Applications of patching the coherent cohomology of modular curves

Abstract: We apply the Taylor--Wiles--Kisin patching method to study certain partial normalizations of crystalline deformation rings associated with two-dimensional representations \bar{r} : G_{\Q_p} \to \GL_2(\F), where $\F$ is a finite field of characteristic $p \ge 5$. Using the $q$-expansion principle, we obtain a multiplicity-one result, which implies that the partial normalization of the crystalline deformation ring is Cohen--Macaulay. As applications, we give a simple criterion for when a crystalline deformation ring coincides with its partial normalization, thereby establishing new cases where these rings are Cohen--Macaulay. We also prove a Zariski-density result for crystalline points in characteristic $p$, and we apply our method to deduce a multiplicity-one result for Serre's mod-$p$ quaternionic modular forms. Most of these results originated from attempts to explain computational data from my thesis on computing crystalline deformation rings via the Taylor--Wiles--Kisin patching method. I will conclude with some expected properties of crystalline deformation rings suggested by the data that remain open.

number theory

Audience: researchers in the topic

London number theory seminar

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