June 18, 2024
January-June 2012 (Winter/Spring
Thematic Program on Galois Representations

March 19-23, 2012
Workshop on Cohomology of Shimura varieties: Arithmetic Aspects and the Construction of Galois representations

Organizers: Laurent Clozel (Paris 11), Matthew Emerton (U Chicago),
David Geraghty (Princeton and IAS), Sug Woo Shin (MIT)

Supported by the National Science Foundation
Award #1101503

The workshop will have several goals, including: describing the latest results on the cohomology of Shimura varieties;explaining the key ideas and techniques which underly these results; analyzing the applications of these results to the construction of Galois representations attached to automorphic forms, with the goal of describing the state of the art in this problem.


B. Howard (Boston College)
A conjectural extension of the Gross-Zagier theorem

I will talk about ongoing work with Bruinier and Yang toward an extension of the Gross-Zagier theorem to higher weight modular forms. The goal is to find an arithmetic interpretation of the central derivative of the Rankin-Selberg convolution L-function of a cuspidal eigenform of weight n with a theta series of weight n-1. The arithmetic interpretation is in terms of heights of special cycles on unitary Shimura varieties.

A. Caraiani (Harvard University)
Weight spectral sequences and monodromy

Given a cuspidal automorphic representation of GL(n) which is conjugate self-dual and regular algebraic, one can associate to it an l-adic Galois representation which is compatible with local Langlands. I will explain the geometric techniques behind establishing the compatibility of monodromy operators.

M. Emerton (Unversity of Chicago)
Shimura varieties and Galois representations: an overview

The talk will present an overview of recent results related to the construction of Galois representations via the cohomology of Shimura varieties. The talk will include a review of some relevant background concepts, and will provide some motivation for the constructions, including the role of endoscopy. Precise details of the constructions will be the subject of subsequent talks.

L. Fargues (Université de Strasbourg)
The cohomology of the basic locus of Shimura varieties

We will describe Rapoport-Zink uniformization of the tube over the basic locus of PEL type Shimura varieties. We will then explain how to use it to realize local Langlands correspondences in the l-adic cohomology of basic Rapoport-Zink spaces.

T. Gee (Imperial College)
p-adic Hodge-theoretic properties of etale cohomology with mod p coefficients, and the cohomology of Shimura varieties

I will discuss some new results about the etale cohomology of varieties over a number field or a p-adic field with coefficients in a field of characteristic p, and give some applications to the cohomology of unitary Shimura varieties. (Joint with Matthew Emerton.)

A. Iovita (Concordia University)
Overconvergent modular sheaves and p-adic families of Hilbert modular forms

This talk will present joint work with F. Andreatta, V. Pilloni, G. Stevens in which we define p-adic families of overconvergent Hilbert modular forms and construct the cuspidal part of the respective eigenvariety.

M. Kisin (Harvard University)
Integral models for Shimura varieties of abelian type (slides)

We will explain a construction of good integral models for Shimura varieties of abelian type, with hyperspecial level structure.

M. Kisin (Harvard University)
Mod p points on Shimura varieties of abelian type

The Langlands-Rapoport gives a description of the mod p points of a Shimura variety with hyperspecial level structure at p. The ultimate motivation for the conjecture is Langlands' program to describe the zeta function of a Shimura variety in terms of automorphic L-functions. We will explain some results towards the conjecture for Shimura varieties of abelian type.

S. Kudla (University of Toronto)
Generating series for arithmetic cycles on Shimura varieties

In this talk I will describe the construction of special cycles for the Shimura varieties attached to unitary groups and the generating series for their classes in arithmetic Chow groups. I will then review some of the evidence for the conjecture/speculation that such series are the q-expansions of modular forms valued in the Chow groups.

K. Lan (Princeton University and IAS)
Vanishing theorems for torsion automorphic sheaves

The talk will explain what the title means, with some review of background knowledge.

K. Madapusi Pera (Harvard University)
Compactifications of integral models of Shimura varieties of Hodge type

I will explain how to construct good integral models for compactifications of Shimura varieties of Hodge type.

E. Mantovan (California Institute of Technology)
The reduction modulo p of Shimura varieties.

This will be a survey lecture reporting on the geometry of PEL-type Shimura varieties at primes of bad reduction. We will introduce and discuss the Newton polygon stratification and Oort's foliation of the reduction, and their relation to Igusa varieties and Rapoport-Zink spaces. These ideas have immediate application to the study of the Galois representations arising in the cohomology of Shimura varieties.

P. Scholze (Universität Bonn)
A new approach to the local Langlands correspondence for GL_n over p-adic fields

We give a new local characterization of the Local Langlands Correspondence, using deformation spaces of p-divisible groups, and show its existence by a comparison with the cohomology of some Shimura varieties. This reproves results of Harris-Taylor on the compatibility of local and global correspondences, but completely avoids the use of Igusa varieties and instead relies on the classical method of counting points a la Langlands and Kottwitz. Further, we have a new proof of bijectivity of this correspondence, relying on a description of the inertia-invariant nearby cycles in certain situations.

P. Scholze (Universität Bonn)
On the cohomology of compact unitary group Shimura varieties at ramified split places

(joint with Sug Woo Shin) We extend the methods of our proof of the LLC to general (possibly ramified) PEL data. In particular, we formulate a precise conjecture relating the cohomology of deformation spaces of p-divisible groups with PEL (or EL) structure to the Langlands correspondence, and prove this conjecture in all cases of EL type. Using the Langlands-Kottwitz method, one can deduce new results about the cohomology of compact unitary group Shimura varieties at ramified split places, and reprove results of Shin about the existence of Galois representations associated to RACSD cuspidal automorphic representations.

S.W. Shin (Massachusetts Institute of Technology)
Construction of Galois representations

I will explain the construction of Galois representations associated with conjugate self-dual cuspidal automorphic representations of GL(n) over a CM field which are regular and algebraic, building upon the work of Kottwitz, Clozel and Harris-Taylor (under a mild assumption when n is even). The basic approach is to realize the Galois representations in the cohomology of certain compact Shimura varieties for U(1,n-1) if n is odd and U(1,n) if n is even. For the latter one has to understand the endoscopic part of the cohomology, which was envisioned by Langlands and studied by Blasius and Rogawski in the case of U(1,2). To describe the Galois reprsentations at ramified places, one needs inputs from the study of bad reduction of Shimura varieties (to be explained in the lectures by Fargues and Mantovan).

E. Viehmann (Universität Bonn)
Newton strata and EO-strata in PEL Shimura varieties

I explain a group-theoretic approach to study the Ekedahl-Oort stratification for good reductions of Shimura varieties of PEL type. As an application I outline how to prove non-emptiness of Newton strata.

W. Zhang (Columbia University)
L-values and heights on Shimura varieties

Generalizing the Gross-Zagier formula for Rankin L-vaues and heights of CM points, the arithmetic version of a conjecture of Gan-Gross-Prasad relates some L-values and heights on Shimura varieties. I'll describe a relative-trace-formula-like approach to the conjecture. One question arising from this approach is an identity (called "arithmetic fundamental lemma" or AFL) between intersection numbers on unitary Rapoport-Zink space and relative orbital integrals. The lower rank cases were proved earlier; recent progress on the AFL has been made in a joint work with Rapoport and Terstiege.


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