


Number Theory / Théorie des nombres (Org: Damien Roy and/et Kenneth Williams)
 MICHAEL BENNETT, University of British Columbia
Elliptic curves with prescribed torsion and
good reduction outside a small set
[PDF] 
We consider the problem of classifying elliptic curves over
Q with prescribed torsion group and conductor of the form
p^{a} q^{b} where p and q are arbitrary primes. This can be achieved
under certain technical conditions, using results from the theory of
Galois representations and modular forms. This is joint work with Nike
Vatsal and Soroosh Yazdani.
 DOUGLAS C. BOWMAN, Northern Illinois University, USA
Integers n for which the integer parts of
n ×a+ s are not equal to the integer
parts of n ×b+ s
[PDF] 
(Joint work with Alexandru Zaharescu)
Let a and b be positive real numbers and s a real number
satisfying 0 £ s < 1. Let ëxû denote the greatest
integer £ x. Define Y_{k}(a,b;s) to be the kth
positive integer n such that ëna+sû ¹ ënb+sû. For i=1,2 we compute asymptotics for the
probability that Y_{i}(a,b;0) > Q for Q large as a and
b range independently over a subinterval of [0,1). We find the
expected value of Y_{1}(b,a;0) as a and b
range independently over [0,1). When a, b, and s are
fixed, the algebraic structure of the set of natural numbers
{Y_{i}(b,a;s)  i Î Z^{+}} is characterized.
 KWOKKWONG STEPHEN CHOI, Simon Fraser University, Burnaby, British Columbia V5A 1S6
Binary sequences with merit factors greater than 6.34
[PDF] 
For decades the merit factor problem of binary sequences has been stuck
at a value of 6 for the maximum asymptotic merit factor and indeed a
number of authors speculated that this was best possible. We construct
an infinite family of binary sequences that we conjecture have merit
factors greater than 6.34. The numerical experimentation that led to
this example is a significant and interesting part of the story. This
problem is related to finding small L_{4} norm of polynomials with +1
or 1 coefficients.
This is a joint work with Peter Borwein and Jonathan Jedwab.
 ALINA COJOCARU, Fields Institute
The square sieve and the LangTrotter conjecture
[PDF] 
Let E be an elliptic curve defined over the rationals and without
complex multiplication. Let K be a fixed imaginary quadratic field.
We use the square sieve to find nontrivial upper bounds for the number
of primes p of ordinary reduction for E such that Q(p_{p})=K,
where p_{p} is the Frobenius endomorphism of E at p. This
represents progress towards a 1976 LangTrotter conjecture.
(This is joint work with E. Fouvry and M. Ram Murty)
 CHANTAL DAVID, Concordia University
On the vanishing of twisted Lfunctions of elliptic curves
[PDF] 
(joint work with J. Fearnley and H. Kisilevsky)
Let E be an elliptic curve over the rationals with Lfunction
L_{E}(s). Let c be a Dirichlet character, and let L_{E}(s, c)
be the Lfunction of E twisted by the character c. For
quadratic characters c, L_{E}(1, c) vanishes for at least half
of the characters (where the sign of the functional equation is 1),
and Goldfeld conjectured that the density of vanishing is exactly 1/2
in this case. For higher order characters, the functional equation now
relates L_{E}(1, c) and L_{E}(1, [`(c)]), and there is no reason
to predict a positive density of vanishing. We present in this talk
some evidence for the case of twists by cubic character c, based
on empirical computations and random matric theory.
 ERIC FREEMAN, Carleton University, 4302 Herzberg Laboratories, Ottawa,
Ontario K1S 5B6
Systems of cubic diophantine inequalities
[PDF] 
We consider systems of cubic Diophantine inequalities. In particular,
we have that if s is any integer with s ³ (10R)^{g}, where
g = (10R)^{5}, then given any R real cubic forms C_{1},¼,C_{R} in s variables, there is a nonzero integral solution
x of the simultaneous Diophantine inequalities
C_{1}(x) < 1,C_{2}(x) < 1,¼,C_{R}(x) < 1.
 JOHN FRIEDLANDER, Department of Mathematics, University of Toronto,
Toronto Ontario M5S 3G3
On some exponential sums over Mersenne numbers
[PDF] 
Let m be a positive integer, a and g integers relatively prime to
m. We give estimates for the exponential sum

å
n £ N

L(n)exp(2 pi a g^{n}/m), 

where L is the von Mangoldt function, and for a number of
similar sums. In particular, our results yield bounds for exponential
sums of the form

å
p £ N

exp(2 pi a M_{p}/m), 

where p runs through primes and M_{p} is the Mersenne number
M_{p}=2^{p}1. These results are joint work with W. Banks, A. Conflitti,
and I. Shparlinski.
 EYAL GOREN, Department of Mathematics and Statistics, McGill University,
Montreal, Quebec H3A 2K6
Local structure of PEL moduli spaces
[PDF] 
We shall discuss the local structure of PEL moduli spacesthe
moduli spaces parameterizing abelian varieties with polarization,
endomorphism and level structure. Two main techniques will be
exposed: local models and displays. Both very geometric in nature,
though eventually very algebraic.
We shall explain a general theorem, due to Andreatta and the speaker,
that allows calculation of universal displays and show how it
recaptures also previously known cases.
 MANFRED KOLSTER, Department of Mathematics, McMaster University, Hamilton,
Ontario L8S 4K1
Divisibility properties of special values of Lfunctions for
quadratic characters
[PDF] 
For a quadratic character c over Q and an integer n > 0 the values of the Lfunction of c at 1n are nonzero
rational numbers if c has parity (1)^{n}. Most of the time the
values are 2integral, and in these cases one can prove general
divisibility properties by powers of 2. This has been done by Fox,
Urbanowicz and K. S. Willia ms using sophisticated identities for
generalized Bernouilli numbers. We will discuss a purely algebraic
approach a la Gauss, which also allows to generalize the results to
quadratic characters over arbitrary abelian fields.
 WENTANG KUO, Queen's University, Kingston, Ontario
Summatory functions of elements in Selberg's class
[PDF] 
Let F(s) be a Dirichlet series, F(s)=å_{n=1} a_{n} n^{s},
Âs > 1. Define the the summatory function S(x) to be
å_{n £ x} a_{n}. We assume that F(s) satisfies the following
conditions. First, for all e > 0, a_{n}=O(n^{e}). In
addition, it admits analytic continuations and functional equations.
More precisely, there is a function D(s) = Q^{s}ÕG(a_{i}s+g_{i}), Q > 0, a_{i} > 0, Âg_{i} > 0, such that
F(s)D(s)=w[`(F)](1s)[`(D)](1s), w=1.
Furthermore, assume that F(s) is entire. Twice of the summation of
a_{i} is called the degree d_{F} of F. In this talk, I
will derive an estimation of S(x) without extra conditions. The
trivial estimation is S(x)=O(x^{1+e}), "e > 0.
I will provide two estimations of S(x). One is a joint work with Ram
Murty; we prove that for d_{F} ³ 1, S(x) = O(Q^{1q+e}x^{q+e}), where q = d/(d+2). For the larger
d_{F} ³ 2, I can get a better result: S(x) = O(Q^{1q¢+e}x^{q¢+e}), where q = (d1)/(d+1). In
both cases, the implied constants are independent of Q.
 JUNGJO LEE, Queen's University, Kingston, Ontario
Twists of elliptic curves
[PDF] 
Let E be an elliptic curve defined over Q. We construct
cohomology classes from quadratic twists of E and apply the
localglobal duality theorem (a reformulation of the reciprocity law)
to these cohomology classes. As a result, we get a bound for rank of
E. The technique of using the reciprocity law was used by Kolyvagin
to bound the size of Selmer group and study TateShafarevich group.
This work was discussed with V. Kolyvagin, R. Murty and J. Shalika.
 GREG MARTIN, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
Sidon sets and symmetric sets of real numbers
[PDF] 
A set S of integers is called a B^{*}[g] set if for any given
m there are at most g ordered pairs (s_{1},s_{2}) Î S×S with
s_{1}+s_{2} = m; in the case g = 2, these are better known as Sidon
sets. It is trivial to show that any B^{*}[g] set contained in
{1,2,...,n} has at most Ö[(2gn)] elements, but proving a
lower bound of the same order of magnitude is more difficult. This
problem, surprisingly, is intimately related to the following problem
concerning measurable subsets of the real numbers: given
0 < e < 1, estimate the supremum of those real numbers d
such that every subset of [0,1] with measure e contains a
symmetric subset with measure d. Using harmonic analysis and
relationships among L^{p} norms as well as methods from combinatorial
and probabilistic number theory, we establish fairly tight upper and
lower bounds for these two interconnected problems.
 DAVID MCKINNON, University of Waterloo, Waterloo, Ontario
Counting rational points on ruled varieties
[PDF] 
In this talk, I will describe a general result computing the number of
rational points of bounded height on an algebraic variety V which is
covered by lines. The main technical result used to achieve this is an
upper bound on the number of rational points of bounded height on a
line. This bound varies in an pleasantly controllable manner as the
line varies, and hence can be used to sum the counting functions of the
lines which cover V.
 KUMAR MURTY, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Splitting of Abelian varieties
[PDF] 
It is well known that an irreducible polynomial over the integers may
become reducible mod p for every prime p. In this talk, we shall
discuss the analogue for Abelian varieties. Given an absolutely simple
Abelian variety over a number field, does it stay absolutely simple
modulo infinitely many primes?
 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
Irreducible elements and irreducible polynomials
[PDF] 
If f is a polynomial with integer coefficients which is a prime
number for infinitely many specializations, then it is clear that f
must be irreducible over the rational number field. An analogous
result over number fields is not true due to the possible presence of
infinitely many units. However, using Siegel's theorem on integral
points of curves of genus ³ 1, we show that an analogous result is
``almost true'' and the obstruction is the presence of
``Mersennelike'' primes in a number field. We also discuss the case
of a function field over a finite field. (This is joint work with
Jasbir Chahal.)
 YIANNIS PETRIDIS, City University of New York, Lehman College,
New York 104581589, USA
The distribution of modular symbols
[PDF] 
Eisenstein series twisted by modular symbols were introduced by
Goldfeld to study the distribution of modular symbols in connection to
a weak form of the ABC conjecture. I will present distribution results
that follow from the study of the pole at s=1 of such series. The
proof uses families of Eisenstein series twisted by characters and
perturbation methods of the Laplace operator.
 CAMERON L. STEWART, University of Waterloo, Waterloo, Ontario
On sums which are powers
[PDF] 
Erdos and Moser investigated the problem of finding sets of positive
integers A with the property that a+b is a square whenever a and
b are distinct elements of A. With Rivat and Sarkozy we showed
that if A is a subset of the first N positive integers then A has
cardinality at most 37logN provided that N is large enough. We
shall discuss recent joint work with Gyarmati and Sarkozy where we
replace the requirement that a+b be a square with the requirement
that a+b be a pure power.
 JEFFREY LIN THUNDER, Northern Illinois University, USA
Asymptotic estimates for some Diophantine inequalites
[PDF] 
We estimate the number of integer solutions to inequalities of the form
F(x) £ m, where F(X) is a homogeneous polynomial with integer
coefficients which factors completely over the complex field as a
product of linear forms. We give asymptotic estimates as the parameter
m®¥ which have a good dependency on F.
 MICHEL WALDSCHMIDT, Université P. et M. Curie (Paris VI),
Théorie des nombres Case 247, 75013 Paris, France
Algebraic values of analytic functions
[PDF] 
Given an analytic function of one complex variable f, we investigate
the arithmetic nature of the values of f at algebraic points. A
typical question is whether f(a) is a transcendental number for
each algebraic number a. Since there exist transcendental
entire functions f such that f^{(k)}(a) Î Q[a]
for any k ³ 0 and any algebraic number a, one needs to
restrict the situation by adding hypotheses, either on the functions,
or on the points, or else on the set of values.
Among the topics we discuss are recent results due to Andrea Surroca on
algebraic values of analytic functions and Diophantine properties of
special values of polylogarithms.
http://www.institut.math.jussieu.fr/ ~ miw/

