Description of the minicourses
Each minicourse consists of two 90minutes lectures in the mornings.
Study groups will be conducted in the afternoons.
 Analytic spaces and their étale cohomology / Vladimir Berkovich
Lecture 1. Basic notions of analytic geometry; affinoid algebras and affinoid spaces; analytic spaces and morphisms;
analytic spaces associated to algebraic varieties and formal schemes; the padic half plane.
Lecture 2. Étale topology on an analytic space; basic results on étale cohomology; vanishing cycles for formal schemes.
 Rigid cohomology / Elmar GroßeKlönne
Lecture notes (pdf)
Lecture 1. Motivation (Zeta function of a variety over a finite field, Lefschetz trace formula, lifting from characteristic p to characteristic 0).
Basics of deRham cohomology, overconvergence, dagger algebras and rigid cohomology. Functoriality and Frobenius.
Lecture 2. Finiteness of rigid cohomology. Lefschetz trace formula. Fisocrystals. Comparison between deRham and rigid cohomology, Hodge filtration
and filtered φmodules; admissibility. Rigid cohomology without smoothness or affineness.
 padic symmetric domains and uniformization / Ehud de Shalit
Exercises (pdf)
Lecture 1. The padic upper half plane and its coverings:
Its geometry and the reduction to the BruhatTits tree; étale and deRham cohomology;
the padic upper half plane as a moduli space and the Drinfel'd tower; Carayol's program.
Lecture 2. Uniformization and the higher dimensional Drinfel'd domains:
Quotients by discrete cocompact subgroups and algebraization; the CerednikDrinfel'd theorem;
the higher dimensional padic symmetric domains and their quotients.
 Period domains and their cohomology / Sascha Orlik
Lecture notes (pdf)
Lecture 1. Period domains over finite and padic field for GL_{n}:
Filtered vector spaces, semistability, HarderNarasimhan filtration and HNpolygons.
Relation to Geometric Invariant Theory. Isocrystals and period domains over padic fields.
Lecture 2. Cohomology of period domains:
Étale cohomology over finite fields, étale cohomology over padic fields in the "basic case".
Survey of period domains for arbitrary reductive groups.
 padic representations of padic groups / Peter Schneider
Lecture notes (pdf)
Lecture 1. (padic) Banach spaces and Banach space representations of compact padic Lie groups:
examples; duality with Iwasawa modules; admissibility.
Lecture 2. Rational representation theory of GL_{n}(Q_{p}): spherical smooth representations and the Satake isomorphism;
Banach completions of SatakeHecke algebras and a framework for an unramified padic local Langlands correspondence.
 Modp Galois representations / Laurent Berger
Lecture notes (pdf)
Lecture 1. Modp representations of the local Galois group; Fontaine's rings in characteristic p; (Φ,Γ)modules and the correspondence.
Lecture 2. Ψ and the construction of representations of B_{2}(Q_{p}). Representations of GL_{2}(Q_{p}).
Identification of the representations constructed above.
 Applications to number theory: padic families of modular forms / Amnon Besser
Lecture 1 (pdf) Lecture 2 (pdf)
Lecture 1. Motivation (congruences between modular forms and padic zeta functions); overconvergent padic modular forms;
the canonical subgroup and the U_{p} operator; strategies of Hida and Coleman for creating families.
Lecture 2. Completely continuous operators, families of padic modular forms; generalizations to Hilbert modular forms; other approaches.
Special lectures
 Eyal Goren (McGill): "Canonical subgroups over Hilbert modular varieties".
 Lecture notes (pdf)
Abstract: The theory of the canonical subgroup originated with Lubin and Katz, initially motivated by defining the U operator for overconvergent elliptic modular
forms. The power of such results became apparent in work on the Artin conjecture and in results on analytic continuation of overconvergent modular forms.
Many authors have studied the canonical subgroup in various settings, for example, AbbesMokrane, AndreattaGasbarri, B. Conrad, L. Fargues, KisinLai and there are as yet
unpublished results by K. Buzzard, E. Nevens and J. Rabinoff, and the list is not complete.
For applications as mentioned above one needs a very refined theory of canonical subgroups.
In a joint project with Payman Kassaei (King's College) we improve on the available literature in the case of Hilbert modular varieties, using a different technique than used
by other authors; it continues our work in the case of curves and is based on the study of the special fiber of morphisms between the Shimura varieties in question and on
deformation theory for abelian varieties. The specific modulitheoretic description of the varieties is only relevant in the analysis of the geometry of the moduli spaces
involved. Once this information is obtained, only techniques from rigid geometry are used. Our approach thus seems suited to generalization to a wide class of Shimura varieties.
 Urs Hartl (Münster): "The image of the RapoportZink period morphism".
 Rapoport and Zink have constructed moduli spaces for pdivisible groups
and period morphisms from these spaces to Grassmann varieties. We
present their definitions and discuss the image of the period morphisms.
Florian Herzig (Northwestern): "Weight Cycling and Serretype Conjectures".
 Abstract: Suppose that \rho is a threedimensional modular mod p Galois representation whose restriction to the decomposition groups at p is irreducible and generic.
If ρ is modular in some (Serre) weight, then a representationtheoretic argument shows that it also has to be modular in certain other weights
(we can give a short list of possibilities). This goes back to an observation of Buzzard for GL_{2}.
Previously we formulated a Serretype conjecture on the possible weights of ρ. Under the assumption that the weights of ρ are contained in the predicted weight set,
we apply the above weight cycling argument to show that ρ is modular in precisely all the nine predicted weights.
This is joint work with Matthew Emerton and Toby Gee.
David Kazhdan (Hebrew University): "Representation theory of reductive groups over two dimensional local fields".
 Vytautas Paskunas (Bielefeld): "On the image of Colmez' Montreal functor".
 Abstract: To a unitary LBanach space representation Pi of GL_{2}(Q_{p}), satisfying some admissibility conditions,
Colmez can in a functorial way attach a padic L representation V(π) of Gal^{({Q}p/Qp)}. It is expected that if Π is irreducible then
dim_{L} V(Π)≤2.
We show that this is true in many cases.
 Michael Rapoport (Bonn): "Φmodules and coefficient spaces".
 Abstract: I will define stacks of modules equipped with a Frobenius semilinear endomorphism.
These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces used by Kisin in his study of
deformation spaces of local Galois representations. I will discuss the special and the general fiber of these stacks.
This is joint work with G. Pappas.
 Matthias Strauch (Indiana): "Jordan Hoelder series for certain parabolically induced representations".
 Abstract: This talk is about joint work with S. Orlik and concerns locally analytic representations of a (split) $p$adic reductive group $G$,
induced from a parabolic subgroup $P$. Let $V$ be a finitedimensional algebraic representation of $P$ on which the unipotent radical of $P$ acts trivially.
Then we consider the locally analytic induced representation $Ind^G_P(V)$ which consists of locally analytic $V$valued functions on $G$
which are equivariant with respect to $P$.
We explain how to find a JordanHoelder series for $Ind^G_P(V)$, using as input a JordanHoelder series for the Verma module $U(\mathfrak{g}) \otimes_{U(\mathfrak{p})} V'$.
More generally, we start with a $U(\mathfrak{g})$module $M$, all of whose weights are integral, associate to $M$ a locally analytic representation,
and determine a JordanHoelder series for this representations in terms of the JordanHoelder series for $M$.
School Schedule (Word file)
