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.
Abstracts
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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