Arithmetic Geometry of
Orthogonal and Unitary Shimura Varieties
BIRS, June 3-8, 2012
Schedule (PLEASE SEE
"SURVIVAL INFO" AT THE BOTTOM)
Sunday
16:00
Check-in begins (Front Desk – Professional Development Centre - open
24 hours)
17:30 - 19:30 Buffet Dinner
Monday
8:45
Welcome and information
9:00 - 10:00 Steve Kudla: Special cycles for unitary groups: the
unramified case
10:00 - 10:30 Coffee
10:30 - 11:30 Ben Howard: A Gross-Zagier theorem for higher
weight modular forms
11:30 - 13:30 Lunch
13:45
Group Photo
14:00 - 15:00 Sug-Woo Shin: Nonemptiness of Newton strata for PEL
type Shimura varieties
15:00 - 15:30 Coffee
15:30 - 16:30 Keerthi Madapusi: Regular integral models for orthogonal
Shimura varieties and the Tate conjecture for K3 surfaces in
finite characteristic.
Tuesday
9:00 - 10:00 Michael Rapoport: On the geometry of unitary Shimura
varieties in the ramified case
10:00 - 10:30 Coffee
10:30 - 11:30 Torsten Wedhorn: Reductions of PEL Shimura varieties
and group theoretic invariants
11:30 - 13:30 Lunch
13:30 - 14:30 Brian Smithling: Moduli descriptions of some local
models for Shimura varieties
14:30 - 15:00 Coffee
15:00 - 16:00 Adrian Vasiu: Arithmetic properties of good integral
models of Shimura varieties of Hodge type
16:00 - 16:30 Break
16:30 - 17:30 Ulrich Goertz: Affine Deligne-Lusztig varieties in
the Iwahori case
Wednesday
8:30 - 9:30 Jurg Kramer: Arithmetic intersections on modular
curves
9:30 -9:40 Break
9:40 - 10:40 Jose Burgos-Gil: The singularities of the invariant
metric of the sheaf of Jacobi forms on the universal elliptic
curve
10:40 - 11:00 Coffee
11:00 - 12:00 Ulf Kuhn: Modularity
of generating series for arithmetic Hecke correspondences
12:00
Lunch and free
afternoon
Thursday
9:00 - 10:00 Jayce Getz: Twisted relative endoscopy
10:00 - 10:30 Coffee
10:30 - 11:30 Ehud DeShalit: Integral Structures in Locally Algebraic representations
and Kirillov Models
11:30 - 13:30 Lunch
13:30 - 14:30 Chung Pang Mok: Endoscopic classification of automorphic representations
on quasi-split unitary groups
14:30 - 15:00 Coffee
15:00 - 16:00 Ulrich Terstiege: On the arithmetic fundamental lemma in
the minuscule case
16:00 - 16:30 Break
16:30 - 17:30 David Helm: The local Langlands correspondence for GL_n in families
and completed cohomology of Shimura varieties.
Friday
8:30 - 9:30 Matthew Greenberg: Triple product p-adic L-functions for
balanced weights
9:30 -9:40 Break
9:40 - 10:40 Ellen Eischen: p-adic families of Eisenstein series
for unitary groups
10:40 - 11:00 Coffee
11:00 - 12:00 John Voight: Semi-arithmetic
points
12:00
Lunch and departure
======================== ABSTRACTS ============
BURGOS GIL
A theorem by Mumford implies that every automorphic line bundle on a
pure open Shimura variety, provided with an invariant smooth metric,
can be uniquely extended as a line bundle on a toroidal
compactification of the variety, in such a way that the metric
acquires only logarithmic singularities. This result is the key to
being able to compute arithmetic intersection numbers from these
line bundles. Hence it is natural to ask whether Mumford's result
remains valid for line bundles on mixed Shimura varieties. In this
talk we will examine the simplest case, namely the sheaf of Jacobi
forms on the universal elliptic curve. We will show that Mumford's
result can not be extended to this case and that a new interesting
kind of singularities appear. We will discuss some preliminary
results. This is joint work with G. Freixas, J. Kramer and U.
Kühn.
DE SHALIT
The Breuil-Schneider conjecture grew out of an attempt to formulate
a general p-adic local Langlands correspondence. It predicts when a
locally algebraic representation of GL_n over a p-adic field F
should have an integral structure. Despite the elegant formulation,
these integral structures are very elusive. Except for GL_2(Q_p),
where the conjecture is established indirectly, there are only
partial results. We shall describe local methods of Vigneras and of
the speaker (with Kazhdan), which apply to certain smooth
representations of GL_2(F). We shall also mention a recent
breakthrough by Sorensen which uses global methods (unitary Shimura
varieties and eigenvarieties).
EISCHEN
Special values of certain L-functions can be expressed in terms of
values of Eisenstein series at points on the Shimura variety for
U(n,n). One approach to p-adically interpolating values of
these L-functions relies on construction of a p-adic family of
Eisenstein series. In this talk, I will explain how to
construct such a family of Eisenstein series, and I will explain how
to p-adically interpolate certain values of both holomorphic and
non-holomorphic Eisenstein series on U(n,n).
GETZ
The theory of twisted endoscopy relates packets of automorphic
representations on classical groups to automorphic representations
on general linear groups, and is an essential ingredient in
understanding the cohomology of Shimura varieties. We will
introduce a (rudimentary) conjectural theory of twisted relative
endoscopy. The goal of this theory is to relate periods of
automorphic representations on classical groups to periods of
automorphic representations on general linear groups (which are much
better understood). These periods often define special
algebraic cycles on the Shimura variety, and thus the theory of
twisted relative endoscopy should give a means of studying what part
of the cohomology (or arithmetic cohomology) of the Shimura variety
can be explained by these cycles.
GOERTZ
In this talk, I will report on recent joint work with Xuhua He and
Sian Nie on affine Deligne-Lusztig varieties in the Iwahori case, in
particular about non-emptiness and their dimensions. Affine
Deligne-Lusztig varieties are analogues of usual Deligne-Lusztig
varieties in the context of an affine root system, and are a
group-theoretic tool to study the reduction of Shimura varieties (of
PEL type), especially the Newton stratification, the
Kottwitz-Rapoport stratification, and how these to stratifications
are related.
GREENBERG
The conjecture of Gross and Prasad and its refinements constitute a
framework for generalizing the famous formula of Gross and Zagier,
relating the height of a Heegner point on an elliptic curve to the
central derivative of an associated L-function, to the context of
orthogonal and unitary Shimura varieties. In this talk, I
would like to discuss work-in-progress with Marco Seveso dealing
with p-adic analogues of these conjecture in a low-dimensional
(though very rich!) test scenario. Specifically, I'll describe
a construction of a triple product p-adic L-function in the case of
"balanced weights." The key tools are existing classical
special value formulae due to Gross-Kudla, Boecherer-Schulze-Pillot,
and Ichino, and the Ash-Stevens theory of p-adic deformation of
arithmetic cohomology.
HELM
We discuss joint work with Matthew Emerton that describes a
conjectural local Langlands correspondence for families of Galois
representations. Emerton has shown that for two-dimensional Galois
representations such a correspondence arises in his description of
the completed cohomology of the modular tower. We will focus
on some of the representation-theoretic questions that arise from
our results, with an eye towards understanding the completed
cohomology of other Shimura towers.
HOWARD
I'll talk about an extension of the Gross-Zagier theorem to higher
weight modular forms, expressing the height pairings of special
cycles on unitary Shimura varieties to the central derivatives of
Rankin-Selberg L-functions. This is joint work with Jan
Bruinier and Tonghai Yang.
KRAMER
In our talk we will report on asymptotic formulas for the arithmetic
self-intersection of the relative dualizing sheaf equipped with the
Arakelov metric on modular curves attached to congruence subgroups
as the level tends to infinity. In case of the modular curve
$X_{0}(N)$ ($N$ squarefree and not divisible by $2,3$) such results
are due to A.~Abbes and E.~Ullmo. We will present analogous results
for the modular curve $X_{1}(N)$ (for suitable squarefree $N$),
which then enable us to compute the Faltings height of the
associated Jacobian $J_{1}(N)$ asymptotically (as $N$ tends to
infinity).
KUDLA
In the first part of this lecture I will review the definition of
arithmetic special cycles on Shimura varieties for unitary groups of
signature (n-1,1) and explain how the arithmetic 0-cycles arise in
the computation of their height pairings. In the second part of
talk, I will review the structure of the Rapoport-Zink space used in
the p-adic uniformization of the supersingular locus of such a
varieties in the case of an inert prime. Finally, I will give the
definition of the analogous special cycles in this RZ space and
explain their properties.
KUHN
The aim of the talk is to explain our approach to Kudla's
conjectures for the case of the product of two modular curves. The
mayor difficulties in this situation are of analytical nature. We
present a mild modification of this Green function that satisfies
the requirements of being a Green function in the sense of Arakelov
theory on the natural compactification in addition. Only this allows
us to define arithmetic special cycles and to show that the
generating series of those modified arithmetic special cycles is as
predicted by Kudla's conjectures a modular form with values in the
first arithmetic Chow group. Moreover its intersection with the
arithmetic canonical class yields essentialy the derivative of an
Eisenstein series. This is joint work with Rolf Berndt:
http://xxx.uni-augsburg.de/abs/1205.6417
http://www.math.uni-hamburg.de/home/kuehn/berndt-kuehn-part-II.pdf
MADAPUSI
We construct regular integral canonical models for Shimura varieties
of orthogonal type with maximal parahoric level, and we show that
certain moduli spaces of polarized K3 surfaces can be viewed as open
sub-schemes of such integral models. Using a result of Kisin, this
then implies the Tate conjecture for K3 surfaces in odd
characteristic p, as long as they admit a polarization of degree
indivisible by p^2. The same methods also work to prove the Tate
conjecture for cubic fourfolds in odd characteristic.
MOK
Recently Arthur has established the endoscopic classification of
automorphic representations on orthogonal and symplectic groups
(modulo stabilization of the twisted trace formula). In this talk we
report on the current work on extending Arthur's results to unitary
groups.
RAPOPORT
I will explain structure theorems for the formal moduli space of
$p$-divisible groups of Picard type of signature $(1, n-1)$ for a
ramified quadratic extension of $\BQ_p$. The underlying reduced
scheme possesses a stratification by Deligne-Lusztig varieties for
symplectic groups over $\BF_p$; the strata are parametrized by
simplices in the Bruhat-Tits building of a $p$-adic unitary group.
This is joint work with U. Terstiege and S. Wilson; the results are
analogous to the results of I. Vollaard and T. Wedhorn in the
case of an unramified quadratic extension of $\BQ_p$.
SHIN
The Newton strata for Shimura varieties arising from unramified PEL
data of type A and C are known to be nonempty whenever expected by
Vasiu and Viehmann-Wedhorn based on earlier work by many. I will
explain a different approach to prove the result in the unramified
case via Honda-Tate theory and Galois cohomology. Assuming a result
on the existence of F-crystals more can be shown.
SMITHLING
Local models are certain schemes introduced to model the
\'etale-local structure of p-adic integral models of Shimura
varieties. A general definition of them in the setting of PEL
Shimura varieties was given by Rapoport and Zink; recently Pappas
and Zhu have formulated a general group-theoretic definition of them
for tamely ramified groups and established many good properties for
them. Unfortunately the local models do not in general admit
ready moduli-theoretic interpretations. To facilitate applications,
it is therefore of interest to describe to describe them in
moduli-theoretic terms when possible. In the case of
orthogonal and ramified unitary groups, such a description has been
proposed by Pappas and Rapoport. I will report on some
progress towards proving their conjecture.
TERSTIEGE
The arithmetic fundamental lemma conjecture of Wei Zhang connects
the derivative of a certain orbital integral with an intersection
number on a Rapoport-Zink space parameterizing p-divisible groups of
unitary type of signature (1,n-1). It arises in the relative trace
formula approach to the arithmetic Gan-Gross-Prasad conjecture. This
talk is about joint work with M. Rapoport and W. Zhang proving the
AFL-conjecture in the minuscule case.
VASIU
We present an accessible survey of known results on arithmetic
properties of good integral models of Shimura varieties of Hodge
type in arbitrary unramified mixed characteristic (0,p). We will
also include some of those proofs that are the most useful for the
further studies of such arithmetic properties.
VOIGHT
We present a method for constructing algebraic points on elliptic
curves defined over totally real fields, combining the theory of
Belyi maps and quaternionic Shimura varieties; our method
generalizes the construction of Heegner points arising from
classical modular curves. In particular, we report on some
computational investigations of these points.
WEDHORN
It is explained how to attach to a point in the parahoric reduction
of Shimura varieties of PEL type (i.e. those reductions that are
studied in the book of Rapoport and Zink) a group theoretic
invariant which encodes the isomorphism class of the (multi)chain of
p-divisible groups with additional structures associated to that
point. This invariant spawns several other invariants which lead to
the Newton stratification, the Ekedahl-Oort stratification and the
Kottwitz-Rapoport stratification. In the case of good reduction
several properties of these stratifications
============
MEALS
*Breakfast (Buffet): 7:00 – 9:30 am, Sally Borden Building, Monday –
Friday
*Lunch (Buffet): 11:30 am – 1:30 pm, Sally Borden Building, Monday –
Friday
*Dinner (Buffet): 5:30 – 7:30 pm, Sally Borden Building, Sunday –
Thursday
*Coffee Breaks: As per daily schedule, in the foyer of the
TransCanada Pipeline Pavilion (TCPL)
*Please remember to scan your meal card at the host/hostess station
in the dining room for each meal.
MEETING ROOMS
All lectures will be held in the new lecture theater in the
TransCanada Pipelines Pavilion (TCPL). LCD projector and blackboards
are available for presentations.
GROUP PHOTO
The meeting place for the group photograph is the foyer of
TCPL. However, the photo is taken outside so it may be
necessary for people to bring a jacket.
CHECKOUT
Friday by 12 noon.
Participants are welcome to use BIRS facilities (BIRS Coffee Lounge,
TCPL and Reading Room) until 3 pm on Friday, although participants
are still required to checkout of the guest rooms by 12 noon.