


Arithmetic Geometry / Géométrie arithmétique Org: Eyal Goren (McGill) and/et Adrian Iovita (Concordia)
 GIL ALON, McGill University, 805 Sherbrook St., Montreal
BruhatTits buildings, padic hyperplane arrangements and
OrlikSolomon algebras

The ddimensional symmetric space of Drinfeld, which serves as a
padic and higher dimensional analog of the complex upper half
plane, comes with a natural map to the BruhatTits building of
GL_{d+1} over a padic field. Under this map, preimages of
stars of simplices are simple rigid padic spaces whose derham
cohomology can be expressed in terms of OrlikSolomon algebras. This
leads to a coefficient system on the building. Calculating the
cohomology of this system is a crucial step in the calculation of the
deRham cohomology of Drinfeld's space, shown by E. de Shalit to be
isomorphic to a certain space of harmonic cochains on the building.
More generally, starting from a padic hyperplane arrangement we can
define a coefficient system on the BruhatTits building, that reflects
the local properties of the arrangement. We calculate the cohomology
of this local system for any finite padic hyperplane arrangement.
 DAVID BOYD, University of British Columbia, Vancouver, BC V6T 1Z2
The Apolynomials of periodic knots

We show that the Apolynomials of periodic knots have a remarkable
factorization into polynomials with coefficients in real cyclotomic
fields and how the Mahler measure of certain of the factors relates to
the hyperbolic volume of the knot complement.
 IMIN CHEN, Simon Fraser University, Department of Mathematics
More on Diophantine equations via Galois representations

The method of Galois representations and modularity has been used to
solve several classes of Diophantine equations. In this talk, I will
discuss some more cases and phenomenon which arise in this method.
 SEBASTIAN CIOABA, Queen's University at Kingston
Eigenvalues, expanders and Ramanujan graphs

The explicit construction of Ramanujan graphs requires deep results
from number theory, representation theory and arithmetic geometry. We
will discuss various aspects regarding the distribution of the
eigenvalues of regular graphs including a new elementary proof of a
theorem of Serre.
 PETE CLARK, McGill
On a question of Lang and Tate

We show that there are genus one curves of every index over the
rational numbers, answering affirmatively an old question of Lang and
Tate. Using Kolyvagin's example of a rational elliptic curve whose
MordellWeil and ShafarevichTate groups are both trivial, we show
that there are infinitely many curves of every index over every number
field.
 HENRI DARMON, McGill University
Exceptional zeros of padic Lfunctions

I will report on work in progress concerning exceptional zeros of
certain padic Lfunctions attached to modular forms and the
socalled Linvariant.
 SAMIT DASGUPTA, Harvard University, Department of Mathematics, One Oxford
Street, Cambridge, MA 02138
StarkHeegner points on modular Jacobians

We present a construction which lifts Darmon's StarkHeegner points
from elliptic curves to certain modular Jacobians. Let N be a
positive integer and let p be a prime not dividing N. Our
essential idea is to replace the modular symbol attached to an
elliptic curve E of conductor Np with the universal modular symbol
for G_{0}(Np).
We then construct a certain torus T over Q_{p} and sublattice L
of T, and prove that the quotient T/L is isogenous to the maximal
toric quotient J_{0} (Np)^{pnew} of the Jacobian of X_{0}(Np).
This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum
on the padic periods of elliptic curves, which was proven by
Greenberg and Stevens; indeed, our proof borrows greatly from theirs.
As a byproduct of our theorem, we obtain an efficient method of
calculating the padic periods of J_{0} (Np)^{pnew}.
 JORDAN ELLENBERG, Princeton University
Elliptic curves over towers of function fields

I will discuss some results on MordellWeil ranks of elliptic curves
over towers of function fields; a typical case is E/k (t^{1/pn}) as
n grows. I will show how to generalize two results of Silverman on
towers with abelian Galois groups to the general case, and I will
discuss the circumstances over which E can be shown to have finite
MordellWeil rank over the whole tower.
 ALEXANDRU GHITZA, McGill University, Montreal
Cuspidality of Hecke eigensystems (\operatornamemodp)

A system of Hecke eigenvalues (modp) is said to be cuspidal
if it is given by an eigenform which is also a cusp form. We prove
that all Hecke eigensystems are in fact cuspidal; although the proof
is given only for Siegel modular forms (modp), it can easily be
extended to other kinds of Shimura varieties for which an arithmetic
compactification à la FaltingsChai is known (e.g., Hilbert
modular forms).
 YOSHITAKA HACHIMORI, CICMA, Concordia University, 1455 de Maisonneuve Blvd. W.,
Montreal, Quebec H3G 1M8
On Iwasawa theory for padic Lie extensions

We will discuss about the structure of the Selmer groups of elliptic
curves and the Galois group of the maximal unramified abelian
pextensions over padic Lie extensions as modules over the
Iwasawa algebras.
 ERNST KANI, Dept. of Math. & Stats, Queen's University, Kingston,
Ontario K7L 2N6
Endomorphisms of Jacobians of Modular Curves and an
Application

The purpose of this talk is to present some results on the structure
of the algebra of Qrational endomorphisms of the
Jacobian J_{X} of a modular curve X/Q of arbitrary
level N.
These results are then applied to determine the NeronSeveri group of
Qrational divisors of the modular diagonal quotient
surface Z_{N,e} in the case that N=p, where is p is a
prime.
 PAYMAN KASSAEI, McGill University
A "subgroupfree" approach to canonical subgroups

I will discuss joint work with Eyal Goren on "canonical subgroups".
Beyond the crucial role they play in the foundations of the theory of
overconvergent modular forms, canonical subgroups have found new
applications to the problem of analytic continuation of overconvergent
modular forms (Buzzard (JAMS 03), K(Preprint 04)). One seeks to
generalize these results to other Shimura varieties.
We use formal and rigid geometry to study the precise extent of
overconvergence of the canonical subgroup and its other properties
(for all PEL shimura curves). In our approach, we use the common
geometric features of these Shimura curves (rather than their specific
modulitheoretic description) to prove a general canonical subgroup
theorem.
 HERSHY KISILEVSKY, Concordia University, Montreal, Quebec
Vanishing twists of elliptic Lfunctions

We continue our study of the central values of Lfunctions of
elliptic curves defined over the rationals. We study also the
vanishing and nonvanishing values of the central values of twists of
these Lfunctions by Dirichlet characters and in some cases by
certain Artin characters.
 MARK KISIN, University of Chicago
Modularity of Galois representations

I will explain progress towards proving the conjecture of
FontaineMazur on modularity of two dimensional potentially
semistable Galois representations. These hinge on developments in
integral padic Hodge theory in the presence of wild ramification,
and a new technique for understanding local deformation rings.
 MANFRED KOLSTER, McMaster University, 1250 Main St. W, Hamilton, ON L8S 4K1
On Iwasawa linvariants of number fields

linvariants attached to pparts of eigenspaces of class
groups are related to étale cohomology groups, and the behaviour of
the linvariants under pextensions is related to Euler
characteristics for these groups. We discuss the consequences for
"Kidatype" formulas and Greenberg's Conjecture for totally real
fields.
 GILLES LACHAUD, Institut de Mathématiques de Luminy, Case 907,
13288 Marseille Cedex 9, France
Sails and Klein polyhedra

This is a chapter in geometry of numbers. We present a generalization
of continued fractions to higher dimensions, already introduced by
F. Klein and recently considered by V. Arnold. If C is a simplicial
cone in the euclidean space of dimension d, the Klein hull of
C is the convex hull of the integer points contained in C, and the
sail V of C is the boundary of its Klein hull. We first
discuss whether the Klein hull of a cone is a generalized
polyhedron. Then we develop a generalization of Lagrange's theorem to
higher dimensions: the walls (resp. the generators) of C are
algebraic if and only if they admit a periodic approximation by a path
of chambers (resp. of vertices) in V. If C is defined over a
totally real field of degree d, we define a group stabilizing the
sail with a fundamental set which is the union of a finite number of
simplexes, and there is an algorithm for the construction of that
fundamental set. This is a refinement of results of Shintani.
Finally, some examples taken from the case of simplest cubic fields
will be given.
 JAMES LEWIS, University of Alberta, Dept. of Math., Edmonton, AB T6G 2G1
Algebraic Cycles and Mumford Invariants

Let X be a projective algebraic manifold and let CH^{r} (X) be the
Chow group of algebraic cycles of codimension r on X, modulo
rational equivalence. Working with a motivical filtration
{F^{n}}_{n ³ 0} on CH^{r} (X) ÄQ with graded
piece Gr_{F}^{n} CH^{r} (X) ÄQ, we construct a space of
arithmetical Hodge theoretic invariants ÑJ^{r,n} (X) and a
corresponding map f_{X}^{r,n} : Gr_{F}^{n} CH^{r} (X)ÄQ ® ÑJ^{r,n} (X), and determine conditions
for which the kernel and image of f_{X}^{r,n} is `large'. This
is joint work with Shuji Saito.
 ALVARO LOZANOROBLEDO, Colby College, 8800 Mayflower Hill, Waterville, ME 04901
Elliptic Units and Galois Representations

Let K be a quadratic imaginary number field with discriminant D_{K} ¹ 3,4 and class number one. Fix a prime p ³ 7 which is not
ramified in K and write h_{p} for the class number of the ray class
field of K of conductor p. Given an elliptic curve A/K with
complex multiplication by K, let [`(r_{A})] :Gal ([`(K)]/ K(m_{p¥}) )® SL (2,Z_{p}) be the representation which arises
from the action of Galois on the Tate module of A. We show that if
p\nmid h_{p} then the image of a certain deformation r_{A} :Gal ([`(K)]/ K(m_{p¥}) )® SL (2,Z_{p} [[X]]) of [`(r_{A})] is
"as big as possible", that is, it is the full inverse image of a
Cartan subgroup of SL (2,Z_{p}). The proof rests
on the theory of Siegel functions and elliptic units as developed by
Kubert, Lang and Robert.
 MARCHUBERT NICOLE, McGill University, Montréal
A Geometric Interpretation of Eichler's Basis Problem for
Hilbert Modular Forms

Let p be a prime and B_{p,¥} be the quaternion algebra
ramified at p and ¥. Eichler proved that the rational vector
space M_{2} ( G_{0}(p) ) of weight 2 modular forms of
level p is generated by the theta series Q(I) associated to
left ideals I of a maximal order in B_{p,¥}. Using results
of Deuring, the theta series Q(I) can all be retrieved from the
quadratic modules ( Hom(E_{1},E_{2}), deg), where E_{i},
i=1,2 are supersingular elliptic curves, and deg is the quadratic
degree map. The JacquetLanglands correspondence also implies that
the space of newforms of weight 2 and squarefree level N is
spanned by theta series coming from quadratic forms in four variables
and thence, the analogous result holds for Hilbert modular forms.
Under some restrictive hypotheses, we show that the theta series
spanning the vector space of Hilbert modular newforms of weight 2
and level p can be constructed via totally definite quaternion
algebras from superspecial points on a Hilbert modular variety.
 KEN ONO, University of Wisconsin
Hilbert class polynomials and traces of singular moduli

In this lecture I will describe joint work with Jan Bruinier and Paul
Jenkins on the arithmetic and asymptotic properties of singular
moduli. Following up on recent works of Borcherds and Zagier, we
obtain exact formulae for traces and Hecke traces of singular
moduli. As a consequence, we also obtain formulae for Hilbert class
polynomials. We use these results to obtain some results (also
independently obtained by Duke in recent work) on the `average'
distribution of singular moduli.
 DAVID SAVITT, McGill University
Computing with Galois representations arising from Jacobians
of genus 2 curves

Given a curve of genus 2 whose Jacobian has GL(2) type, we will
explain how one can use a computer to extract a great deal of
information about certain Galois representations arising on the
Jacobian. In particular, one can often prove that the Jacobian is
modular. This is joint work with W. Stein.
 ROMYAR SHARIFI, McMaster University, Hamilton, Ontario L8S 4K1
Iwasawa theory over Kummer extensions

We consider Iwasawa modules over the compositum of the cyclotomic
Z_{p}extension of a number field with an abelian prop
extension of it such that the entire extension is Galois. We will
describe aspects of a relationship between the Galois group of the
maximal unramified abelian prop extension such a field, cup
products, and padic Lvalues.
 YE TIAN, CICMA, Montreal
Euler Systems of CM points on Shimura curves

We discuss Euler systems of CM points on Shimura curves over totally
real fields and application to some Diophantine equations.
 MAK TRIFKOVICH, McGill Math. Dept.
Rational Points on Elliptic Curves over Imaginary Quadratic
Fields

Henri Darmon has proposed a conjectural padic analytic construction
of rational points on elliptic curves over Q. In this
talk, we will present the analogs of his conjectures for curves
defined over imaginary quadratic fields, along with some experimental
results. The most interesting feature of the new setting is the
highly conjectural nature of modularity for such curves: the analog of
X_{0}(N) in this setting is a 3dimensional real manifold, so a
modular parametrization is too much to hope for. Coincidence of
Lfunction, however, is sufficient for computations.
 HUI JUNE ZHU, McMaster University, Canada
Pointcounting divisibility on totally ramified covers

The classical Warning's theorem studies divisibility of the number of
rational points on an algebraic variety over finite fields. Recently
much progress in this direction has been made, with interesting
applications. In this talk we shall discuss a padic approach to
this problem. Using Dwork's method and a new transformation lemma we
prove a divisibility result for ArtinSchreier covers in
characteristic p > 0, that generalizes a result of Scholten and myself
in the characteristic 2 case, obtained a few years ago using a
completely different method (Katz's sharp slope estimates). We will
discusss its application to the Newton polygon stratification of the
moduli space of curves in characteristic p > 0.

