




Number Theoryin Honour of David Boyd / Théorie des
nombresen l'honneur de David Boyd (Peter Borwein and Michael Bennett, Organizers)
 JIM ARTHUR, Toronto
To be announced

 IMIN CHEN, Simon Fraser University, British Columbia
On relations between induced representations for GL_{2}(Z/p^{2}Z)

We will exhibit a relation between induced representations for
GL_{2}(Z/p^{2}Z) analogous to the one for GL_{2}(Z/pZ) which relates
the induced representations from split and nonsplit Cartan subgroups.
Some possible applications to modular curves will be discussed. This
work is joint with B. deSmit and M. Grabitz.
 KWOK KWONG (STEPHEN) CHOI, Simon Fraser University, Burnaby, British Columbia V5A 1S6
Small prime solutions of quadratic equations

In this talk, we will study the small prime solutions of certain
diophantine equations by using the HardyLittlewood (circle) method.
In particular, we prove the following. If b_{1},...,b_{5} are nonzero
integers, then the quadratic equation b_{1}p_{1}^{2}+¼+b_{5}p_{5}^{2}=n has
prime solutions satisfying p_{j} << Ö[(n)]+max{b_{j}}^{20+e}. In contrast to the earlier works which treat
the enlarged major arc by the DeruringHeilbronn phenomenon about the
Siegel zero. We will explain the possible existence of Siegel zero
does not have special influence and hence the DeuringHeilbronn
phenomenon can be avoided. This observation enables us to get better
results without numerical computations.
 KARL DILCHER, Dalhousie University, Halifax, Nova Scotia
Bernoulli numbers and confluent hypergeometric functions

If the reciprocal of the confluent hypergeometric function
M(s+1;r+s+2;z) is taken as an exponential generating function, the
resulting sequences of numbers have many properties resembling those of
the Bernoulli numbers, which arise when r=s=0. Other special cases
include van der Pol numbers and generalized van der Pol numbers (for
positive integers r=s), as well as various sequences of combinatorial
numbers and linear recurrence sequences of arbitrary orders. Explicit
expressions resembling Euler's formula involve sums of powers of the
zeros of the corresponding confluent hypergeometric functions.
 RON FERGUSON, Simon Fraser University, Burnaby, British Columbia
Units in integral cyclic group rings for order 4p

For a finite abelian group A, the group of units in the integral
group ring Z A may be written as the direct product of its
torsion units ±A with a free group U_{2}A. Of finite index in
U_{2}A is the group WA, the elements of U_{2} which are mapped
to cyclotomic units by each character of A. The order of
U_{2}A/WA depends on class numbers h_{d}^{+} in real cyclotomic
rings Z [z_{d}+z_{d}^{1}].
Of finite index in WA is the group of constructible units YA,
for which a multiplicative basis may be explicitly written. The order
of WA/YA is the circular index c(A). In many cases, for
example where A is a pgroup with p a regular prime, this index
is trivial.
The theory is developed for determining c(C_{n}) where C_{n} is a
cyclic group of order n=4p, for p prime, and finding a
multiplicative basis for the group WC_{n}. This is a
continuation of the work of Hoechsmann for the case n=lp. The
extension for a higher power, however, does require the introduction of
new methods.
 JOHN FRIEDLANDER, Toronto
Exponential sums over thin sets

We discuss results with H. Iwaniec and with I. Shparlinski
on character and exponential sums over thin sets of integers.
 KEVIN HARE, Simon Fraser University, British Columbia
Spectra of Pisot numbers

Here we study the spectrum that is the result of evaluating certain
classes of polynomials at a value q > 1. Some classes of polynomials
that have been looked at included those of height m, or those with
coefficients restricted to ±. This talk with give a history of
problems relating to these spectrums, the problems interrelationship
with Pisot numbers, and some recent results in this area. Lastly this
talk will describe some possible future directions for work in this
area.
 HERSHY KISILEVSKY, Concordia University, Montreal, Quebec
Vanishing and nonvanishing of twisted Lfunctions

Given an elliptic curve E over the rational field, Q, we
compare the MordellWeil groups E(K) and E(Q) for cyclic
cubic extensions K/Q. We expect that the rank E(K) is
equal to the rank of E(Q) for almost all such
K/Q, but that the rank actually increases for an infinite
number of K. We describe our results on this problem, with
particular reference to the rank 0 curve of conductor 37 which is
unusually rich in this regard.
 GREG MARTIN, University of Toronto, Toronto, Ontario M5S 3G3
Squarefree values of trinomial discriminants

The discriminant of a trinomial of the form x^{n}±x^{m}±1 has the
form ±n^{n} ±(nm)^{nm}m^{m} if n and m are relatively prime.
We investigate when these discriminants have nontrivial square factors.
We explain various unlikelyseeming parametric families of square
factors of these discriminant values: for example, when
n º 2 mod 6 we have

æ è


n^{2}n+1 3


ö ø

2


ê ê


æ è

n^{n}(n1)^{n1} 
ö ø

. 

In addition, we discover many other square factors of these
discriminants that do not fit into these parametric families. The set
of primes whose squares can divide these sporadic values as n varies
seems to be independent of m, and this set can be seen as a
generalization of the Wieferich primes, those primes p such that
2^{p1} º 1 mod p^{2}. The truth about the density of squarefree
values of these discriminants, and about the density of these
``sporadic'' primes, is still unclear. (Joint work with David Boyd)
 KUMAR MURTY, Toronto
To be announced

 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
Pair correlation conjecture and the Chebotarev density theorem

We will formulate a paircorrelation conjecture for the zeros of Artin
Lfunctions analogous to a similar conjecture formulated by
Montgomery in the theory of the Riemann zeta function. We will then
investigate the consequences of such a conjecture for the error term in
the effective version of the Chebotarev density theorem. (This is
joint work with Kumar Murty.)
 NATHAN NG, University of British Columbia, Vancouver, British Columbia
The summatory function of the Mobius function

We present some conditional results concerning the summatory function
of the Mobius function, M(x). These results depend on a conjecture
by Steve Gonek and Dennis Hejhal regarding negative discrete moments of
the zeta function. The purpose of this talk is to show how this
conjecture allows one to obtain much better results about M(x) than
were previously known. Assuming the Riemann Hypothesis and the
GonekHejhal conjecture, we can show the weak Mertens conjecture is
true and construct a limiting distribution attached to M(x). The
constructed limiting distribution then reveals the behaviour of
M(x). For example, we speculate that
M(x)=W_{±} 
æ è

x^{[1/2]}(logloglogx)^{[5/4]} 
ö ø



based on properties of the constructed limiting distribution.
 CHRISTOPHER PINNER, Kansas State University, Manhattan, Kansas 66506, USA
A bound for a complete exponential sum modp

We give an upper bound for the exponential sum å_{x=1}^{p} e(f(x)/p), p a prime and f a nonzero integer polynomial, of
interest in cases where the Weil bound is worse than trivial. The
result extends a technique of Konyagin for monomials to a general
polynomial. Such bounds readily yield estimates for the corresponding
polynomial Waring problem mod p; namely the smallest g such
that f(x_{1})+¼+f(x_{g}) º N mod p) is
solvable in integers for any N.
This is joint work with Todd Cochrane.
 DAMIEN ROY, University of Ottawa, Ottawa, Ontario
Interpolation on perturbations of cartesian products

In 1970, E. Bombieri and S. Lang used analytic results of P. Lelong to
establish a Schwarz lemma for a well distributed set of points in
C^{n}. Their result was extended to an interpolation lemma,
first by D. W. Masser in the case of polynômials, then by
M. Waldschmidt for analytic fonctions. J.C. Moreau gave an analog of
it over the reals and P. Robba in the padic realm. Robba also
conjectured a padic interpolation lemma for the case where the set
of points of interpolation is what he calls a perturbation of a product
set, a situation which includes both the case of a well distributed set
and the case of a cartesian product. In this talk, we present an
algebraic proof of Robba's conjecture together with a generalization of
it over the complex numbers.
 CAM STEWART, University of Waterloo, Waterloo, Ontario
Multivariate Diophantine equations with many solutions

We shall discuss recent joint work with J.H. Evertse, P. Moree and
R. Tijdeman on Sunit equations and norm polynomial equations with
many solutions.
 MICHEL WALDSCHMIDT, Institut de Mathematiques de Jussieu
Polylogarithms and multiple zeta values

For a tuple s=(s_{1},¼,s_{k}) of positive integers the
power series
Li_{s}(z)= 
å
n_{1} > ¼ > n_{k} ³ 1


z_{1}^{n1}¼z_{k}^{nk} n_{1}^{s1}¼n_{k}^{sk}



defines an analytic function in the unit polydisk z_{i} < 1 (1 £ i £ k). When s_{1} ³ 2 this function is continuous on the closed
polydisk z_{i} £ 1 (1 £ i £ k) and its value at the point
z_{1}=¼ = z_{k}=1 is the multiple zeta value z(s).
Several shuffle relations connect the different
Li_{s} and give rise to quadratic or linear algebraic
dependence relations between the multiple zeta values. The main
Diophantine conjecture is that all algebraic dependence relations
between these numbers are consequences of these.
 HUGH WILLIAMS, Department of Computer Science, University of Manitoba,
Winnipeg, Manitoba R3T 2N2
Some computational results concerning Yokoi's conjecture

Let d = m^{2}+4 be the fundamental discriminant of the real quadratic
field K = Q(Öd). Yokoi's Conjecture states
that if the class number of K, written h(d), is 1, then
m £ 17. It is known that there can be at most one exceptional value
of m ( > 17) such that h(d) = 1, and that this exceptional value
must exceed 10^{300000}; it is also known that Yokoi's conjecture
must hold if the extended Riemann hypothesis is true. In spite of this
evidence for the truth of the conjecture, it seems that proving it is a
very difficult problem, indeed.
In this paper we show how earlier arguments of Beck can be refined and
implemented on a computer to restrict further those values of m which
would produce a counter example to Yokoi's conjecture For example, we
were able to use a computer to show that there are at most 36 possible
residue classes for m (modulo 179) such that h(d) might be 1.
This is joint work with Jozsef Beck (Rutgers) and Edlyn Teske
(Waterloo).

