Talks
A. Abbes (CNRS, IHÉS),
Title : The padic Simpson correspondence
Abstract : The padic Simpson correspondence, initiated recently by Faltings,
aims at describing all padic representations of the fundamental group of a (proper)
smooth variety over a padic field in terms of linear algebra (Higgs modules).
I will present a new approach for this correspondence generalizing that of Faltings.
This is a joint work with Michel Gros.
T. Abe (IPMU, Tokyo),
Title : Langlands program for padic coefficients and the product formula for epsilon factors
Abstract : We will show that Deligne's "petits camarades conjecture"
is equivalent to Langlands program for function fields for padic coefficients.
The main ingredient of the proof is the product formula for epsilon factors, in whose proof
we use the theory of arithmetic Dmodules of Berthelot. A large part of this talk is a joint work with A. Marmora.
K. Bannai (Keio, Yokohama),
Title : On the Eisenstein class for Hilbert modular surfaces
Abstract : This is a work in progress with G. Kings. The Eisenstein class for Hilbert modular
variety is defined to be the pullback of the motivic polylogarithm class on the universal abelian variety
with respect to torsion points. In this talk, we will give a conjectural formula for the Hodge realization
of the Eisenstein class for Hilbert modular surfaces, and discuss the padic analogue.
C. Breuil (CNRS, Orsay),
Title : Ordinary representations of GLn(Qp) and fundamental algebraic representations.
Abstract : Motivated by a possible padic Langlands correspondence for GLn(Qp),
we associate to an ndimensional uppertriangular representation rho of
Gal(Qpbar/Qp) over Qpbar a unitary Banach space representation
Pi(rho)^ord of GLn(Qp) over Qpbar that is built out of principal series representations.
The construction of Pi(rho)^ord is guided by the structure of the "ordinary part" of the tensor product of
the fundamental algebraic representations of GLn. There is an analogous construction over Fpbar.
In the latter case we show (under suitable hypotheses) that Pi(rho)^ord occurs in the rhopart of the cohomology
of a compact unitary group. This is joint work with Florian Herzig.
G. Chenevier (CNRS, Palaiseau),
Title : The infinite fern of padic Galois representations of the absolute Galois group of Q_p
Abstract : Let X_d be the rigid analytic space over
Q_p parameterizing all the padic semisimple Galois representations of the absolute Galois group of
Q_p in dimension d. We will discuss certain properties of X_d and of its subset of points parameterizing
crystalline representations. We shall show that this subset is Zariski dense in many irreducible components of
X_d, including all the "residually irreducible" ones when p>d+1.
A prominent role is played by the moduli space of families of triangular (phi,gamma)modules over the Robba ring of
Q_p.
K. Fujiwara (Nagoya),
Title : Valuations for topological spaces
Abstract : For a valuation field K of height 1,
Tate defined the notion of rigid analytic spaces over K in 1960's.
The category of rigid analytic spaces over K admits many equivalent definitions.
On the other hand, for general K, the category of Berkovich analytic spaces over K has more objects.
In this talk, we introduce a category of topologically ringed spaces over K,
and show that the above categories are seen as full subcategories. This is a joint work with Fumiharu Kato.
E. Hellmann (Bonn),
Title : Families of trianguline representations and finite slope spaces
Abstract : We show how to construct families of Galoisrepresentations on rigid spaces using families
of (phi,Gamma)modules and apply this to (a variant of) Chenevier's space of trianguline (phi,Gamma)modules.
This yields a new definition of Kisin's finite slope subspace as well as higher dimensional analogues.
Especially we show that these finite slope spaces contain eigenvarieties for unitary groups as closed subspaces.
This implies that the Galoisrepresentations on eigenvarieties for certain unitary groups form a trianguline family
over a dense Zariskiopen subset.
N. Imai (RIMS, Kyoto),
Title :
Cohomology of crystalline loci of open Shimura varieties of PEL type
Abstract : We define a crystalline locus of a Shimura variety of PEL type, and compare its cohomology with a cohomology of the Shimura variety.
Actually, we prove that they coincide up to noncuspidal parts.
This is a joint work with Yoichi Mieda.
P. Kassaei (London),
Title : Modularity lifting in weight (1,1,...,1)
Abstract : We show how padic analytic continuation of overconvergent Hilbert modular forms can be used to prove modularity lifting results in parallel weigh one. Combined with modp modularity results, these results can be used to prove certain cases of the strong Artin conjecture over totally real fields.
Y. Mieda (Kyoto),
Title : ladic cohomology of the RapoportZink tower for GSp(4)
Abstract : RapoportZink spaces are certain moduli spaces of quasiisogenies of
pdivisible groups with additional structures and can be regarded as local
analogues of Shimura varieties.
Each RapoportZink space has a natural system of rigid étale coverings,
which is called the RapoportZink tower.
It is conjectured that the ladic cohomology of this tower partially
realizes the local Langlands correspondence and the local JacquetLanglands
correspondence for rather general reductive groups.
In this talk, I will investigate the ladic cohomology of the RapoportZink
space for GSp(4) by rigidgeometric techniques (Lefschetz trace formula,
padic uniformization,...), and give a result on the conjecture mentioned
above.
J. Nekovar (Paris),
Title : Some remarks on cohomology of quaternionic Shimura varieties
Abstract : We are going to explain what information about étale cohomology
of quaternionic Shimura varieties can be obtained from congruence relations,
without using the IharaLanglandsKottwitz method. This is a joint work with
C. Cornut.
A. Shiho (Tokyo),
Title : On restriction of overconvergent isocrystals
Abstract : We explain some results on the relation between certain properties
of a given overconvergent isocrystal and certain properties of the restriction of it to curves.
A. Tamagawa (RIMS, Kyoto),
Title : Variation of ladic Galois representations (joint work with Anna Cadoret)
Abstract : An ladic representation of the (arithmetic) fundamental group of a variety
over a number field (or, more generally, a finitely generated field) can be viewed as a family of
\elladic Galois representations parametrized by the variety.
The main subject of a series of joint work of Cadoret and myself is to understand how
the images of those ladic Galois representations vary and, in particular,
to show that they have certain rigidity properties.
In this talk, I will present a summary of our invesitigation of this problem, mainly in the case where
the variety in question is a curve, with special empahasis on recent progress for general ladic
representations and on some open questions.
Y. Tian (Beijing),
Title : Analytic continuation of overconvergent Hilbert modular forms of parallel weight 1
Abstract : The method of analytic continuation was introduced by Buzzard and Taylor
to treat the icosahedral case of the Artin conjecture over Q.
In this talk, we will explain an analog of this result in the Hilbert case.
More precisely, we'll show that a finite slope overconvergent Hilbert modular form of level
Gamma_1(Np) extends automatically to a very large region in the rigid analytic Hilbert modular variety.
Combined with the theory of companion forms developed by Toby Gee,
this will allow us to show a modularity lifting theorem for Hilbert modular forms in weight 1.
This is a joint work with Payman Kassaei and Shu Sasaki.
T. Tsuji (Tokyo),
Title : padic perverse sheaves and arithmetic Dmodules
with singularities along a simple normal crossing divisor
Abstract : I will discuss a generalization of the theory
of crystalline padic representations/sheaves and
filtered Frobenius modules by G. Faltings and O. Brinon
in the relative case, to padic perverse sheaves for
the stratification given by a simple normal crossing divisor.
We use log étale cohomology and log arithmetic Dmodules.
T. Tsushima (Kyushu),
Title : Geometric realization of the local Langlands correspondence for representations of conductor three
Abstract :
A purely local proof of the local
JacquetLanglands correspondence for
$?textit{GL}_2$ is known.
But a purely local proof of a geometric realization of the local Langlands correspondence for $?textit{GL}_2$ is not known.
In this talk, we introduce
the stable reduction of the LubinTate
curve with $K_1(?mathfrak{p}^3)$level
structure. Our talk contains
the even characteristic case.
By directly computing the epsilon factors and using the local JacquetLanglands correspondence for $?textit{GL}_2$, we prove that, in the cohomology, the local Langlands correspondece is actually realized for representations of conductor three.
This is a joint work with Naoki Imai.
S. Wewers (Ulm),
Title : Cyclic extensions and the local lifting problem
Abstract : The (local) Oort conjecture states that any finite cyclic Galois extension
k[[z]]/k[[t]], where k is an algebraically closed field of characteristic p>0,
can be lifted to a cyclic Galois extension R[[z]]/R[[t]], where R is a dvr of characteristic zero with residue field k.
I will report on recent joint work with Andrew Obus in which we prove a substantial part of the conjecture.
(In a more recent preprint, Florian Pop has given a proof of the full conjecture, using our results.)
In my talk, I will focus on the use of differential Swan conductors in our proof and how they lead to
a certain nonhomogenous linear differential equation.
L. Xiao (Chicago).
Title : Global triangulation over eigenvarieties
Abstract : In Coleman and Mazur's ground breaking paper, they introduced a rigid analytic curve,
called the eigencurve, parametrizing the padic overconvergent modular forms.
The associated family of Galois representations is crystalline at p
form a Zariski dense subset of points on the eigencurve.
It was first observed by Kisin that the crystalline periods of this family of Galois representations vary continuously;
this fact is one of the crucial ingredients of his proof of FontaineMazur conjecture.
Following Colmez, one may interpret Kisin's construction as the existence of a triangulation for the associated family
of (phi, Gamma)modules. We will prove that such triangulation extends to the whole eigencurve,
generalizing Kisin's result on local affinoid neighborhoods of classical points.
We will show that the same argument generalizes for eigenvarieties.
This is a joint work with Jay Pottharst and Kiran Kedlaya.
