves at minimal level. We establish a multiplicity-one result for the patched module by the $q$-expansion principle and show that a certain partial normalization of the crystalline deformation ring is Cohen–Macaulay. As applications, we prove new cases where crystalline deformation rings are Cohen–Macaulay, establish a Zariski density result for crystalline points in characteristic $p$, and prove a multiplicity-one result for Serre’s mod-$p$ quaternionic modular forms.