Applications of patching the coherent cohomology of modular curves
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910129419452416 |
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| author | Bao, Chengyang |
| author_facet | Bao, Chengyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12830 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Applications of patching the coherent cohomology of modular curves Bao, Chengyang Number Theory 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. |
| title | Applications of patching the coherent cohomology of modular curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.12830 |