Arithmetic Geometry week in Tokyo June 4 to 8, 2012 Department of Mathematical Sciences University of Tokyo, Japan Japanese version

Talks

A. Abbes (CNRS, IHÉS),
Title : The p-adic Simpson correspondence
Abstract : The p-adic Simpson correspondence, initiated recently by Faltings, aims at describing all p-adic representations of the fundamental group of a (proper) smooth variety over a p-adic 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 p-adic 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 p-adic coefficients. The main ingredient of the proof is the product formula for epsilon factors, in whose proof we use the theory of arithmetic D-modules 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 pull-back 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 p-adic analogue.

C. Breuil (CNRS, Orsay),
Title : Ordinary representations of GLn(Qp) and fundamental algebraic representations.
Abstract : Motivated by a possible p-adic Langlands correspondence for GLn(Qp), we associate to an n-dimensional upper-triangular representation rho of Gal(Qp-bar/Qp) over Qp-bar a unitary Banach space representation Pi(rho)^ord of GLn(Qp) over Qp-bar 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 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 Florian Herzig.

G. Chenevier (CNRS, Palaiseau),
Title : The infinite fern of p-adic 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 p-adic 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 Galois-representations 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 Galois-representations on eigenvarieties for certain unitary groups form a trianguline family over a dense Zariski-open 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 non-cuspidal parts. This is a joint work with Yoichi Mieda.

P. Kassaei (London),
Title : Modularity lifting in weight (1,1,...,1)
Abstract : 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.

Y. Mieda (Kyoto),
Title : l-adic cohomology of the Rapoport-Zink tower for GSp(4)
Abstract : Rapoport-Zink spaces are certain moduli spaces of quasi-isogenies of p-divisible groups with additional structures and can be regarded as local analogues of Shimura varieties. Each Rapoport-Zink space has a natural system of rigid étale coverings, which is called the Rapoport-Zink tower. It is conjectured that the l-adic cohomology of this tower partially realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for rather general reductive groups. In this talk, I will investigate the l-adic cohomology of the Rapoport-Zink space for GSp(4) by rigid-geometric techniques (Lefschetz trace formula, p-adic 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 Ihara-Langlands-Kottwitz 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 l-adic Galois representations (joint work with Anna Cadoret)
Abstract : An l-adic 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 \ell-adic 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 l-adic 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 l-adic 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 : p-adic perverse sheaves and arithmetic D-modules with singularities along a simple normal crossing divisor
Abstract : I will discuss a generalization of the theory of crystalline p-adic representations/sheaves and filtered Frobenius modules by G. Faltings and O. Brinon in the relative case, to p-adic perverse sheaves for the stratification given by a simple normal crossing divisor. We use log étale cohomology and log arithmetic D-modules.

T. Tsushima (Kyushu),
Title : Geometric realization of the local Langlands correspondence for representations of conductor three
Abstract : A purely local proof of the local Jacquet-Langlands 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 Lubin-Tate 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 Jacquet-Langlands 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 p-adic 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 Fontaine-Mazur 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.

The conference will take place in the conference hall of the Department of Mathematical Sciences, University of Tokyo, Komaba 153-8914 Tokyo, Japan.
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