# THEMATIC PROGRAMS

May  5, 2016

## Workshop on Shimura varieties and related topics March 4-8, 2003

Organizers: T. Haines (Maryland) and G. Pappas (Michigan State)

 James Arthur Elena Mantovan Don Blasius Sascha Orlik Ching-Li Chai Michael Rapoport Laurent Fargues Jeremy Teitelbaum Jens Funke Torsten Wedhorn Ulrich Goertz Uwe Weselmann Eyal Goren Tonghai Yang Haruzo Hida Chai-Fu Yu Tetsushi Ito Gerard Laumon Ron Livne

James Arthur, University of Toronto
On the universal groups in automorphic forms and algebraic geometry

It is believed that there are universal groups that govern fundamental processes in number theory. One of these is the Langlands group, which is conjectured to classify automorphic representations. Another is the motivic Galois group of Grothendieck, which is thought to classify parallel objects in algebraic geometry. We shall discuss some questions on the possible structure of these groups.

Don Blasius,UCLA
Weight-monodromy conjecture for some simple Shimura varieties

We will prove Ramanujan Conjecture and Weight-Monodromy conjecture for Shimura varieties of arbitrary dimension attached to forms of GU(2) and GU(3).

Ching-Li Chai, University of Pennsylvania
The Hecke orbit problem: Hecke symmetries and Oort foliation

One striking feature of the moduli space $\mathcal{A}_g$ is that it has a lot of symmetries, usually attributed to Hecke. These symmetries are algebraic correspondences on $\mathcal{A}_g$ coming from the symplectic group $GSp(2g)$. Over the complex numbers these Hecke correspondences are usually studied using the complex uniformization of $\mathcal{A}_g$ described above. Over a field of positive characteristic, for instance the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$, the moduli space $\mathcal{A}_g$ has fine structures not coming from characteristic $$0$$, because the local structure of abelian varieties in characteristic $$p$$ may not all look alike''. Typical examples of such fine structures include the Newton polygon stratification and the Ekedahl-Oort stratification. These two fine structures are both preserved by the Hecke symmetries, but in general they are not enough to characterize the orbit of the Hecke symmetries. \smallbreak

Recently Oort defined a foliation'' structure on the moduli space $\mathcal{A}_g$ in characteristic $$p$$. Conjecturally the Zariski closure of the orbits of the Hecke symmetries are exactly the closures of leaves'' in Oort's foliation. We will explain some recent progress on the Hecke orbit problem, including the local structure of the leaves. The latter generalizes a theorem of Serre and Tate announced in the 1964 Summer Institute of Algebraic Geometry at Woods Hole.

Laurent Fargues, Universite de Paris VII (Jussieu)
Boundary cohomology of Lubin-Tate spaces

I will explain how a generalized theory of canonical subgroups applies to exhibit a stratification of Lubin-Tate spaces outside sufficiently great compact subset of the open p-adic ball. To each of this stratum is associated a parabolic subgroup of $GL_n$ and the cohomology (in the limit) of this stratum is parabolicaly induced. This applies to a conjecture of Prasad on autodual representations of division algebras. I will discuss how this should apply to finish M. Strauch program on proving part of Harris Taylor results by purely local methods.

Jens Funke, Fields Institute
(Singular) theta lifts and the construction of Green currents and differential characters for cycles in locally symmetric spaces of orthogonal and unitary type

In this talk, we establish a duality result between the singular theta lift introduced by Borcherds and the Kudla-Millson theta lift. For O(p,2), this gives rise to the construction of Green functions for certain special cycles, while for arbitrary signature, one obtains differential
characters in the sense of Cheeger and Simons. We also discuss the extension of these results to unitary groups.
This is joint work with Jan Bruinier.

Ulrich Goertz, Universitaet Koeln
The Jordan-Hoelder series of nearby cycle sheaves on some Shimura varieties and affine flag varieties

We consider sheaves of nearby cycles on certain Shimura varieties with Iwahori level structure and affine flag varieties, respectively. Describing the Jordan-Hoelder series of these sheaves reveals interesting combinatorial patterns, which can be interpreted in terms of the cohomology of certain intersections of Schubert cells and opposite Schubert cells in the affine flag variety for the corresponding group. This is joint work with Thomas Haines.

Eyal Goren, McGill University
Local models and displays

I would like to start with some basics about local models - in fact with no level p involved - to give some examples and illustrate some phenomena. I would then like to explain what are displays, how one uses them to study deformation problems and a technical lemma that allows one to calculate universal displays. This will then be applied to calculating the universal display of Hilbert modular varieties in the case of bad reduction, and time permitting, in cases of quaternion algebras ramified at p.

Haruzo Hida, UCLA
Automorphism Groups of Shimura Varieties mod $p$

We try to prove that any scheme automorphism of each geometrically irreducible component of a certain Shimura variety of PEL type modulo $p$ (of prime-to-$p$ level) is given by an isogeny action (generalizing a result of Shimura in characteristic 0). This can be generalized also to the naive Igusa tower over the variety. If time allows, we would be able to discuss some application of this result.

Tetsushi Ito, Max-Planck Institut (Bonn)
Weight-monodromy conjecture for p-adically uniformized varieties

In this talk, I will give a proof of the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties with p-adic uniformization by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply an argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining this result with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants.

Gerard Laumon, Universite de Paris-Sud (Orsay)
On the fundamental lemma for unitary groups

The Fundamental Lemma'' is a series of combinatorial identities which have been discovered by Langlands and which have been precisely formulated by Langlands and Shelstad. Over a non archimedian local field of equal characteristics, the orbital integrals which enter in the statement of the Fundamental Lemma'' for unitary groups are directly related to the affine Springer fibers for the general linear groups and it is not difficult to formulate a geometric conjecture which implies the Fundamental Lemma''. In this talk, I would like to explain a natural link between those affine Springer fibers and some compactified Jacobians of singular curves. Moreover, using this link, I would like to show that the above geometric conjecture follows from the purity conjecture for the cohomology of the affine Springer fibers of Goresky, Kottwitz and MacPherson.

Ron Livne, The Hebrew University of Jerusalem
Local points and parity of jacobians

Poonen and Stoll had observed that the Cassels-Tate pairing on the Shafarevich-Tate group of a principally polarized abelian variety over a global field need not be alternating. Thus the order of $\Sha$ need not be a square.

For a jacobian of a curve there is an explicit criterion to decide when this happens in terms of the local points at primes of bad reduction. We will apply this to Shimura curves. The analysis of these local points uses the p-adic uniformization, and can be done in much greater generality.

The work described is joint with B. Jordan and Y. Varshavsky.

Elena Mantovan, Berkeley
On certain unitary group Shimura varieties

We shall describe the geometry of a certain class of PEL type Shimura varieties. In particular, we shall study the Newton polygon stratification of the reduction in positive characteristic p of these Shimura varieties. We shall show that each stratum can be described in terms of the products of some smooth varieties (we call Igusa varieties) with the reduced fibers of the pertinent Rapoport-Zink spaces, and of the action on them of a certain p-adic group (which depends on the stratum). As a result of this analysis, weobtain a description of the l-adic cohomology of the Shimura varieties in terms of the l-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces, in the appropriate Grothendieck group.

Sascha Orlik, Universitaet Leipzig
The cohomology of period domains

We compute the \'etale cohomology of period domains over local fields for quasisplit reductive groups. The period domains, which were introduced by Rapoport and Zink, are open admissible rigid-analytic subsets of generalized flag varieties. They parametrize (weakly) admissible filtrations of a given isocrystal with additional structure of a reductive group.

Michael Rapoport, Universitaet Koeln
An update on affine Deligue-Luszlig varieties

After defining the objects in my title, I will survey recent results on the due to Kottvitz, Leigh, Reuman and Wintenberger.

Jeremy Teitelbaum, University of Illinois (Chicago)
P-adic analytic representation theory and inklings of a p-adic analytic Langlands correspondenc

Is there a "p-adic analytic Langlands correspondence" which gives an automorphic interpretation to the families of overconvergent modular forms and p-adic Galois representations that have been constructed by Hida and Coleman? In this talk I will survey some recent developments suggesting that such a correspondence exists. In particular I will give an overview of the theory of p-adic analytic representation theory I have been developing with Peter Schneider and discuss briefly how it has been related to arithmetic by Breuil and Emerton.

Torsten Wedhorn, Universitaet Koeln
A generalization of the Ekedahl-Oort stratification

In this joint work with Ben Moonen a generalization of the Ekedahl-Oort stratification is described: For every abelian scheme over a base scheme $S$ of characteristic $p$, Ekedahl and Oort defined a stratification of $S$. This has been in particular succesfully applied to universal abelian schemes over moduli spaces of abelian varieties. We generalize this to arbitrary proper and smooth schemes of Hodge type. It turns out that we get stratifications which are indexed by certain quotients of a symmetric group. There is also the notion of this stratification with $G$-structure for $G$ a reductive group and using results of Lusztig we get a stratification indexed by a certain quotient
of the Weyl group of $G$.

Uwe Weselmann, University of Heidelberg
A twisted topological trace formula and liftings of automorphic representations

For the cohomology groups of locally symmetric spaces attached to a connected reductive group $G$ we developed a topological trace formula describing the action of Hecke operators twisted by an outer automorphism $\eta$. In some cases this twisted trace formula is stable and may be compared with the untwisted trace formula for the stable endoscopic group $G_1$. The fundamental lemma for this comparison in the case $G=PGl_5$, $G_1=Sp_4$ can be shown (joint work with J. Ballmann and R. Weissauer) to be equivalent to the fundamental lemma in the case $G=GL_4\times GL_1$ and $G_1=GSp_4=GSpin_5$ (proved by Flicker). Using character identities between local representations and properties of the $\theta$-lift one gets multiplicity results for automorphic representations contributing to the cohomology of Siegel modular 3-folds.

Tonghai Yang, University of Wisconsin
Doubling integral and a variant of the Gross-Zagier formula

This is part of joint work with Steve Kudla and Michael Rapoport. We will first describe how to write the L-function of a modular form as an doubling integral against some Eisenstein series on Sp(4), following Piateski-Shapiro and Rallis. We will then explain how to relate the central derivative of the Eisenstein series with Height pairing of Heegner divisors. Finally we put them together to obtain a variant of the Gross-Zagier formula, and also a variant of Gross's formula on central L-value.

Chai-Fu Yu, National Tsing-Hua University
Fine structures and Hecke orbit problems of Hilbert-Blumenthal varieties

We will report current progress on the reduction of Hilbert-Blumenthal varieties. We determine the dimension and singularities of the strata defined by various p-adit invariants. The invariants include Lie types, a-numbers and leaves. We will also explain the connection with the Hecke orbit problems and the irreducibility of the non-supersingular strata. We can prove the irreducibility of these non-supersingular strata in many cases, including the cases when p is unramified or totally ramified. For the supersingular strata, the number of irreducible components is determined and is expressed in terms of special values of the zeta function. This is a joint work with Ching-Li Chai.