**Abstracts**
**Noriyuki Abe**, Hokkaido University

*On a structure of modulo p compact induction of split p-adic
groups*

We consider a compact induction from a modulo p irreducible representation
of a hyperspecial maximal compact subgroup of a split p- adic
group. This representation is related to a parabolic induction.
As an application, we give a classification of modulo p irreducible
admissible representations in terms of supercuspidal representations.

**Konstantin Ardakov**, Queen Mary University of London

*Localisation of affinoid enveloping algebras *

I will explain how to relate representations of affinoid enveloping
algebras to Berthelot's arithmetic D-modules on the flag variety,
and sketch some applications to the p-adic representation theory
of semisimple compact p-adic Lie groups.

**Joël Bellaïche**, Brandeis University

*On Hecke algebras modulo $p$*

We will present, motivate, and discuss special cases of a conjecture
on the dimension and structure of the local components of the
algebra of Hecke operators acting on the space of modular forms
modulo a prime $p$ of a fixed level and all weights.

**Francesc Castella**, McGill University

*On the p-adic Euler system of big Heegner points*

Attached to a newform f of weight 2 and an imaginary quadratic
field K, the Kummer images of Heegner points give rise to an anticyclotomic
Euler System for the p-adic Galois representation associated with
f. In this talk I will try to explain, assuming that the prime
p splits in K, how the extension by Ben Howard of this construction
to Hida families can be seen as producing a two-variable p-adic
L-function in the spirit of Perrin-Riou. In the weight variable,
we can thus show that Howard's construction interpolates the étale
Abel-Jacobi images of Heegner cycles; in the anticyclotomic variable,
this leads to new cases of the Bloch-Kato conjecture. The extension
of some of these results to general CM fields in which p splits
completely seems to be within reach.

**Samit Dasgupta**, University of California - Santa Cruz

*$p$-arithmetic cohomology and completed cohomology of totally
indefinite Shimura varieties*

This is a report on work in progress with Matthew Greenberg.
Let $F$ be a totally real field and let $p$ be a rational prime
that for simplicity we assume is inert in $F$. Let $f$ be a modular
form for a totally indefinite quaternion algebra $A$ over $F$
whose local representation at $p$ is Steinberg. We define a Darmon
$\mathcal{L}$-invariant attached to $f$, which is a vector of
$p$-adic numbers indexed by the embeddings of $F$ into $\mathbf{C}_p$.
This $\mathcal{L}$-invariant is defined using the cohomology of
a $p$-arithmetic subgroup of $A^*$ and is modeled after Darmon's
definition of the $L$-invariant in the case

$F=\mathbf{Q}, A=\mathrm{M}_2(\mathbf{Q})$.

Next we consider certain $p$-adic Banach space representations
of $\mathrm{GL}_2(F_p)$ denoted $B(k,\mathcal{L})$, where $k$
is the vector of weights of $f$ and $\mathcal{L}$ is a vector
as before. These Banach space representations generalize the construction
of Breuil in the case $F=\mathbf{Q}$, and build on work of Schraen.
Our main result is that there exist $\mathrm{GL}_2(F_p)$-interwining
operators from $B(k,\mathcal{L})$ to the $f$-isotypic component
of the completed cohomology of $A$ if and only if the vector $\mathcal{L}$
is the negative of the Darmon $\mathcal{L}$-invariant. This generalizes
a result of Breuil in the case $F=\mathbf{Q}, A=\mathrm{M}_2(\mathbf{Q})$.

**Gabriel Dospinescu**, École Polytechnique

*Extensions of de Rham representations and locally algebraic vectors*

In this joint work with Vytautas Paskunas, we extend Colmez'
results on locally algebraic vectors in the p-adic Langlands correspondence.
The main result is the following: if p>3 and if Pi is a finite
length admissible unitary Banach space representation of GL_2(Q_p),
for which the locally algebraic vectors are dense in Pi, then
the image of Pi by the Montreal functor is a potentially semi-stable
p-adic Galois representation. We will explain how the theory of
phi-gamma modules can be applied to prove this theorem.

**Matthew Emerton**, University of Chicago

*Density of Galois representations having prescribed types*

If X is a deformation space of p-adic Galois representations
(either local or global), and S is a set of types, then one can
consider the subset X(S) of points in the rigid analytic generic
fibre of X whose associated Galois representations are potentially
Barsotti--Tate at p of type lying in the set S. We give conditions
on the set S which imply that the set X(S) is Zariski dense in
X. As one application we conclude that the set of representations
which are both potentially Barsotti--Tate and crystabelline at
p is Zariski dense in X. This is joint work with Vytautus Paskunas.

**Laurent Fargues**, Université de Strasbourg

*Beyond the curve*

I will give results and conjectures I never had time to speak
about concerning the curve I defined with J.M.-Fontaine.

**Toby Gee**, Imperial College

*The Breuil-Mézard Conjecture for potentially Barsotti-Tate
representations*

I will discuss the proof of many cases of the Breuil-Mézard
conjecture for two-dimensional potentially Barsotti-Tate representations
(joint with Mark Kisin).

**David Geraghty**, Princeton University

*Modularity lifting in non-regular weight *

Modularity lifting theorems were introduced by Taylor
and Wiles and formed a key part of the proof of Fermat's Last Theorem.
Their method has been generalized successfully by a number authors
but always with the restriction that the Galois representations
and automorphic representations in questions have regular weight.
I will describe a method to overcome this restriction in certain
cases. I will focus mainly on the case of weight 1 elliptic modular
forms. This is joint work with Frank Calegari.

**Florian Herzig**, University of Toronto

*Ordinary representations of GLn(Qp) and fundamental algebraic
representations*

Motivated by a hypothetical p-adic Langlands correspondence for
GLn(Qp) we associate to an n-dimensional ordinary (i.e. upper-triangular)
representation rho of Gal(Qp-bar/Qp) over E a unitary Banach space
representation Pi(rho)^ord of GLn(Qp) over E that is built out
of principal series representations. (Here, E is a finite extension
of Qp.) There is an analogous construction over Fp-bar. In the
latter case we show under suitable hypotheses that Pi(rho)^ord
occurs in the rho-part of the cohomology of a compact unitary
group. This is joint work with Christophe Breuil.

**Payman Kassaei**, King's College London

*Modularity lifting in weight (1,1,...,1) *

We show how p-adic analytic continuation of overconvergent Hilbert
modular forms can be used to prove modularity lifting results
in parallel weigh one. Combined with mod-p modularity results,
these results can be used to prove certain cases of the strong
Artin conjecture over totally real fields.

**Ruochuan Liu**, University of Michigan

*Crystalline periods of eigenfamilies of p-adic representations*

We will explain a (phi,Gamma)-module variant of Kisin's finite
slope subspace technique and its applications to eigenfamilies
of p-adic representations

**Jonathan Pottharst**, Boston University

*Triangulation of eigenvarieties*

In the 1980s, Wiles showed the Galois representation over a Hida
family to be reducible when restricted to a decomposition group
at p. This result is the basis for the study of variation of Selmer
groups of modular forms in the family. In joint work with K. S.
Kedlaya and L. Xiao, we prove the analogous result over the eigencurve,
using a strong finiteness result for Galois cohomology of rigid
analytic families of (phi,Gamma)-modules over the Robba ring.
Applications to Iwasawa theory are then possible by our previous
work. (A similar result has been found recently by R. Liu, using
a significant strengthening of Kisin's method of interpolation
of crystalline periods.)

**David Savitt**, University of Arizona

*The Buzzard-Diamond-Jarvis conjecture for unitary groups*

We will discuss the proof of the weight part of Serre's conjecture
for rank two unitary groups in the unramified case (that is, the
Buzzard-Diamond-Jarvis conjecture for unitary groups). This
is joint work with Toby Gee and Tong Liu. More precisely,
we prove that any Serre weight which occurs is a predicted weight;
this completes the analysis begun by Barnet-Lamb, Gee, and Geraghty,
who proved that all predicted weights occur. Our methods
are purely local, using Liu's theory of (phi,G-hat) modules to
determine the possible reductions mod p of certain two-dimensional
crystalline Galois representations.

**Benjamin Schraen**, Centre national de la recherche scientifique

*On the presentation of supersingular representations*

Let F be a quadratic extension of the field of p-adic numbers
and k an algebraic closed field of characteristic p. We say that
a smooth representation of GL2(F) on a k-vector space is of finite
presentation if it is a of finite presentation in the category
of smooth representations of GL2(F). In this talk we prove that
if F is different of Qp, irreducible supersingular representations
of GL2(F) on k-vector spaces are not of finite presentation.

**Claus Sorensen**, Princeton University

*Eigenvarieties and invariant norms*

By a slight modification of the classical local Langlands correspondence,
one can attach a locally algebraic representation of GL(n) to
any n-dimensional potentially semistable Galois representation
(with distinct Hodge-Tate weights). A conjecture of Breuil and
Schneider asserts that the former admits an invariant norm. We
will prove this when the latter comes from a classical point on
an eigenvariety. More generally, for any definite unitary group
G, we will explain how its eigenvariety (of some fixed tame level)
mediates part of a global correspondence between Galois representations
of the CM field, and Banach-Hecke modules B with a unitary G-action.
For any regular weight W, we express the locally W-algebraic vectors
of B in terms of the Breuil-Schneider representation on the Galois
side.

**Yichao Tian**, Morningside Center of Mathematics

*Analytic continuation of weight 1 overconvergent Hilbert modular
forms in the tamely ramified case*

Abstract: The method of analytic continuation was initiated by
Buzzard-Taylor to treat the icosahedral case of the Artin conjecture
over Q. In this talk, I will explain how to extend this approach
to the Hilbert case. Let p be an odd prime number, and F be a
totally real field in which p is unramified. We prove that a p-adic
Galois representation over F, which is residually ordinarily modular
and saitsifies certain local conditions at p, comes from actually
a Hilbert modular form of weight 1. For the moment, we only know
how to treat the case where the Galois representation is tamely
ramified at p. This is a joint work with Payman Kassaei and Shu
Sasaki. I hope Payman will have explained the general principle,
then I will focus on the details of the analytic continuation
process.

**Gergely Zábrádi**, Eötvös Loránd

*From (phi,Gamma)-modules to G-equivariant sheaves on G/P*

Let G be the Q_p-points of a Q_p-split connected reductive group
with Borel subgroup P=TN. For any simple root alpha of T in N,
we associate functorially to a finitely generated etale (phi,Gamma)-module
D over Fontaine's ring (equipped with an additional linear action
of the group Ker(alpha)) a G-equivariant sheaf on the flag variety
G/P. This functor is faithful. In case of G=GL_2(Q_p) the global
sections of this sheaf coincide with the representation D\boxtimes
P^1 constructed by Colmez. This is joint work with Peter Schneider
and Marie-France Vigneras.

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