## Schedule

Monday Tuesday Wednesday Thursday Friday
9:15 Registration Lau Lau Lau Lau
10:45 Lau Fargues Fargues Fargues Dospinescu
12:00 Fargues Dospinescu Dospinescu Dospinescu Hu
15:00 Fargues Ludwig Excursion Schraen
16:30 Dospinescu Knight Excursion Hellmann

All talks will go for one hour. There will be coffee breaks from 10:15 to 10:45, and (on Monday, Tuesday, Thursday) from 16:00 to 16:30.

Useful local information: pdf

## Mini-courses

Gabriel Dospinescu: An introduction to the p-adic local Langlands correspondence for $GL_2(\mathbb Q_p)$.

Laurent Fargues: $p$-adic Hodge theory, vector bundles and their modifications.

Eike Lau: Displays and $p$-divisible groups.

## Research talks

### Tuesday, March 24, 2015

 15:00 Judith Ludwig $p$-adic Langlands Functoriality 16:30 Erick Knight A $p$-adic Jacquet-Langlands Correspondence

### Thursday, March 26, 2015

 15:00 Benjamin Schraen Classicality on eigenvarieties via patching I 16:30 Eugen Hellmann Classicality on eigenvarieties via patching II

### Friday, March 27, 2015

 12:00 Yongquan Hu On the Cohen-Macaulayness of crystabelline Galois deformation rings

## Abstracts

#### Eugen Hellmann: Classicality on eigenvarieties via patching II

The classicality criterion of Coleman asserts that an overconvergent $p$-adic modular form of weight $k$ is classical if the valuation of its $U_p$-eigenvalue is less than $k-1$. This criterion generalizes to numerical criteria for classicality on higher dimensional eigenvarieties. In these two talks we want to explain how one can use the patching method to obtain finer classicality criteria on eigenvarieties for definite unitary groups. In the first talk we will introduce a so called space of trianguline representations and compare it to eigenvarieties via the big patching module introduced in the work of Carainai-Emerton-Gee-Geraghty-Paskunas-Shin. This machinery translates the problem to a local analysis of the space of trianguline representations which we explain in the second talk in the 2-dimensional case. This is joint work with C. Breuil.

#### Yongquan Hu: On the Cohen-Macaulayness of crystabelline Galois deformation rings

In this talk, we will prove that certain crystabelline Galois deformation rings of two dimensional residual representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ are Cohen-Macaulay, when $p>3$ and the residual representation has only scalar endomorphisms. This is joint work with V. Paskunas.

#### Erick Knight: A $p$-adic Jacquet-Langlands Correspondence

I will construct a $p$-adic Jacquet-Langlands correspondence, which is a correspondence between Banach space representations of $\text{GL}_2(\mathbb{Q}_p)$ and Banach space representations of the unit group of the quaternion algebra $D$ over $\mathbb{Q}_p$. The correspondence satisfies local-global compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Langlands correspondence, in the sense that the $D^\times$ representations can often be shown to have the expected locally algebraic vectors.

#### Judith Ludwig: $p$-adic Langlands Functoriality

We will study an example of p-adic Langlands functoriality: Let $B$ be a definite quaternion algebra over the rationals, $G$ the algebraic group defined by the units in $B$ and $H$ the subgroup of $G$ of norm one elements. Then the classical transfer of automorphic representations from $G$ to $H$ is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of $H(\mathbb A)$ in the discrete automorphic spectrum. After recalling some aspects of the classical story (e.g. what $L$-packets are) we will prove a $p$-adic version of this transfer. More precisely we will extend the classical transfer to $p$-adic families of automorphic forms as parametrized by eigenvarieties. We will prove the $p$-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

#### Benjamin Schraen: Classicality on eigenvarieties via patching I

The classicality criterion of Coleman asserts that an overconvergent $p$-adic modular form of weight $k$ is classical if the valuation of its $U_p$-eigenvalue is less than $k-1$. This criterion generalizes to numerical criteria for classicality on higher dimensional eigenvarieties. In these two talks we want to explain how one can use the patching method to obtain finer classicality criteria on eigenvarieties for definite unitary groups. In the first talk we will introduce a so called space of trianguline representations and compare it to eigenvarieties via the big patching module introduced in the work of Carainai-Emerton-Gee-Geraghty-Paskunas-Shin. This machinery translates the problem to a local analysis of the space of trianguline representations which we explain in the second talk in the 2-dimensional case. This is joint work with C. Breuil.