# Abstact Stefano Morra

**Stefano Morra**will speak on

**POTENTIALLY CRYSTALLINE DEFORMATION RINGS IN THE ORDINARY CASE**

**Abstract:** Potentially crystalline deformation rings play a crucial role in the $p$-adic Langlands program and in local to global compatibility results. According to the Breuil-Mézard conjecture, the relation between $p$-adic Hodge theory constraints on the deformations side and Serre type conjectures on the automorphic side should provide us with a precise description of the geometry of such spaces.

In this talk we address to the problem of an ordinary Galois representation $\bar{\rho} : G{_Q}_p \rightarrow {\textbf{GL}}_3 {(\textbf{F}}_p)$. We show that, for a carefully chosen inertial type and in Hodge-Tate weights $(0, 1, 2)$, the potentially crystalline deformation space is formally smooth over $ \textbf{Z}_p$ and any potentially crystalline lift of $\bar{\rho}$ is ordinary. This is in accordance with the Breuil-Mézard conjecture.

The proof relies on integral $p$-adic Hodge theory, using a p-adic convergence argument to determine explicitly the finite height lifts of the residual $\bar{\rho}$. The key phenomenon for formal smoothness is then behavior of the monodromy operator on Breuil modules, which is in this case accessible to computation.

This is joint work with Brandon Levin.

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Stefano Morra, Université de Montpellier, place Eugène Bataillon, 34095 Montpellier, France

*E-mail* : stefano.morra@univ-montp2.fr