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arXiv · 2604.12830

Applications of patching the coherent cohomology of modular curves

Chengyang Bao
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Abstract

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.

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Chengyang Bao. 2026-04-14. Applications of patching the coherent cohomology of modular curves. https://arxiv.org/abs/2604.12830

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