




Number Theory / Théorie des nombres (A. Akbary and O. Kihel, Organizers)
 E. BENJAMIN, Maine
To be announced

 D. BRADLEY, Department of Mathematics and Statistics, University of Maine,
Orono, Maine 044695752, USA
Shuffles and multiple zeta values

Multiple zeta values are values of multiple polylogarithms at positive
integer arguments. They can also be viewed as a multiply nested sum
extension of the Riemann zeta function. As noted by Pierre Cartier in
his recent Bull. Amer. Math. Soc. article commemorating the 40th
anniversary of the IHES, classifying the relations amongst multiple
zeta values connects with some of the deepest problems in
transcendental number theory. It is wellknown that multiple zeta
values satisfy certain ``shuffle'' relations. We describe here a new
technique for proving certain shuffle convolution formulae.
Applications include a short proof of a former conjecture of Zagier and
some new results for multiple zeta values as well.
 S. CHOI, Simon Fraser University, Burnaby, British Columbia
Norm of Littlewood polynomials

We are interested in studying the norm of Littlewood polynomials,
i.e., polynomials with coefficients +1 or 1. The problem is to
find polynomials having small L_{4} norms over the unit circle. The
best known polynomials are Turyn polynomials whose coefficients are the
shifted legendre symbols. Some old and new results about this problem
will be discussed in this talk, especially for polynomials constructed
by cyclic difference sets (Turyn polynomial is one of the examples).
 A. COJOCARU, Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario K7L 3N6
Squarefree orders for nonCM elliptic curves modulo p

Let E be an elliptic curve defined over Q and without complex
multiplication. We consider the problem of determining an effective
asymptotic formula for the number of primes p < x of good reduction
for E and such that the group of points of E modulo p has
squarefree order. This problem is related to the one regarding the
cyclicity of E modulo p, but more challenging. The situation of an
elliptic curve with complex multiplication can also be considered,
however it is very different from the noncomplex multiplication case
and so has to be discussed on a different occasion.
 CLIFTON CUNNINGHAM, University of Calgary, Calgary, Alberta
The geometry of depthzero representations

It has been observed in a number of cases that the number of points on
certain hyper elliptic curves over a finite field may be recovered as
the values of depth zero representations of padic groups, but the
nature of this relationship remains a mystery. In this talk I will
describe a new geometric approach to these representations which
promises to shed light on this and other related problems.
 HENRI DARMON, McGill
To be announced

 CHANTAL DAVID, Concordia
To be announced

 L. DAVISON, Laurentian University
Levy constants for classes of continued fractions

Suppose (u_{n})_{n ³ 1} is a bounded sequence of positive integers
and a = [0,u_{1},u_{2},...,u_{n}¼] is the corresponding
continued fraction with convergents ([(p_{n})/(q_{n})])_{n ³ 0}. If
the infinite word u = u_{1},u_{2},... has the property that any
finite subword of u occurs with a frequency, then we prove
that lim_{n®¥}[1/(n)]lnq_{n} exists. An
application of this result is given.
 JEANMARIE DE KONINCK, Université de Laval, Québec G1K 7P4
The median value of the kth prime factor of an integer

Let p_{1}(n) < p_{2}(n) < ¼ be the distinct prime factors of an integer
n ³ 2. Given a positive integer k, we investigate the distribution
function of the arithmetic function n® p_{k}(n). As an
application of our result, we prove the estimate loglogp^{*}_{k} = kc + O(1/Ök), where p^{*}_{k} stands for the median value of the
kth prime factor of an integer and where c=[1/3] +gå_{n ³ 2} å_{p} [1/(np^{n} )] » 0.59483. This is
joint work with Professor Gérald Tenenbaum.
 SAÏD EL MORCHID, Casablanca
Base des unités cyclotomiques de certains composés
de trois corps abéliens

On considère trois extensions ab\' eliennes réelles finies de
Q, k_{1}, k_{2} et k_{3} de conducteurs respectifs
p_{1}^{e1}, p_{2}^{e2} et p_{3}^{e3}, avec p_{1}, p_{2} et p_{3} des
nombres premiers distinsts deux à deux et e_{1}, e_{2} et e_{3} des
entiers naturels non nuls. L'objet de cette communication est de
déterminer une base du groupe des unités cyclotomiques du composé
k=k_{1}k_{2}k_{3}. Ceci est un travail conjoint avec Hugo Chapdelaine.
 J.G. HUARD, Canisius College, Buffalo, New York 142081098, USA
Sums of twelve and sixteen squares

For a positive integer k, let r_{k}(n) denote the number of
representations of the positive integer n as the sum of k squares.
In 1987, using the theory of modular functions, Ewell proved a formula
for r_{16}(n) in terms of realdivisor functions and established a
result for r_{12}(n). In 1996, Milne obtained formulas for r_{k}(n)
where k is any integer of the form 4m^{2} or 4m(m+1). Milne
combined methods from the theory of elliptic functions, continued
fractions, hypergeometric functions, Schur functions, and Hankel and Tur
án determinants. Using a recent elementary arithmetic identity of
Huard, Ou, Spearman and Williams, we give elementary proofs of Ewell
s formulae and of Milne s formula for k=16. This is joint work
with K. S. Williams.
 C. INGALLS, Department of Mathematics and Statistics,
University of New Brunswick, Fredericton, New Brunswick E3B 5A3
Birational classification of orders

We classify orders over surfaces up to birational and Morita
equivalence. We use the geometric ramification data of a maximal order
on a surface to define a class of terminal orders. We compute all
possible étale local structures of terminal orders. We use the ideas
of Mori's minimal model program for log surfaces to show that terminal
orders with nonnegative Kodaira dimension have unique minimal models
up to Morita equivalence. We describe the possible centres and
ramification divisors of the minimal models of orders of negative
Kodaira dimension.
 ERNST KANI, Queen's University, Kingston, Ontario K7L 3N6
The class number relations of Kronecker, Gierster and Hurwitz

Let H(n) denote the (weighted) number of positive definite binary
quadratic forms of discriminant n. It is wellknown that H(n) can
be expressed in terms of the class number h(n) of the field
Q(Ö[(n)]). For an integer D ³ 1, let
H_{D}(n) = 
å
x Î S_{D}(n)

H 
æ è


4nx^{2} D


ö ø

, 

where S_{D}(n) = {x Î Z : x^{2} £ 4n, x^{2} º 4n mod D.
In 1857 Kroecker found a closed expression for H_{1}(n), and this was
generalized by Gierster(1880) and Hurwitz(1883) to H_{p2}(n), where
p is a prime. In this lecture I will explain how to sharpen the
Gierster/Hurwitz results and also how to generalize them to arbitrary
D = N^{2}.
 JOHN LABUTE, Department of Mathematics, McGill University, Montreal,
Quebec H3A 2K6
Central series of Galois groups over Q with
restricted ramification

Let p be a prime ¹ 2 and let q_{1},¼,q_{n} be primes º 1 mod p. Let S={q_{1},¼,q_{n},p,¥} and let G_{S} be the
galois group of the maximal pextension of Q which is
unramified outside of S. Under certain congruence conditions on the
the finite primes in S, we give an explicit description in terms of
generators and relations of the Lie algebra associated to the lower
central series and the pcentral series of the group G_{S}.
 RAM MURTY, Queen's University
Generalized Hurwitz zeta functions

We will discuss generalizations of the classical Hurwitz zeta
functions, obtain analytic continuations for them and show that certain
special values (namely at negative integers) are given by generalized
Bernoulli polynomials. (This is joint work with Kaneenika Sinha.)
 A. OZLUK, University of Maine, Orono, Maine 04469, USA
Distribution of zeros of Dedekind Zeta functions of
quadratic extensions of imaginary quadratic number fields

This talk will consider the distribution of nontrivial zeros close to
the real axis of the family of all Dedekind Zeta functions of quadratic
extensions of a given imaginary quadratic number field. The main focus
will the case where the ground field has class number one.
 D. ROY, Universite d'Ottawa, Ottawa, Ontario
Approximation of real numbers by cubic algebraic integers

In their 1968 seminal paper, Davenport and Schmidt studied the
approximation of a given real number by algebraic integers of a fixed
degree d. They did so by resorting to the dual problem of
approximating the d1 consecutive powers of this number by rational
numbers with the same denominator. In this talk, we show that, for
d=3, their exponent of approximation for the dual problem is optimal,
against natural heuristics, and we discuss consequences on the original
problem.
 CHIP SNYDER, University of Maine, Orono, Maine 04469
On a class number formula for real quadratic

For an even Dirichlet character y, we obtain a formula for
L(1,y) in terms of a sum of Dirichlet Lseries evaluated at
s=2 and s=3 and a rapidly convergent numerical series involving the
central binomial coefficients. We then derive a class number formula
for real quadratic number fields by taking L(s,y) to be the
quadratic Lseries associated with these fields.
 CAMERON STEWART, University of Waterloo, Waterloo, Ontario
On shifted products which are powers

Let V denote the set of positive integers which are kth powers
of a positive integer for some integer k larger than one. We shall
discuss the problem of estimating the size of a set A of positive
integers with the property that ab+1 is in V whenever a and b
are distinct elements of A.
 F. THAINE, Concordia University
Cyclic polynomials and the multiplication matrices of their roots

Let D be an integrally closed, characteristic zero domain, K its
field of fractions, m ³ 2 an integer and P(x)=å_{k=0}^{m1}c_{k}x^{k} = Õ_{i=0}^{m1}(xq_{i}) Î D[x] a cyclic polynomial. Let
t be a generator of Gal(K(q_{0})/K) and suppose
the q_{i} are labeled so that t(q_{i})=q_{i+1}
(indices mod m). Suppose that the discriminant
discr_{K(q0)/K} (q_{0},q_{1},...,q_{m1}) is
nonzero. For 0 £ i, j £ m1, define the elements a_{i,j} Î K
by q_{0}q_{i} = å_{j=0}^{m1} a_{i,j}q_{j}. Let A=[a_{i,j}]_{0 £ i,j < m}. We call A the multiplication matrix of the
q_{i}. We have that P(x) is the characteristic polynomial of
A. We study the relations between P(x) and A. We show how to
factor P(x) in the field K[A] and how to construct A in terms of
the coefficients c_{i}. We give methods to construct matrices A,
with entries in K, such that the characteristic polynomial of A
belongs to D[x], is cyclic, and has A as the multiplication matrix
of its roots. One of these methods derives from a natural composition
of multiplication matrices. The other method gives matrices A, that
are generalizations of matrices of cyclotomic numbers of order m,
whose characteristic polynomials have roots that are generalizations of
Gaussian periods of degree m.
 ALAIN TOGBÉ, Greenville College, Illinois, USA
On related cubic Thue equations

We will discuss Baker's method for solving certain related cubic
families of Thue equations. First we will give details for solving a
first equation and then use changes of variables to obtain the
solutions to the other equations.
 GARY WALSH, University of Ottawa, Ottawa, Ontario K1N 6N5
On the product of likeindexed terms in
binary recurrence sequences

In recent work, Corvaja and Zannier have
used the subspace theorem to prove an upper bound
for gcd(a^{k}1,b^{k}1) for a fixed pair of positive
integers a,b. Also, it has recently been conjectured
by Rudnick that this gcd is 1 infinitely often
provided that a and b are multiplicatively
independent. In joint work with Florian Luca,
we use the methods of Corvaja and Zannier to prove
that for fixed a,b, the equation
(a^{k}1)(b^{k}1)=z^{n} has finitely many integer
solutions k,x and n > 1. We further describe a
computational method to determine all integer solutions
(k,x) to the equation (a^{k}1)(b^{k}1)=x^{2}, and
solve this equation for all 1 < a < b < 100 provided
that (a1)(b1) is not a square.

