Chengyang Bao | Searcharxiv

arXiv2026

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.

math.NT↗

arXiv2022

Locally induced Galois representations with exceptional residual images

In this paper, we classify all continuous Galois representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside $\{p,\infty\}$ and locally induced at $p$, under the assumption that $\overline{\rho}$ is exceptional, that is, has image of order prime to $p$. We prove two results. If $f$ is a level one cuspidal eigenform and one of the $p$-adic Galois representations $\rho_f$ associated to $f$ has exceptional residual image, then $\rho_f$ is not locally induced and $a_p(f)\neq 0$. If $\rho$ is locally induced at $p$ and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of $\overline{\rho}$ are assumed to have class numbers prime to $p$, then $\rho$ has finite image up to a twist.

math.NT↗