Main

Program

 The Hebrew University of Jerusalem Einstein Institute of Mathematics

# Minerva School on P-adic Methods in Arithmetic Algebraic Geometry

## Description of the mini-courses

Each mini-course consists of two 90-minutes lectures in the mornings.
Study groups will be conducted in the afternoons.

1. 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 p-adic half plane.

Lecture 2. Étale topology on an analytic space; basic results on étale cohomology; vanishing cycles for formal schemes.

2. Rigid cohomology / Elmar Große-Klönne
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 de-Rham cohomology, overconvergence, dagger algebras and rigid cohomology. Functoriality and Frobenius.

Lecture 2. Finiteness of rigid cohomology. Lefschetz trace formula. F-isocrystals. Comparison between de-Rham and rigid cohomology, Hodge filtration and filtered φ-modules; admissibility. Rigid cohomology without smoothness or affine-ness.

3. p-adic symmetric domains and uniformization / Ehud de Shalit
Exercises (pdf)

Lecture 1. The p-adic upper half plane and its coverings: Its geometry and the reduction to the Bruhat-Tits tree; étale and de-Rham cohomology; the p-adic 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 Cerednik-Drinfel'd theorem; the higher dimensional p-adic symmetric domains and their quotients.

4. Period domains and their cohomology / Sascha Orlik
Lecture notes (pdf)

Lecture 1. Period domains over finite and p-adic field for GLn: Filtered vector spaces, semi-stability, Harder-Narasimhan filtration and HN-polygons. Relation to Geometric Invariant Theory. Isocrystals and period domains over p-adic fields.

Lecture 2. Cohomology of period domains: Étale cohomology over finite fields, étale cohomology over p-adic fields in the "basic case". Survey of period domains for arbitrary reductive groups.

Lecture notes (pdf)

Lecture 1. (p-adic) Banach spaces and Banach space representations of compact p-adic Lie groups: examples; duality with Iwasawa modules; admissibility.

Lecture 2. Rational representation theory of GLn(Qp): spherical smooth representations and the Satake isomorphism; Banach completions of Satake-Hecke algebras and a framework for an unramified p-adic local Langlands correspondence.

6. Mod-p Galois representations / Laurent Berger
Lecture notes (pdf)

Lecture 1. Mod-p 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 B2(Qp). Representations of GL2(Qp). Identification of the representations constructed above.

7. Applications to number theory: p-adic families of modular forms / Amnon Besser
Lecture 1 (pdf)      Lecture 2 (pdf)

Lecture 1. Motivation (congruences between modular forms and p-adic zeta functions); overconvergent p-adic modular forms; the canonical subgroup and the Up operator; strategies of Hida and Coleman for creating families.

Lecture 2. Completely continuous operators, families of p-adic 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, Abbes-Mokrane, Andreatta-Gasbarri, B. Conrad, L. Fargues, Kisin-Lai 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 moduli-theoretic 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 Rapoport-Zink period morphism".
Rapoport and Zink have constructed moduli spaces for p-divisible 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 Serre-type Conjectures".
Abstract: Suppose that \rho is a three-dimensional 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 representation-theoretic 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 GL2. Previously we formulated a Serre-type 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 L-Banach space representation Pi of GL2(Qp), satisfying some admissibility conditions, Colmez can in a functorial way attach a p-adic L- representation V(π) of Gal({Q}p/Qp). It is expected that if Π is irreducible then dimL V(Π)≤2. We show that this is true in many cases.

Michael Rapoport (Bonn): "Φ-modules and coefficient spaces".
Abstract: I will define stacks of modules equipped with a Frobenius semi-linear 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 finite-dimensional 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 Jordan-Hoelder series for $Ind^G_P(V)$, using as input a Jordan-Hoelder 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 Jordan-Hoelder series for this representations in terms of the Jordan-Hoelder series for $M$.

School Schedule (Word file)