Abstracts

Alina Cojocaru (UIC)
Twin primes for elliptic curves
For an elliptic curve E over Q and for a prime p, we let E_p denote
the reduction of E modulo p. In 1988, Neal Koblitz asked for the
number of primes p < x for which E_p(F_p) forms a group of prime
order. In many ways, this question may be viewed as an analogue of
the classical twin prime conjecture. In my talk, I will discuss an
average version of Koblitz's question. This is joint work with Antal
Balog and Chantal David.

Nathan Dunfield (UIUC)
Increasing the number of fibered faces of arithmetic hyperbolic 3manifolds.
I will exhibit a closed hyperbolic 3manifold which satisfies a very strong form of Thurston's Virtual Fibration
Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many fundamentally
distinct ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball
goes to infinity, in fact faster than any power of the logarithm of the degree of the cover. The example manifold M is
arithmetic, and the proof uses detailed numbertheoretic information, at the level of the Hecke eigenvalues, to drive a
geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M
over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(7)].
We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(3)], and then transfer over to a
quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal
congruence subgroup of level 7 of the multiplicative group of a maximal order of D. This is joint work with Dinakar
Ramakrishnan.

Emmanuel Breuillard (Paris 11)
Diophantine geometry and the Tits alternative
I will discuss some recent work about heights of rational
points on semisimple algebraic groups and their character varieties. The
main theorem can be viewed as a semisimple analog of the classical Lehmer
conjecture. We will then explain how it can be used to prove a uniform
version of the Tits alternative for linear groups. Finally I will describe
some consequences for subgroups of GL(d,F_p).

Ben Howard (Boston College)
Intersection theory on Shimura surfaces
Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura
varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension
one cycles on the integral model of a Shimura curve, has been completed by KudlaRapoportYang. I will describe results in a
higher dimensional setting: on the integral model of a Shimura surface one can consider the intersection of an embedded
Shimura curve with a family of codimension two cycles of complex multiplication points. The intersection numbers of these
cycles are related to Fourier coefficients of a Hilbert modular form of halfintegral weight.

SuhHyun Choi (Harvard)
Deformation lifting spaces of mod l representations and their properties
Having appeared in several papers dealing with modularity
problems, it is reasonable to investigate the deformation lifting spaces
of Galois representations. Let our underlying residual representation be
rho: G_K > GL_n(\bar F_l), where K is over Q_p, and assume that l is not
equal to p. In this case, I will introduce results on the dimension and
the characterization of some irreducible components of the deformation
lifting space of rho.

Kiran Kedlaya (MIT)
Towards a slope filtration theorem with nondiscrete coefficients
The slope filtration theorem for Frobenius modules over the Robba ring
has proved to be an important tool in padic Hodge theory, via the
construction of (phi, Gamma)modules. However, up to now it has always
been formulated in terms of a ring of power series over a discretely
valued field. This restriction looks artificial when one starts studying
BergerColmez's relative (phi, Gamma)modules, and especially when one
compares what we know about such objects to what we know in the function
field setting (by recent work of Hartl). I will explain the preceding
remarks and then discuss a prospective approach to the nondiscrete slope
filtration theorem.

Jay Pottharst (Boston College)
Selmer groups over eigenvarieties
Greenberg defined Selmer groups over padic deformations of motives that are "ordinary". This leaves open the
question of how to define a Selmer group over, for example, the eigencurve of ColemanMazur. We present a conjectural program
to do this, using a direct generalization of Greenberg's hypothesis in the setting of (phi,Gamma)modules, and give evidence
by comparing our local conditions to BlochKato's.

Michael Harris (Paris 7)
Automorphic realization of residual Galois representations

Sug Woo Shin (Chicago)
Construction of automorphic Galois representations
According to the conjectural global Langlands correspondence, there should be a bijection between the set of cuspidal automorphic representations of GL(n) and the set of ndimensional global Galois representations, over any number field, if suitable conditions are imposed on both sets. We report on a recent progress on attaching Galois representations to the cuspidal automorphic representations arising from unitary groups (via base change). Our work is built upon previous work of Kottwitz, Clozel, HarrisTaylor and TaylorYoshida. Morel and ClozelHarrisLabesse have obtained results in a similar direction.

Kannan Soundararajan (Stanford)
Recent progress on Quantum Unique Ergodicity
An important special case
of the QUE conjecture of Rudnick and Sarnak
asserts that Maass cusp forms for SL_2(Z)\H
become equidistributed on the fundamental domain,
as the eigenvalue goes to infinity. An analogous
conjecture can be formulated for holomorphic
Hecke eigenforms of large weight. I'll discuss
recent work that has led to a resolution of
these conjectures.
