


Number Theory / Théorie des nombres Org: Michael Bennett (UBC), Peter Borwein (Simon Fraser), David Boyd (UBC), Imin Chen (Simon Fraser), and/et Stephen Choi (Simon Fraser)
 AMIR AKBARY, University of Lethbridge, Lethbridge, Alberta T1K 3M4
On nonvanishing of convolution of Dirichlet series

We study the nonvanishing on the line Â(s)=1 of the convolution
series associated to two Dirichlet series in a certain class of
Dirichlet series. The nonvanishing of various Lfunctions on the
line Â(s)=1 will be simple corollaries of our general theorems.
This is joint work with Shahab Shahabi.
 CHANTAL DAVID, Department of Mathematics, Concordia University,
Montreal, Quebec H3G 1M8
Average Frobenius distribution for inert primes in Q(i)

For any number field K which is not totally imaginary, Elkies showed
that any elliptic curve E over K has infinitely many supersingular
primes. The result is also believed to be true for number fields which
are totally imaginary, but the proof does not seem to generalise to
this case. One reason may be that over totally imaginary fields, we can
build examples where the supersingular primes must be inert primes, and
then the set of supersingular primes is much thinner than what is
expected for elliptic curves over Q. We then consider the problem of
enumerating the inert primes of the quadratic imaginary field Q(i)
for which a given elliptic curve E has supersingualr reduction. We
prove that on average, the number of such primes up to x is in
accordance with the standard conjectures. This gives further evidence
that there should be infinitely many supersingular primes in this case
as well. (joint work with F. Pappalardi)
 KARL DILCHER, Dalhousie University, Halifax, Nova Scotia B3H 3J5
Resultants of Chebyshev and related polynomials

Resultants and discriminants have always been important tools in number
theory. Departing from a theorem of Emma Lehmer on resultants of
cyclotomic polynomials, we make the following observation: When one
takes the resultant of a certain pair of polynomials closely related to
the Chebyshev polynomials (of the second kind) and involving a
parameter, one surprisingly obtains again a Chebyshev polynomial in
this parameter. We prove an identity that contains this observation and
others as special cases. The proofs use a "chain rule" for resultants
and an identity related to the Euclidean algorithm for polynomials;
these last results, though not new, do not appear to be widely known in
the literature. (Joint work with K. B. Stolarsky).
 JOHN FRIEDLANDER, University of Toronto, Toronto, Ontario M5S 3G3
On a zerosum problem

We discuss recent joint work with S. Adhikari, S. Konyagin, and
F. Pappalardi concerning a zerosum problem in the residue class ring
of integers modulo n.
 HERSHY KISILEVSKY, Concordia University
Twisting elliptic Lfunctions by Dirichlet characters

We continue our examination of the vanishing of the central values of
Lfunctions of elliptic curves twisted by Dirichlet characters.
 CLAUDE LEVESQUE, University of Laval, Quebec G1K 7P4
Fundamental systems of units for some families of number
fields of degree 12 over Q

We explicitly exhibit the unit group of certain infinite families of
fields L of degree 12 over Q which are composita of real
quadratic fields and pure sextic fields. This is achieved with the
help of the unit groups of the three subfields of L which are of
degree 6.
 DAVID MCKINNON, University of Waterloo, Waterloo, Ontario
Rational approximation and counting points

When computing upper bounds for the number of rational points of
bounded height on a (compact) projective algebraic variety defined over
Q, it is an easy observation that there either must be few
points, or those points must be packed closely together. In this talk,
I will describe a few results describing how close rational points on
an algebraic variety can get to one another, and the implications these
results have for the density of rational points on these varieties.
 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
Modular forms and Dirichlet series

We will use the theory of modular forms and Dirichlet series, more
specifically, those attached to Hecke grossencharacters of imaginary
quadratic fields to settle a recent conjecture of Borwein and Choi.
This is joint work with R. Osburn.
 NATHAN NG, Universite de Montreal
Discrete moments of the Riemann zeta function

In this talk we consider the square of the Riemann zeta function times
a Dirichlet polynomial averaged over the zeros of the zeta function.
We give an asymptotic evaluation of this moment, thus generalizing
earlier work of Conrey, Ghosh, and Gonek. As a consequence, we will
deduce that there are infinitely many consecutive zeros of the zeta
function on the critical line whose gaps are larger than the average.
 DAMIEN ROY, Département de mathématiques, Université d'Ottawa,
Ottawa, Ontario K1N 6N5
Simultaneous approximation and small value estimates

We present results of simultaneous approximation of several numbers
from a field of transcendence degree one by conjugate algebraic
numbers. In some cases, we also derive, by duality, new Gel'fond type
criterions for polynomials taking small values at distinct points.
 CAM STEWART, University of Waterloo
On a family of pseudorandom sequences

In this talk we plan to discuss some joint work with A. Sarkozy
concerning pseudorandom binary sequences. We shall construct a large
family of such sequences with a remarkable uniformity within the
family.
 GARY WALSH, University of Ottawa, Ottawa, Ontario
Ternary recurrence sequences and torsion points on Williams curves

We describe an algorithm to compute all occurrences of a given integer
in a family of ternary recurrence sequences which arise from units in
pure cubic fields. The algorithm involves computing all rational
integer points on a parametric family of elliptic curves, which just
happen to contain a parametric point of order three. Rational torsion
points of higher order on these curves correspond to integer solutions
of certain diophantine equations F(x,y)=0, where the height and
degrees of each F are quite large. Fortunately, these polynomials
satisfy some unexpected properties, allowing for the complete
description of the rational torsion subgroup of the original family of
curves. This is joint work with Emanuel Herrmann.
 HUGH WILLIAMS, University of Calgary
Periodic continued fractions with short periods

One of the most important invariants of an order of an algebraic number
field is the regulator of that order. This object is often very
difficult to evaluate, even in a field as seemingly simple as a real
quadratic field. Indeed this is one of the oldest problems of number
theory, and it can be traced as far back as Archimedes. A very useful
technique for evaluating the regulator of a real quadratic order is the
method of continued fractions. It has been known for over two hundred
years that the fundamental unit of such an order can be determined from
the periodic continued fraction expansion of the generator of the
order. In this case the regulator is simply the logarithm of the
fundamental unit, and this is roughly the same as the length of the
period of the corresponding continued fraction.
For most real quadratic orders the regulator tends to be rather large,
but in certain, unusual and infrequent cases it is small, ensuring that
the ideal class number will be large. Such orders are of very great
interest to number theorists because their ideal class groups often
have exotic structures. The search for such orders is a very old
problem in number theory; indeed, Dickson's History of the Theory of
Numbers devotes several pages to the work of many authors concerning
this problem. In this talk I will describe some families of radicands
D(X), given by D(X)=AX2+BX+C such that the value of the period
length l(D(X)) of the regular continued fraction expansion
of vD(X) tends to be small.

