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Arithmetic Geometry / Géométrie arithmétique
Org: Eyal Goren (McGill) and/et Adrian Iovita (Concordia)

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GIL ALON, McGill University, 805 Sherbrook St., Montreal
Bruhat-Tits buildings, p-adic hyperplane arrangements and Orlik-Solomon algebras

The d-dimensional symmetric space of Drinfeld, which serves as a p-adic and higher dimensional analog of the complex upper half plane, comes with a natural map to the Bruhat-Tits building of GL_d+1 over a p-adic field. Under this map, pre-images of stars of simplices are simple rigid p-adic spaces whose de-rham cohomology can be expressed in terms of Orlik-Solomon 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 p-adic hyperplane arrangement we can define a coefficient system on the Bruhat-Tits building, that reflects the local properties of the arrangement. We calculate the cohomology of this local system for any finite p-adic hyperplane arrangement.

DAVID BOYD, University of British Columbia, Vancouver, BC V6T 1Z2
The A-polynomials of periodic knots

We show that the A-polynomials 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 Mordell-Weil and Shafarevich-Tate 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 p-adic L-functions

I will report on work in progress concerning exceptional zeros of certain p-adic L-functions attached to modular forms and the so-called L-invariant.

SAMIT DASGUPTA, Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138
Stark-Heegner points on modular Jacobians

We present a construction which lifts Darmon's Stark-Heegner 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 G₀(Np).

We then construct a certain torus T over Q_p and sub-lattice L of T, and prove that the quotient T/L is isogenous to the maximal toric quotient J₀ (Np)^p-new of the Jacobian of X₀(Np). This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens; indeed, our proof borrows greatly from theirs. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J₀ (Np)^p-new.

JORDAN ELLENBERG, Princeton University
Elliptic curves over towers of function fields

I will discuss some results on Mordell-Weil 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 Mordell-Weil 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 Faltings-Chai 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 p-adic Lie extensions

We will discuss about the structure of the Selmer groups of elliptic curves and the Galois group of the maximal unramified abelian p-extensions over p-adic 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 Q-rational endomorphisms of the Jacobian J_X of a modular curve X/Q of arbitrary level N.

These results are then applied to determine the Neron-Severi group of Q-rational 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 "subgroup-free" 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 moduli-theoretic description) to prove a general canonical subgroup theorem.

HERSHY KISILEVSKY, Concordia University, Montreal, Quebec
Vanishing twists of elliptic L-functions

We continue our study of the central values of L-functions of elliptic curves defined over the rationals. We study also the vanishing and non-vanishing values of the central values of twists of these L-functions 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 Fontaine-Mazur on modularity of two dimensional potentially semi-stable Galois representations. These hinge on developments in integral p-adic 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 l-invariants of number fields

l-invariants attached to p-parts of eigenspaces of class groups are related to étale cohomology groups, and the behaviour of the l-invariants under p-extensions is related to Euler characteristics for these groups. We discuss the consequences for "Kida-type" 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 {Fⁿ}_{n ³ 0} on CH^r (X) ÄQ with graded piece Gr_Fⁿ 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_Fⁿ 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 LOZANO-ROBLEDO, 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<sub>A</sub>)] :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<sub>p<sup>¥</sup></sub>) )® SL  (2,Z<sub>p</sub> [[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.

MARC-HUBERT 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 M₂ ( G₀(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(E₁,E₂), deg), where E_i, i=1,2 are supersingular elliptic curves, and deg is the quadratic degree map. The Jacquet-Langlands 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 pro-p 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 pro-p extension such a field, cup products, and p-adic L-values.

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 p-adic 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 X₀(N) in this setting is a 3-dimensional real manifold, so a modular parametrization is too much to hope for. Coincidence of L-function, however, is sufficient for computations.

HUI JUNE ZHU, McMaster University, Canada
Point-counting 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 p-adic approach to this problem. Using Dwork's method and a new transformation lemma we prove a divisibility result for Artin-Schreier 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.