Difference between revisions of "NTS/Abstracts Spring 2011"

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== Anton Gershaschenko  ==
 
== Anton Gershaschenko  ==
  
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==  Keerthi Madapusi ==
== Shuichiro Takeda, Purdue ==
 
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: On the regularized Siegel-Weil formula for the second terms and
+
| bgcolor="#DDDDDD" align="center"| Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactifications of Shimura varieties
non-vanishing of theta lifts from orthogonal groups
 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract: In this talk, we will discuss (a certain form of) the
+
Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semi-stable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (l-adic, p-adic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́.
Siegel-Weil formula for the second terms (the weak second term
+
Using the theory of Shimura varieties and the Faltings-Chai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain sub-scheme of a torus embedding is again a torus embedding. In the situation where the Mumford-Tate group of A has a reductive model over Zp, for v|p (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively.
identity). If time permits, we will give an application of the
+
A by-product of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the Mumford-Tate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places v|p of E.
Siegel-Weil formula to non-vanishing problems of theta lifts. (This is
+
The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the Faltings-Chai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the Faltings-Kisin method.
a joint with W. Gan.)
 
 
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<br>
  
== Xinyi Yuan ==
+
== Bei Zhang ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Volumes of arithmetic line bundles and equidistribution
+
| bgcolor="#DDDDDD" align="center"| Title: p-adic L-function of automorphic form of GL(2)
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
In this talk, I will introduce equidistribution of small points in
+
Abstract: Modular symbol is used to construct p-adic L-functions
algebraic dynamical systems. The result is a corollary of the
+
associated to a modular form. In this talk, I will explain how to
differentiability of volumes of arithmetic line bundles in Arakelov
+
generalize this powerful tool to the construction of p-adic L-functions
geometry. For example, the equidistribution theorem on abelian varieties
+
attached to an automorphic representation on GL_{2}(A) where A is the ring
by Szpiro-Ullmo-Zhang is a consequence of the arithmetic Hilbert-Samuel
+
of adeles over a number field. This is a joint work with Matthew Emerton.
formula by Gillet-Soule.
 
 
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<br>
  
 
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== David Brown ==
== Jared Weinstein, IAS  ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Resolution of singularities on the tower of modular curves
+
| bgcolor="#DDDDDD" align="center"| Title: Explicit modular approaches to generalized Fermat equations
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract:  
+
Abstract: TBA
The family of modular curves X(p^n) provides the geometric link between two types of objects:  On the one hand, 2-dimensional representations of the absolute Galois group of Q_p, and on the other, admissible representations of the group GL_2(Q_p).    This relationship, known as the local Langlands correspondence, is realized in the cohomology of the modular curves.  Unfortunately, the Galois-module structure of the cohomology of X(p^n) is obscured by the fact that integral models have very bad reduction.  In this talk we present a new combinatorial picture of the resolution of
 
singularities of the tower of modular curves, and demonstrate how this
 
picture encodes some features of the local Langlands correspondence.
 
 
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</center>
 
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 +
  
 
<br>
 
<br>
  
== David Zywina, U Penn  ==
+
== Tony Várilly-Alvarado ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Bounds for Serre's open image theorem
+
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract.
+
Abstract: TBA
 
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|}                                                                         
 
</center>
 
</center>
  
 
<br>
 
<br>
 
+
== Wei Ho ==
== Soroosh Yazdani, UBC and SFU ==  
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Local Szpiro Conjecture
+
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
For any elliptic curve E over Q, let N(E) and Delta(E)    denote it's conductor and minimal discriminant. Szpiro conjecture    states that for any epsilon>0, there exists a constant C    such that    Abs(Delta(E)) < C (N(E))^{6+\epsilon}    for any elliptic curve E. This conjecture, if true, will have    applications to many Diophantine equations.    Assuming Szpiro conjecture, one expects that there are    only finitely many semistable elliptic curves    E such that    min_{p|N(E)} v_p(\Delta(E)) >6.    We conjecture that, in fact, there are none. In this talk    we study this conjecture in some special cases, and provide    some evidence towards this conjecture.
+
Abstract: TBA
 
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</center>
 
</center>
  
 
<br>
 
<br>
 
+
== Rob Rhoades ==
== Zhiwei Yun, MIT ==  
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: From automorphic forms to Kloosterman sheaves (joint work with J.Heinloth and B-C.Ngo)
+
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract: Classical Kloosterman sheaves are rank n local systems on
+
Abstract: TBA
the punctured line (over a finite field) which incarnate Kloosterman
 
sums in a geometric way. The arithmetic properties of the Kloosterman
 
sums (such as estimate of absolute values and distribution of angles)
 
can be deduced from geometric properties of these sheaves. In this
 
talk, we will construct generalized Kloosterman local systems with an
 
arbitrary reductive structure group using the geometric Langlands
 
correspondence. They provide new examples of exponential sums with
 
nice arithmetic properties. In particular, we will see exponential
 
sums whose equidistribution laws are controlled by exceptional groups
 
E_7,E_8,F_4 and G_2.
 
 
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|}                                                                         
 
</center>
 
</center>
  
 
<br>
 
<br>
 +
== TBA ==
  
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 +
|-
 +
| bgcolor="#DDDDDD"| 
 +
Abstract: TBA
 +
|}                                                                       
 +
</center>
  
 
+
<br>
 
+
== Chris Davis ==
== David Brown, UW Madison ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title
+
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract.
+
Abstract: TBA
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 
<br>
 
<br>
 
+
== Andrew Obus ==
== Bryden Cais, UW Madison ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title:  On the restriction of crystalline Galois representations
+
| bgcolor="#DDDDDD" align="center"| Title: Cyclic Extensions and the Local Lifting Problem  
 
|-
 
|-
| bgcolor="#DDDDDD"| Abstract: We formulate a generalization of a conjecture of Breuil (now a theorem of Kisin) on the restriction of crystalline p-adic Galois
+
| bgcolor="#DDDDDD"|  
representations to a general class of infinite index subgroups of the Galois groupFollowing arguments of Breuil, we will explain the proof of our generalization in the Barsotti-Tate case.
+
Abstract: The "local lifting problem" asks: given a G-Galois extension A/k[[t]], where k is algebraically closed of characteristic p, does there exist a G-Galois extension A_R/R[[t]] that reduces to A/k[[t]], where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclicThis is basic Kummer theory when p does not divide |G|, and has been proven when v_p(|G|) = 1 (Oort, Sekiguchi, Suwa) and when v_p(|G|) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(|G|) = 3 and many extensions where v_p(|G|) is arbitrarily high. This is joint work with Stefan Wewers.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
  
== Tom Hales, University of Pittsburg ==
+
== Bianca Viray ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title
+
| bgcolor="#DDDDDD" align="center"| Title: Descent on elliptic surfaces and transcendental Brauer element 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
At the International Congress of Mathematicians in India in
+
Abstract: Elements of the Brauer group of a variety X are hard to
August, Ngo Bao Chau was awarded a Fields medal for his proof of the
+
compute. Transcendental elements, i.e. those that are not in the
"Fundamental Lemma."  This talk is particularly intended for students
+
kernel of the natural map Br X --> Br \overline{X}, are notoriously
and mathematicians who are not specialists in the theory of
+
difficult. Wittenberg and Ieronymou have developed methods to find
Automorphic Representions.  I will describe the significance and some
+
explicit representatives of transcendental elements of an elliptic
of the applications of the "Fundamental Lemma." I will explain why
+
surface, in the case that the Jacobian fibration has rational
this problem turned out to be so difficult to solve and will give some
+
2-torsion. We use ideas from descent to develop techniques for general
of the key ideas that go into the proof.
+
elliptic surfaces.
 
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</center>
 
</center>
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<br>
 
<br>
  
== Jay Pottharst, Boston University ==
+
== Frank Thorne ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Iwasawa theory at nonordinary primes
+
| bgcolor="#DDDDDD" align="center"| Title: TBA 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract.
+
Abstract: TBA
 
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</center>
 
</center>
  
 
<br>
 
<br>
 
+
== Rafe Jones ==
== Melanie Matchett Wood, Stanford and AIM ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Geometric parametrizations of ideal classes
+
| bgcolor="#DDDDDD" align="center"| Title: Galois theory of iterated quadratic rational functions 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
In a ring of algebraic integers, the ideal class group measures the
+
 
failure of unique factorizationA classical correspondence due to
+
Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree-2 rational function, focusing on the case where the function commuteswith a non-trivial Mobius transformationIn a sense this is a dynamical systems analogue to the p-adic Galois representation attached to an elliptic curve, with particular attention to the CM caseIn joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case.
Dirichlet and Dedekind allows us to work with ideal classes of
 
quadratic rings concretely in terms of binary quadratic forms with
 
integer coefficientsA recent result of Bhargava gives an analogous
 
correspondence between ideal classes of cubic rings and certain
 
trilinear forms.  From another point of view, the ideal class group is
 
the group of invertible modules of a ring, whose geometric analog is
 
the Picard group of line bundles on a space.  We discuss how we can
 
view these correspondences between ideal classes and forms
 
geometrically, and give new results on parametrizations of ideal
 
classes of certain rank n rings (e.g. orders in degree n number
 
fields) by trilinear forms.
 
 
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</center>
 
</center>
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<br>
 
<br>
  
 +
== Jonathan Blackhurst  ==
  
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#DDDDDD" align="center"| Title: Polynomials of the Bifurcation Points of the Logistic Map 
 +
|-
 +
| bgcolor="#DDDDDD"| 
 +
Abstract: The logistic map f(r,x)=rx(1-x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the long-term behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single non-zero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boom-and-bust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this period-doubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots.
 +
|}                                                                       
 +
</center>
  
== Samit Dasgupta, UC Santa Cruz ==
+
== Liang Xiao ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| On Greenberg's conjecture on derivatives of p-adic L-functions with
+
| bgcolor="#DDDDDD" align="center"| Title: Computing log-characteristic cycles using ramification theory
trivial zeroes
 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
In 1991, Ralph Greenberg stated a conjecture about p-adic L-functions
+
Abstract: There is an analogy among vector bundles with integrable
that have a trivial zero at s=1. Here "trivial" means that the zero
+
connections, overconvergent F-isocrystals, and lisse l-adic sheaves.
arises from the vanishing of an Euler factor that must be removed in
+
Given one of the objects, the property of being clean says that the
order to state the interpolation property of the p-adic L-function.
+
ramification is controlled by the ramification along all generic
Greenberg's conjecture concerns the value of the derivative of the
+
points of the ramified divisors. In this case, one expects that the
p-adic L-function at s=1.  An example of this conjecture is the case
+
Euler characteristics may be expressed in terms of (subsidiary) Swan
of the p-adic L-function of an elliptic curve E/Q with split
+
conductors; and (in first two cases) the log-characteristic cycles may
multiplicative reduction at p. In this case, Greenberg's conjecture
+
be described in terms of refined Swan conductors. I will explain the
reduces to an earlier conjecture by Mazur, Tate, and Teitelbaum, and
+
proof of this in the vector bundle case and report on the recent
was proven by Greenberg himself in joint work with Glenn Stevens.  In
+
progress on the overconvergent F-isocrystal case if time is permitted.
this talk, we will describe a strategy to prove new cases of
 
Greenberg's conjecture.  We will concentrate on the case of the
 
symmetric square of an elliptic curve with good reduction at p.  The
 
strategy is a generalization of my previous work with Darmon and
 
Pollack proving certain cases of the Gross--Stark conjecture (which
 
can also be viewed as a special case of Greenberg's conjecture). The
 
method involves studying explicit p-adic families of modular forms on
 
GSp_4 and their associated Galois representations.
 
 
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|}                                                                         
 
</center>
 
</center>
  
<br>
+
== Winnie Li ==
 
 
== David Geraghty, Princeton and IAS ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Potential automorphy for compatible systems
+
| bgcolor="#DDDDDD" align="center"| Title: Modularity of Low Degree Scholl Representations
 
|-
 
|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#DDDDDD"|  To the space of d-dimensional cusp forms of weight k > 2 for
Abstract: I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l.
+
a noncongruence
 +
subgroup of SL(2, Z), Scholl has attached a family of 2d-dimensional
 +
compatible l-adic
 +
representations of the Galois group over Q. Since his construction is
 +
motivic, the associated
 +
L-functions of these representations are expected to agree with
 +
certain automorphic L-functions
 +
according to Langlands' philosophy. In this talk we shall survey
 +
recent progress on this topic.
 +
More precisely, we'll see that this is indeed the case when d=1. This
 +
also holds true when d=2,
 +
provided that the representation space admits quaternion
 +
multiplications. This is a joint work
 +
with Atkin, Liu and Long.
 +
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
<br>
 
  
== Toby Gee, Northwestern ==  
+
 
 +
== Avraham Eizenbud ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Potential automorphy for compatible systems
+
| bgcolor="#DDDDDD" align="center"| Title: Multiplicity One Theorems - a Uniform Proof 
 
|-
 
|-
 
| bgcolor="#DDDDDD"|   
 
| bgcolor="#DDDDDD"|   
Abstract: I will continue David Geraghty's talk, and discuss a number of applications.
+
 
 +
Abstract: Let F be a local field of characteristic 0.
 +
We consider distributions on GL(n+1,F) which are invariant under the adjoint action of
 +
GL(n,F). We prove that such distributions are invariant under
 +
transposition. This implies that an irreducible representation of
 +
GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.
 +
 
 +
 
 +
 
 +
Such property of a group and a subgroup is called strong Gelfand property.
 +
It is used in representation theory and automorphic forms. This property
 +
was introduced by Gelfand in the 50s for compact groups. However, for
 +
non-compact groups it is much more difficult to establish.
 +
 
 +
For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for
 +
non-Archimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this
 +
lecture we will present a uniform for both cases.
 +
This proof is based on the above papers and an additional new tool. If time
 +
permits we will discuss similar theorems that hold for orthogonal and
 +
unitary groups.
 +
 
 +
[AG] A. Aizenbud,  D. Gourevitch, Multiplicity one theorem for (GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT]
 +
 
 +
[AGRS] A. Aizenbud,  D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, arXiv:0709.4215v1 [math.RT], to appear in the Annals of Mathematics.
 +
 
 +
 
 +
[SZ] B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, preprint available at http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf
 +
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 +
 +
  
 
<br>
 
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 +
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Latest revision as of 17:31, 31 May 2011

Anton Gershaschenko

Title: Moduli of Representations of Unipotent Groups

Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have non-trivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples.


Keerthi Madapusi

Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactifications of Shimura varieties

Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semi-stable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (l-adic, p-adic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́. Using the theory of Shimura varieties and the Faltings-Chai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain sub-scheme of a torus embedding is again a torus embedding. In the situation where the Mumford-Tate group of A has a reductive model over Zp, for v|p (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively. A by-product of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the Mumford-Tate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places v|p of E. The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the Faltings-Chai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the Faltings-Kisin method.


Bei Zhang

Title: p-adic L-function of automorphic form of GL(2)

Abstract: Modular symbol is used to construct p-adic L-functions associated to a modular form. In this talk, I will explain how to generalize this powerful tool to the construction of p-adic L-functions attached to an automorphic representation on GL_{2}(A) where A is the ring of adeles over a number field. This is a joint work with Matthew Emerton.


David Brown

Title: Explicit modular approaches to generalized Fermat equations

Abstract: TBA



Tony Várilly-Alvarado

Title: TBA

Abstract: TBA


Wei Ho

Title: TBA

Abstract: TBA


Rob Rhoades

Title: TBA

Abstract: TBA


TBA

Title: TBA

Abstract: TBA


Chris Davis

Title: TBA

Abstract: TBA


Andrew Obus

Title: Cyclic Extensions and the Local Lifting Problem

Abstract: The "local lifting problem" asks: given a G-Galois extension A/kt, where k is algebraically closed of characteristic p, does there exist a G-Galois extension A_R/Rt that reduces to A/kt, where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclic. This is basic Kummer theory when p does not divide |G|, and has been proven when v_p(|G|) = 1 (Oort, Sekiguchi, Suwa) and when v_p(|G|) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(|G|) = 3 and many extensions where v_p(|G|) is arbitrarily high. This is joint work with Stefan Wewers.


Bianca Viray

Title: Descent on elliptic surfaces and transcendental Brauer element

Abstract: Elements of the Brauer group of a variety X are hard to compute. Transcendental elements, i.e. those that are not in the kernel of the natural map Br X --> Br \overline{X}, are notoriously difficult. Wittenberg and Ieronymou have developed methods to find explicit representatives of transcendental elements of an elliptic surface, in the case that the Jacobian fibration has rational 2-torsion. We use ideas from descent to develop techniques for general elliptic surfaces.


Frank Thorne

Title: TBA

Abstract: TBA


Rafe Jones

Title: Galois theory of iterated quadratic rational functions

Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree-2 rational function, focusing on the case where the function commuteswith a non-trivial Mobius transformation. In a sense this is a dynamical systems analogue to the p-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. In joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case.


Jonathan Blackhurst

Title: Polynomials of the Bifurcation Points of the Logistic Map

Abstract: The logistic map f(r,x)=rx(1-x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the long-term behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single non-zero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boom-and-bust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this period-doubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots.

Liang Xiao

Title: Computing log-characteristic cycles using ramification theory

Abstract: There is an analogy among vector bundles with integrable connections, overconvergent F-isocrystals, and lisse l-adic sheaves. Given one of the objects, the property of being clean says that the ramification is controlled by the ramification along all generic points of the ramified divisors. In this case, one expects that the Euler characteristics may be expressed in terms of (subsidiary) Swan conductors; and (in first two cases) the log-characteristic cycles may be described in terms of refined Swan conductors. I will explain the proof of this in the vector bundle case and report on the recent progress on the overconvergent F-isocrystal case if time is permitted.

Winnie Li

Title: Modularity of Low Degree Scholl Representations
To the space of d-dimensional cusp forms of weight k > 2 for

a noncongruence subgroup of SL(2, Z), Scholl has attached a family of 2d-dimensional compatible l-adic representations of the Galois group over Q. Since his construction is motivic, the associated L-functions of these representations are expected to agree with certain automorphic L-functions according to Langlands' philosophy. In this talk we shall survey recent progress on this topic. More precisely, we'll see that this is indeed the case when d=1. This also holds true when d=2, provided that the representation space admits quaternion multiplications. This is a joint work with Atkin, Liu and Long.


Avraham Eizenbud

Title: Multiplicity One Theorems - a Uniform Proof

Abstract: Let F be a local field of characteristic 0. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that such distributions are invariant under transposition. This implies that an irreducible representation of GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.


Such property of a group and a subgroup is called strong Gelfand property. It is used in representation theory and automorphic forms. This property was introduced by Gelfand in the 50s for compact groups. However, for non-compact groups it is much more difficult to establish.

For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for non-Archimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this lecture we will present a uniform for both cases. This proof is based on the above papers and an additional new tool. If time permits we will discuss similar theorems that hold for orthogonal and unitary groups.

[AG] A. Aizenbud, D. Gourevitch, Multiplicity one theorem for (GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT]

[AGRS] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, arXiv:0709.4215v1 [math.RT], to appear in the Annals of Mathematics.


[SZ] B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, preprint available at http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf



Organizer contact information

David Brown:

Bryden Cais:




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