INVITED SPEAKER ABSTRACTS
Stephen Choi, Simon Fraser University
On the Norms of Littlewood Polynomials
In this talk, we will discuss the norms over the unit circles
of Littlewood polynomials, that is, polynomials with coefficients
+1 or -1. In particular, we are interested in the $L_4$ norm and
Mahler measure of Littlewood polynomials. Some current results
and related conjectures, such as Barker sequences conjecture will
be discussed in the talk.
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Todd Cochrane, Kansas State University
Waring's Problem over Finite Fields
For prime p and positive integer k we define Waring's number
g(k,p) to be the minimal s such that every integer is a sum
of s k-th powers (mod p). Equivalently, letting A denote the
set of k-th powers in the finite field F in p elements, g(k,p)
is the minimal s such that sA=F. We discuss various estimates
for g(k,p) and the variety of methods used to obtain them including
the estimation of exponential sums, additive combinatorics and
the sum-product phenomenon, the geometry of numbers, and heights
and zeros of integer polynomials. The strength of the method
depends on the size of A.
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Karl Dilcher, Dalhousie University
A mod p^3 analogues of a theorem of Gauss on binomial coefficients
The theorem of Gauss that gives a modulo p evaluation of a certain
central binomial coefficient was extended modulo p^2 by Chowla,
Dwork, and Evans. In this talk I present a further extension to
a congruence modulo p^3, with a similar extension of a theorem
of Jacobi. This is done by first obtaining congruences to arbitrarly
high powers of p for certain quotients resembling binomial coefficients
and related to the p-adic gamma function. These congruences are
of a very simple form and involve Catalan numbers as coefficients.
As another consequence we obtain complete p-adic expansions for
certain Jacobi sums. (Joint work with John B. Cosgrave).
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Matthew Greenberg, University of Calgary
Computing with automorphic forms
Like all of scientic research, the study of automorphic forms was
fundamentally altered by the development of the computer. Landmarks
in the computational theory of automorphic forms are Cremona's systematic
enumeration of elliptic curves via their associated modular forms
and Stein's sophisticated software packages for computing with modular
forms. Many students and researchers use these now ubiquitous resources
on a daily basis. In this talk, I will discuss the evolution of
the computational methods for automorphic forms, current developments
in the field, and prospects for future development.
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Matilde Lalín, Université de Montréal
Higher Mahler measure
The classical Mahler measure of an $n$-variable nonzero polynomial
P is the integral of $\log |P|$ over the $n$-dimensional unit
torus $T^n$ with the Haar measure. We consider, more generally,
the integral of $\log^k |P|$. Specific examples yield special
values of zeta functions, Dirichlet L-series, and polylogarithms.
Moreover, one can ask the equivalent to Lehmer's question, and
explore what happens at cyclotomic polynomials. This talk includes
joint work with N. Kurokawa
and H. Ochiai and with K. Sinha.
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Hugh L. Montgomery, University of Michigan
Families of polynomials
This talk falls into two parts. Recently Mauduit and Rivat solved
a problem posed by Gelfond in 1968, which in particular asserts
that the sum of the binary digits of a prime is odd asymptotically
half the time. Their work depends on estimates of moments of an
associated exponential polynomial. We find that as the order increases,
the sequence of moments satisfies a linear recurrence, arising
from polynomials with interesting properties.
Let f_n denote the density of the sum of n independent uniformly
distributed random variables. A formula for this density was derived
independently by Lagrange and Stirling, roughly 250 years ago.
It is piecewise polynomial, and at its transition points is proportional
to eulerian numbers. The eulerian numbers satisfy a Pascal-like
recurrence.
We find that f_n satisfies such a recurrence not just at the transition
points, but for all arguments. This allows us to define a one-parameter
family of polynomials of which the eulerian polynomials are a
special case.
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Kumar Murty, University of Toronto
Lifting elliptic curves to characteristic zero
Let S be a set of primes and for each prime p in S, suppose we
are given an elliptic curve E(p) over the field of p elements.
Under what conditions does there exist an elliptic curve E over
the rationals such that for each p in S, the reduction of E mod
p is E(p)? If S is a finite set, the existence of E follows from
the Chinese remainder theorem. If the complement of S is finite,
we will give a criterion in terms of "minimal lifting conductors".
This is joint work with Sanoli Gun.
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Damien Roy, University of Ottawa
Rational approximation to real points on algebraic curves
Let C be a projective algebraic curve defined over Q. Assume
that the set of real points of C with Q-linearly independent
coordinates is infinite, and define lambda(C) to be the supremum
of the uniform exponents of approximation of those points by
rational points. Although Dirichlet's box principle simply shows
that lambda(C) is at least 1/n, it is tempting to conjecture
that it is always greater, i.e. that there always exist such
points which are constantly much better approximated by rational
points then expected from the box principle. At the moment,
the only curves for which this is known to hold is the curve
defined by the polynomial xz - y^2 and those which derive from
it by a linear automorphism. For most curves, we only have an
upper bound for lambda(C). In the talk, we will discuss this
and also the case of the curve defined by the polynomial x^2
z - y^3 for which recent joint work with Stéphane Lozier
has shown that lambda(C) is at most 107/151.
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Renate Scheidler, University of Calgary
Infrastructure of Function Fields
The infrastructure of a global field has been used for a variety
of important applications, including computing the regulator and
the class number of a global field, and even for cryptography.
Originally proposed by D. Shanks in 1972 in the setting of real
quadratic fields, the concept has since been generalized to number
fields of higher degree and to function fields. Strictly speaking,
every field extension has multiple infrastructures, one for each
ideal class, each containing a certain finite subset of "small"
ideals in that class. Of particular interest is the infrastructure
belonging to the principal class. Geometrically, this infrastructure
is a torus whose dimension is the unit rank of the field extension.
It is possible to equip this torus with a binary operation that
is akin to multiplication and is called a giant step. The resulting
structure behaves "almost" like an Abelian monoid --
and in quadratic extensions even almost like an Abelian group
-- failing only associativity, and just barely. In the unit rank
one case, a second addition-like operation, called a baby step,
imposes an ordering on the infrastructure ideals according to
a natural distance which is "almost" additive under
giant steps. We present the baby step giant step arithmetic framework
of the infrastructure and explain what it means to "just
barely" fail associativity.
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Edlyn Teske-Wilson, University of Waterloo
Homomorphic Cryptosystems
Over the past few years, several solutions have been proposed
that address the problem of homomorphic encryption and homomorphic
signing. This talk highlights selected features of homomorphic
cryptosystems.
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Gary Walsh, University of Ottawa and the Tutte Institute
Rational and integral points on families of elliptic
curves
Though not exhaustive, the goal of this lecture is to present
some recent developments on the computation and existence of rational
and integral points on ellptic curves. This will be exhibited
by way of focussing on certain specific families of elliptic curves,
thereby, hopefully, providing some illuminating examples of the
methods that have been developed for these purposes.
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Lawrence C. Washington, University of Maryland
Class numbers of real cyclotomic fields
Computation of class numbers of real subfields of cyclotomic
fields by standard methods is very difficult because of the presence
of units. Over the past several years, alternative methods have
been developed. This talk will describe some of these methods
and discuss results and conjectures that have arisen from some
of this work.
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CONTRIBUTED TALKS
Timothy Caley, University of Waterloo
The Prouhet-Tarry-Escott problem for Gaussian integers
The Prouhet-Tarry-Escott (PTE) problem is a classical number
theoretic problem which asks for integer solutions to sums of
equal powers. Solutions to the PTE problem give improved bounds
for the "Easier" Waring problem, but they are difficult
to find using conventional methods.
We will describe how solutions can be found by connecting the
problem to finding rational points on elliptic curves. There will
also be a statement of open questions relating to the PTE problem.
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Michael Dewar, Queen's University
The image and kernel of Atkin's $U_p$ operator modulo $p$
We determine the image of Atkin's $U_p$ operator acting on $\pmod
p$ reduced modular forms. In 1972, Serre showed that for level
1 modular forms, $U_p$ was contractionary (i.e. the image has
lower weight than the preimage.) We determine the exact weight
of the space of images and generalize to all levels not divisible
by $p$. As a consequence, we determine the dimension of the kernel
of $U_p \pmod p$ for large weights. This contrasts with the situation
for small weights, which is stil confoundingly mysterious.
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Greg Doyle, Carleton University
A Recursive Formula for the Convolution Sum of Divisor Functions
For fixed arbitrary positive integers $a$ and $b$, we are interested
in determining the value of the convolution sum $\sum_{m=1}^{n-1}\sigma_a(m)\sigma_b(n-m)$,
for an arbitrary positive integer $n$, where $\sigma_a(n) = \sum_{d
\mid n} d^a$. Using an identity given by Alaca, Alaca, McAfee
and Williams, we derive a recursive formula for this convolution
sum for all pairs of odd positive integers $a$ and $b$. In a similar
fashion, we discuss how we might derive a similar recursive formula
for the twisted convolution sum $\sum_{m<n/k}\sigma_a(m)\sigma_b(n-km)$.
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Himadri Ganguli, Simon Fraser University
On the behaviour of the Liouville function on polynomials with
integer coefficients
Let $\lambda (n)$ denote the Liouville function. Complementary
to the prime number theorem, Chowla conjectured that {\bf Conjecture
(Chowla).} $\sum_{n\le x} \lambda (f(n)) = o(x)$ for any polynomial
$f(x)$ with integer coefficients, not in the form of $bg^2(x)$,
where $b$ is a constant. Chowla's conjecture is proved for linear
functions but for the degree greater than 1, the conjecture seems
to be extremely hard and still remains wide open. One can consider
a weaker form of Chowla's conjecture, namely, {\bf Conjecture
1 (Cassaigne, et al).} If $f(x) \in \Z [x]$ and is not in the
form of $bg^2(x)$ for some $g(x)\in \Z[x]$ and constant $b$, then
$\lambda (f(n))$ changes signs infinitely often. Although it is
weaker, Conjecture 1 is still wide open for polynomials of degree
$>1$. In this talk, I will describe some recent progress made
while studying Conjecture 1 for the quadratic polynomials. This
is joint work with Peter Borwein and Stephen Choi.
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Hester Graves, Queen's University
Euclidean Ideal Classes and Hilbert Class Fields
Lenstra generalized the concept of the Euclidean algorithm to
Euclidean ideals. If a domain has a Euclidean ideal, then its
class group is cyclic and the Euclidean ideal's class generates
the class group. Lenstra showed that, assuming GRH, that for every
finite extension of Q that is not imaginary quadratic, said field's
class group is cyclic if and only if every generator is a Euclidean
ideal class. In this talk, we will remove prove Lenstra's result
for certain classes of fields without assuming GRH.
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Behzad Omidi Koma, Carleton University
The Number of Irreducible Polynomials of Even Degree $n$ over
$\F_2$ with Four Given Coefficients
The problem of estimating the number of irreducible polynomials
of degree $n$ over the finite field $F_q$ with some prescribed
coefficients has been largely studied. This is a study of the
number of irreducible polynomials of even degree $n$ over the
finite field $\F_2$ where the coefficients of the terms $x^{n-1}\cdots,x^{n-r}$
are given, for $r \ge 4$. This number is represented by $N(n,t_1,\cdots,t_r)$.
In this paper an approximation for these numbers is given and
also experimentally is shown how good is the approximation. [Joint
work with D. Panario]
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Rob Noble, Dalhousie University
Asymptotics of the weighted Delannoy numbers
The weighted Delannoy numbers give a weighted count of lattice
paths starting at the origin and using only minimal east, north
and northeast steps. Full asymptotic expansions exist for various
diagonals of the weighted Delannoy numbers. In the particular
case of the central weighted Delannoy numbers, certain weights
give rise to asymptotic coefficients that lie in a number field.
In this talk we apply a generalization of a method of Stoll and
Haible to obtain divisibility properties for the asymptotic coefficients
in this case. We also provide a similar result for a special case
of the diagonal with slope 2.
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Fabien Pazuki, University Bordeaux 1
Lower bounds on heights and applications.
Abstract : Let k be a number field and A/k an abelian variety.
We will explain how, from a lower bound on the Neron-Tate height
of the rationnal points P on A, one can give precise information
about the arithmetic of the variety itself or about diophantine
aspects of algebraic curves inside the variety. We will describe
recent results in the case of abelian surfaces.
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Ethan Smith, CRM
Average Frobenius distribution for the degree two primes of a
number field
Let $K$ be a fixed number field and $E$ an elliptic curve defined
over $K$. Given a fixed integer $t$, we consider the problem of
counting the number of degree two prime ideals $\mathfrak p$ of
$K$ such that the trace of Frobenius of $E$ at $\mathfrak p$ is
equal to $t$.
Under some assumptions on the number field, we show that on average,
over the elliptic curves defined over $K$, the number of such
prime with norm less than or equal to $x$ satisfies an asymptotic
that is in accordance with standard heuristics. This is joint
work with Kevin James.
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David Thomson, Carleton University
Swan-like results for low-weight polynomials over finite fields
of odd characteristic.
Joint with B. Hanson (Toronto) and D. Panario (Carleton)
The study of low-weight polynomials is important for implementations
of fast finite field arithmetic using a polynomial basis. Swan
(1962) applies a theory of Stickelberger to determine the parity
of the number of irreducible factors of trinomials (polynomials
with exactly three nonzero terms) over the binary field. Vishne
(1992) extends Swan's work to all finite fields of even characteristic.
In this work, we give the parity of the number of irreducible
factors of all binomials and on a number of classes of trinomials
over finite fields of odd characteristic. We indicate the bottleneck
of this method in the trinomial case, which depends on additive
properties of the quadratic character over finite fields.
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Colin Weir, University of Calgary
A method for constructing cubic function fields of fixed discriminant
This is joint work with R. Scheidler (Calgary).
Research into the construction of certain low degree function
fields has surged in recent years, in part due to the cryptographic
significance of elliptic and hyperelliptic curves. However there
is comparatively little data available for higher degree function
fields, leaving open many questions about the number of non-conjugate
function fields of fixed degree and given discriminant. We will
present an algorithm for tabulating a complete list of cubic function
fields over a fixed finite field of bounded discriminant degree.
Our methods are based on those of H. Cohen (1999) who tabulates
cubic extensions of arbitrary number fields. The algorithm uses
the tools of Kummer Theory and Class Field Theory, allowing for
a natural transition to the function field setting. However, we
are able to utilize the additional automorphisms of algebraic
function fields to make significant improvements. Moreover, the
algorithm is constructive in nature, allowing us to easily generate
function fields of a specified discriminant as well. We will present
the algorithm, our improvements, and a summary of the data we
able able to generate.
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