## Speakers

Ahmed Abbes
Christophe Breuil
Pierre Colmez
Hélène Esnault
Laurent Fargues
Ofer Gabber
Dennis Gaitsgory
Jochen Heinloth
Moritz Kerz
Mark Kisin
Kai-Wen Lan
Martin Olsson
George Pappas
Vytautas Paškūnas
Michael Rapoport
Peter Scholze
Sug Woo Shin
Eva Viehmann

## Titles of the talks, schedule

### Monday, Oct. 7, 2013

 09:30 Christophe Breuil On the locally analytic socle and local-global compatibility 11:00 Pierre Colmez Locally analytic representations of $GL_2$ and $(\varphi,\Gamma)$-modules 15:30 Hélène Esnault A few remarks on isocrystals and the étale fundamental group 17:00 Jochen Heinloth GIT-like stability criteria on algebraic stacks

### Tuesday, Oct. 8, 2013

 09:00 Ofer Gabber The purity theorem for a generalization of perfectoid rings 10:30 Peter Scholze On torsion in the cohomology of locally symmetric varieties 15:30 Eva Viehmann Connected components of Rapoport-Zink spaces and of related moduli spaces 17:00 Dennis Gaitsgory Categorical geometric Langlands conjecture

### Wednesday, Oct. 9, 2013

 09:00 Ahmed Abbes The $p$-adic Simpson correspondence 10:30 Moritz Kerz Deformation of cycle classes 11:45 Michael Rapoport On formal moduli spaces of $p$-divisible groups

### Thursday, Oct. 10, 2013

 09:00 Laurent Fargues $\varphi$-modules and modifications of vector bundles 10:30 Mark Kisin mod $p$ points on Shimura varieties 15:30 George Pappas Integral models of Shimura varieties at tame primes 17:00 Martin Olsson On the calculation of local terms

### Friday, Oct. 11, 2013

 09:00 Vytautas Paškūnas Patching and the $p$-adic local Langlands correspondence, Part 1 10:30 Sug Woo Shin Patching and the $p$-adic local Langlands correspondence, Part 2 11:45 Kai-Wen Lan Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci

## Abstracts

#### Ahmed Abbes: The $p$-adic Simpson correspondence

In 2005, Gerd Faltings laid the foundations of a correspondence aimed at describing all $p$-adic representations of the geometric fundamental group of a smooth algebraic variety over a $p$-adic field in terms of linear algebra, specifically of Higgs bundles. His construction is for $p$-adic Hodge theory what the complex Simpson correspondence is for classical Hodge theory. I will present a new approach for the $p$-adic Simpson correspondence, closely related to Faltings's original approach, and inspired by the work of Ogus and Vologodsky on an analogue in characteristic $p > 0$. This is a joint work with Michel Gros.

#### Christophe Breuil: On the locally analytic socle and local-global compatibility

I state a conjecture on some irreducible locally analytic representations of $GL_n(\mathbb Q_p)$ which should occur inside certain Hecke-isotypic spaces of p-adic automorphic forms. I then prove a few partial results towards this conjecture using (1) recent results due to several people on Eigenvarieties and (2) a new adjunction formula for Emerton's locally analytic Jacquet functor.

#### Pierre Colmez: Locally analytic representations of $GL_2$ and $(\varphi,\Gamma)$-modules

I will explain the construction of functors from locally analytic representations to $(\varphi, \Gamma)$-modules and give some applications.

#### Hélène Esnault: A few remarks on isocrystals and the étale fundamental group

Johan de Jong formulated the following conjecture: a simply connected smooth projective variety over an algebraically closed characteristic $p>0$ field has no non-trivial isocrystal. The analog over the complex numbers is a theorem by Malcev-Grothendieck, the analog with isocrystals replaced by $\mathcal{O}$-coherent $\mathcal{D}$-modules is Gieseker's conjecture (1975, now a theorem by Esnault-Mehta). We make a few little remarks on the problem, under the assumption that the variety lifts to the Witt vectors. If so, Grothendieck's specialization homomorphism, or more precisely a form of it lifts. This enables one to conclude assuming that the geometric generic fiber has a finite étale fundamental group.

#### Laurent Fargues: $\varphi$-modules and modifications of vector bundles

We prove the existence of an equivalence between a category of Kisin type $\varphi$-modules and modifications of vector bundles over the curve. As a particular case, for minuscule modifications, we find a classification of $p$-divisible groups over the ring of integers of a complete algebraically closed extension of the $p$-adic numbers and moreover find back Scholze-Weinstein classification.

#### Ofer Gabber: The purity theorem for a generalization of perfectoid rings

If $I$ is a finitely generated ideal in a ring $A$ such that the $I$-adic completion of $A$ is perfectoid in a sense to be discussed, one can show that finite étale coverings of $\mathop{\rm Spec}(A)-V(I)$ extend to finite étale $A^a$-algebras. This is an adaptation of Scholze's proof of the Faltings purity theorem to be included in the book with Ramero.

#### Dennis Gaitsgory: Categorical geometric Langlands conjecture

Let $X$ be a projective curve and $G$ a reductive group. Naively, the geometric Langlands conjecture aims to compare the category $\mathscr D-{\rm mod}({\rm Bun}(G))$ of $\mathscr D$-modules on the moduli space of $G$-bundles on $X$ with the category ${\rm QCoh}({\rm LocSys}(G^L))$ of quasi-coherent sheaves on the moduli space of $G^L$-local systems on $X$. However, this statement does not hold unless $G$ is a torus, and a correction is needed that has to do with Arthur parameters. In this talk I'll give a formulation of the conjecture that takes into account the Arthur parameters. The latter appear through the study of singularities of the space ${\rm LocSys}(G^L)$. This is a joint work with D. Arinkin.

#### Jochen Heinloth: GIT-like stability criteria on algebraic stacks

To find stability conditions for moduli problems one usually proceeds in two steps: first one makes an educated guess and second one proves that this guess coincides with a stability criterion coming from GIT. We would like to give a criterion that can help to avoid the guessing procedure and illustrate in examples how this can be used to find substacks of moduli problems that admit separated coarse moduli spaces. This was motivated by questions on variants of moduli spaces of principal bundles arising from arithmetic problems for function fields.

#### Moritz Kerz: Deformation of cycle classes

(joint work with S. Bloch and H. Esnault) We study the deformation properties of $K_0$-classes of vector bundles for varieties in characteristic zero. The aim is to relate the relevant obstruction theory with Hodge theory in order to gain a better understanding of Grothendieck's variational Hodge conjecture. In particular we study the case of abelian schemes.

#### Mark Kisin: mod $p$ points on Shimura varieties

As part of his program to understand the zeta function of a Shimura variety in terms of automorphic $L$-functions, Langlands conjectured a group theoretic description of the set of mod $p$ points on the Shimura variety. This conjecture was refined by Kottwitz and Langlands-Rapoport, and proved by Kottwitz in some PEL cases. I will explain a result proving this conjecture for Shimura varieties which parametrize abelian varieties equipped with Hodge cycles, and quotients of these. Along the way one shows that any mod $p$ isogeny class on such a variety contains a point which lifts to a special point.

#### Kai-Wen Lan: Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci

I will report on the construction of $p$-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing arbitrary ramification (including deep levels) at $p$, with good behaviors over the loci where certain (multiplicative) ordinary level structures are defined.

#### Martin Olsson: On the calculation of local terms

Let $k$ be an algebraically closed field and let $c:C\rightarrow X\times X$ be a correspondence. Let $\ell$ be a prime invertible in $k$ and let $K\in D^b_c(X, \overline {\mathbb Q}_\ell )$ be a complex. An action of $c$ on $K$ is by definition a map $u:c_1^*K\rightarrow c_2^!K$. For such an action one can define for each proper component $Z$ of the fixed point scheme of $c$ a local term $\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell$. In this talk I will discuss various techniques for studying these local terms and some independence of $\ell$ results for them. I will also discuss consequences for traces of correspondences acting on cohomology.

#### George Pappas: Integral models of Shimura varieties at tame primes

We will present some recent results about integral models of Shimura varieties at primes where the group is tamely ramified and the level subgroup is parahoric in the sense of Bruhat-Tits.

#### Vytautas Paškūnas: Patching and the $p$-adic local Langlands correspondence, Part 1

This is the first talk of two on a joint work with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty and Sug Woo Shin. We will review the $p$-adic Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and the Breuil-Schneider conjecture. We will then give a very rough sketch of the proof of the conjecture in many cases. The preprint is available at arXiv:1310.0831.

#### Michael Rapoport: On formal moduli spaces of $p$-divisible groups

By a well-known general existence theorem one can construct formal schemes that represent a moduli problem of $p$-divisible groups with auxilliary structure within a fixed isogeny class. I will address the question of how to identify these formal schemes.

#### Peter Scholze: On torsion in the cohomology of locally symmetric varieties

We will discuss the $p$-adic geometry of Shimura varieties with infinite level at $p$: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

#### Sug Woo Shin: Patching and the $p$-adic local Langlands correspondence, Part 2

This is the second half of our report on joint work with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, and Vytautas Paskunas. We will explain the key construction of the so-called patched module for a global definite unitary group and prove a local-global compatibility. This provides a global input for the Breuil-Schneider conjecture in the first talk as well as (if premature) a candidate for the p-adic local Langlands correspondence.

#### Eva Viehmann: Connected components of Rapoport-Zink spaces and of related moduli spaces

Affine Deligne-Lusztig varieties are a generalisation of the underlying reduced subscheme of Rapoport-Zink spaces, or of moduli spaces of local $G$-shtukas. I will explain joint work with M. Chen and M. Kisin on how to define and determine their sets of connected components. These results have applications on the realisation of local Langlands correspondences in the cohomology of Rapoport-Zink spaces (due to Chen), and on the study of the mod $p$ points of Shimura varieties of Hodge type (as in Kisin's talk).