


Number Theory / Théorie des nombres Org: Chantal David (Concordia) and/et Andrew Granville (Montreal)
 MARK BAUER, University of Calgary, Dept. of Math. & Stats.,
2500 University Drive NW, Calgary, AB T2N 1N4
Construction of Hyperelliptic Function Fields of High
ThreeRank

In this talk, we will discuss several explicit constructions of
hyperelliptic function fields whose Jacobian have high 3rank. Some
of the methods are analogues of techniques for generating quadratic
number fields of high 3rank, while others are unique to the function
field setting. In particular, a method for increasing the 3rank by
extending the field of constants will be emphasized, which may also be
used to increase the lrank.
 ALINA C. COJOCARU, Princeton University, Dept. of Mathematics, Fine Hall,
Washington Road, Princeton, NJ 08544, USA
Exceptional primes for elliptic curves

Let E be an elliptic curve defined over the field of rationals, and
for a prime l, let r(l, E) be the ladic Galois
representation associated to E. If E is without Complex
Multiplication, an important result of Serre dating from 1972 asserts
that r(l,E) is surjective for large enough l. But how large is
large enough? This is still an open question, raised by Serre, and
investigated, over the years, by Serre, Mazur, Masser and Wüstholz,
Kraus, Duke, A. C. Cojocaru, and others.
We say that l is an exceptional prime for E if r(l, E) is not
surjective. In this talk, we will show that any oneparameter family
of elliptic curves over the rationals has very few exceptional primes.
This is joint work with Chris Hall (Univ. Texas at Austin).
 JEANMARIE DE KONINCK, Université Laval, Dept. de mathématiques et de
statistique, Quebec G1K 7P4
Esthetic numbers and game theory

Paul and Mary decide to toss a coin 15 times, for money. If HEADS
comes up, Paul gives Mary one dollar. If TAILS comes up, Mary gives
Paul one dollar. Before they start playing, Paul announces that he
only has $2 and therefore that he cannot afford to lose any more than
$2, meaning that if he ever reaches a loss of $3, he will have to
quit the game. Mary is more audacious, revealing that she is willing
to lose as much as $4. What's the probability that, under these
conditions, they can complete the game? As we shall see, the problem
boils down to counting qesthetic numbers, namely those positive
integers whose consecutive digits in base q differ by exactly one.
 KARL DILCHER, Dalhousie University, Halifax, Nova Scotia
On finite patternfree sets of integers

For n Î N let the set A Í {1,...,n} have the
property that it does not contain any solution x,y,z to the
equation s:ax+by=cz, where a,b,c ¹ a+b are fixed positive
integers. Such a set A is called sfree. We investigate
maximal sfree sets and determine the upper maximal density
bound

D

(s) : = 
limsup
 
æ è


1
n


max
 
ì í
î

A:A Í {1,...,n} sfree 
ü ý
þ


ö ø



for a=1 £ b, and c large. As a byproduct we obtain the upper
asymptotic density bound for infinite sfree sets
A Ì N defined by

d

(s) : = 
sup
 
ì í
î


limsup
 
æ è


1
n

AÇ{1,...,n} 
ö ø

: A Ì N sfree 
ü ý
þ

, 

for all coprime a, b, and c large, which may be considered the
limiting case of finite sfree sets.
Joint work with Lutz G. Lucht, Technische Universität Clausthal,
Germany.
 JACK FEARNLEY, Université de Montréal
The Fundamental Critical Points of Modular Elliptic Curves

The number of fundamental critical points of the modular form
associated with an elliptic curve is an upper bound for the analytic
rank of the curve. In their paper "Arithmetic of Weil Curves" Mazur
and SwinnertonDyer show that this bound is sharp for all but 16
curves of conductor less than 424.
The talk will cover aspects of the computation of the fundamental
critical points for all curves of conductor less than 4000. In this
range the bound is sharp for approximately eighty per cent of the
curves.
 HARALD HELFGOTT, Université de Montréal, CP 6128, Succ. CentreVille,
Montreal, QC H3C 3J7
Powerfree values of polynomials with prime arguments

Let f Î Z [x] be a polynomial of degree
r ³ 3 without roots of multiplicity r or (r1). Assume that
f(x) \not º 0 mod p^{r1} has a solution in
(Z/p^{r1})^{*} for every p. Erdös conjectured that
f(p) is then free of (r1)th powers for infinitely many
primes p. I prove as much for every f a root of which generates
its splitting field, and for some other f as well.
 HABIBA KADIRI, Université de Montréal
Applications of explicit zerofree regions.

I will discuss number theoretic applications of explicit zerofree
regions of Dirichlet Lfunctions.
 WENTANG KUO, University of Waterloo, 200 University Ave. West, Waterloo,
ON
A Remark on the SatoTate Conjecture

The original SatoTate Conjecture concerns the angle distribution of
the eigenvalues arisen from nonCM elliptic curves. In this talk, we
formulate a modular analogue of the SatoTate Conjecture and prove
that the angles arisen from nonCM holomorphic Hecke eigenforms with
nontrivial central characters are not distributed with respected to
the SateTate measure for nonCM elliptic curves. Furthermore, under
a reasonable conjecture, we prove that the expected distribution is
uniform.
 YOUNESS LAMZOURI, Université de Montréal, Département de mathématiques
et de statistique, CP 6128, Succ. CentreVille, Montréal,
QC H3C 3J7
Smooth numbers and delayed integral equations

Let P be a set of primes and define S(x,P) as the set of integers
less than x having all their prime factors in P. Let Y(x,P) = S(x,P) and p(x,P) be the number of primes less than x for
which p1 is in S(x,P). A difficult but interesting question is
to have estimates for the function p(x,P).
A wellknow conjecture is that [(p(x,P))/(p(x))] ~ [(Y(x,P))/(x)], under some conditions on the set P. Moreover
Granville and Soundararajan proved in 2002 that estimates for
Y(x,P) depend on the solutions of the delayed integral equation
us(u) = ò_{0}^{u} s(ut) c(t) dt. In this talk we
prove a general version of the above conjecture assuming the
ElliotHalberstam conjecture, study solutions of the integral equation
to obtain estimates for p(x,P), and apply our results to get
asymptotics for the set of integers n less than x for which the
kth iterate of the Euler function f_{k}(n) is smooth.
 JUNGJO LEE, Queen's University, Kingston, ON K7L 3N6
Ranks of elliptic curves

There have been many conjectures and results on the ranks of elliptic
curves so far. We will see some of those conjectures and the related
results especially over the family of quadratic twists of an elliptic
curve.
 CLAUDE LEVESQUE, Université Laval
Systèmes fondamentaux d'unités de composés de deux
corps cubiques

Nous exhibons une famille de composés F de deux corps cubiques
pour laquelle nous pouvons exhiber un système fondamental d'unités
à partir de la connaissance des unités des souscorps cubiques
de F.
 YURU LIU, University of Waterloo
A prime analogue of ErdösPomerance's conjecture for
elliptic curves

Let E/Q be an elliptic curve of rank ³ 1 and b Î E(Q) a
rational point of infinite order. For a prime p of good reduction,
let g_{b}(p) be the order of the cyclic group generated by the
reduction [`(b)] of b modulo p. We denote by w(g_{b}(p) ) the number of distinct prime divisors of g_{b}(p).
Assuming the GRH, we show that the normal order of w(g_{b}(p) ) is loglogp. We also prove conditionally that
there exists a normal distribution for the quantity

w 
æ è

g_{b}(p) 
ö ø

 loglogp 

. 

The later result can be viewed as an elliptic analogue of a conjecture
of Erdös and Pomerance about the distribution of w(f_{a}(n) ), where a is a natural number > 1 and f_{a}(n) the
order of a modulo n.
 SAID MANJRA, University of Ottawa
On the Efunctions

The Efunctions were introduced around 1929 by C. L. Siegel who
studied the irrationality and transcendence of values at the algebraic
points. The minimal differential operator annihilating an
Efunction is called Eoperator. In this lecture, we give an
arithmetic characterization of such operator based on a recent results
of Yves André.
 DAVID MCKINNON, University of Waterloo
Rational Approximation on Algebraic Varieties

If two rational points in affine space have small height, then they
must lie on a rational line with small heightthat is, the line
connecting the two points has coefficients with small height. Thus,
if one wishes to approximate a rational point P with other rational
points, the best way to do it is to use rational points which lie on
lines of small height through P.
What happens if one constrains the approximating points to satisfy a
set of polynomial relations? This question is still open, but the
evidence so far points to a somewhat surprising answer.
 ROGER PATTERSON, University of Calgary, 2500 University Drive NW, AB, T2N 1N4
Sequences of rational torsion on Jacobians

We will construct sequences of hyperelliptic curves of genus g whose
Jacobians possess torsion divisors of order Q(g^{2}). These
families generalize the earlier examples of Flynn and Leprévost.
 GUILLAUME RICOTTA, Université de Montréal, Bureau 6213, Département de
mathématiques et de statistique, CP 6128, Succ. CentreVille, Montréal, QC H3C 3J7
On nontrivial real zeros of Lfunctions

In 2002, J. B. Conrey and K. Soundararajan have shown that there are
infinitely many Dirichlet Lfunctions which do not vanish on the
critical segment. Two crucial remarks about their result. On one
hand, the analytic technique used (mollification method) and their
zeros counting method compel them to get strong asymptotic formula
for the mollified second moment of their family of Dirichlet
Lfunctions at a distance of the inverse of the logarithm of the
analytic conductor of their family. On the other hand, all their work
is justified by the random matrix model of their family: its symmetry
type is the symplectic one which entails a repulsion of the first zero
a little away from the critical segment. Following their work, a
similar analytic study of a family of RankinSelberg Lfunctions
which has the same symmetry type has been undertaken, but this time a
deep and tough new problem occurs: namely the resolution of a shifted
convolution problem on average. The two main purposes of this short
talk will be to describe and legitimize the mollification method which
is not only a cunning craftiness and to give some indications on
shifted convolution problems.
 DAMIEN ROY, University of Ottawa
Criteria of algebraic independence

We review some of the present criteria of algebraic independence and
discuss avenues for further research.
 MICHAEL RUBINSTEIN, University of Waterloo, 2000 University Ave. W, Waterloo, ON
N2L 3G1
Announcing the Lfunction Calculator

I'll describe the Lfunction calculator, and present some of the
algorithms that lie behind it.
 RENATE SCHEIDLER, Dept. of Mathematics & Statistics, University of Calgary,
2500 University Drive NW, Calgary, AB T2N 1N4
Class Number Approximation in Cubic Function Fields

A common way of computing the order h of the Jacobian
Jac(F_{q}) of an algebraic function field K over a finite
field F_{q} is to determine an interval ] EL,E+L[ that is known to contain h and then search this interval
using a baby step giant step technique or Pollard's kangaroo method.
In the case where K/F_{q}(x) is a cubic extension, we use
approximations via truncated Euler products to explicitly compute
suitable values of E and L, thereby giving an algorithm for
finding h that has complexity O(q^{(2g1)/5}) where g is the
genus of K/F_{q}.
This is joint work with A. Stein of the University of Wyoming.
 CAM STEWART, University of Waterloo
On prime factors of n!+1

By Wilson's Theorem if p is a prime then p divides (p1)!+1. In
1856 Liouville proved that (p1)!+1 is not a power of p if p
exceeds 5. In 1976 with Erdos we investigated the greatest and
least prime factors of integers of the form n!+1. We shall discuss
these topics and related recent results of Luca and Shparlinski and of
the speaker.
 ZHIHONG SUN, School of Mathematics and Statistics, Carleton University,
Ottawa, ON K1S 5B6 (from Huaiyin Teachers College in China)
Quartic residues and binary quadratic forms

Let p be a prime of the form 4k+1, m Î Z and p\nmid m. In this talk we give a general criterion for m to be a
quartic residue of p in terms of appropriate binary quadratic
forms. Let d > 1 be a squarefree integer such that ([(d)/(p)])=1,
where ([(d)/(p)]) is the Legendre symbol, and let e_{d}
be the fundamental unit in the quadratic field Q (Öd). Since 1942 many mathematicians tried to characterize those
primes p so that e_{d} is a quadratic or quartic residue
of p. Now we completely solve these open problems by determining
the value of (u+vÖd)^{(p([(1)/(p)]))/2} mod p), where p
is an odd prime, u,v,d Î Z, v ¹ 0, gcd(u,v)=1
and ([(d)/(p)])=1. As an application we obtain a general
criterion for p  u_{(p([(1)/(p)]))/4} (a,b), where
{u_{n}(a,b)} is the Lucas sequence defined by u_{0}=0, u_{1}=1 and
u_{n+1} = bu_{n}au_{n1} (n ³ 1). In the talk we also present
the exact value of the number of incongruent residues of x^{4}+bx
modulo an odd prime.
 ALAIN TOGBE, Purdue University North Central, 1401 South U.S. Highway 421,
Westville, IN 46391
Complete Solutions of a Family of Cubic Thue Equations

In this talk, we will consider a family of Thue equations associated
with a family of number fields of degree 3. First we will give
details about the number field and then we will show how we use
Baker's method to completely solve this family of Thue equations.
 GARY WALSH, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario
K1N 6N5
Quantitative results for Diophantine mtuples of
polynomials

A D(n)mtuple is defined to be a set of integers {a_{1}, ...,a_{m}} with the property that for all 1 £ i < j £ m, a_{i}a_{j}+n is a perfect square.
In recent years there have been many new results on the number of
elements in such sets, most notably by A. Dujella, who essentially
settled on old problem, dating back to Diophantus, by proving that
there are no D(1)6tuples, and only finitely D(1)5tuples.
More recently, Dujella, Fuchs and Tichy proved upper bounds in the
case that the a_{i} are polynomials of equal degree, and n is a
linear polynomial.
We will present quantitative improvements to these results, along with
specific examples, which together imply that the new upper bounds are
sharp. This is joint work with Andrej Dujella and Clemens Fuchs.
 HUGH C. WILLIAMS, Mathematics and Statistics, University of Calgary,
2500 University Drive NW, Calgary, AB T2N 1N4
Computing the Regulator of a Real Quadratic Field

One of the most important invariants of a real quadratic number field
is its regulator R. The problem of computing R is very old and
very difficult, particularly when the field discriminant D becomes
large. (Clearly, the actual value of the regulator can never be
computed because it is a transcendental number; we are content with an
approximate value that is within 1 of the actual value.) The best
current method for computing R is Buchmann's subexponential method.
Unfortunately, the correctness for the value of R produced by this
technique is dependent on the truth of a generalized Riemann
hypothesis. The best unconditional algorithm (the value of R is
unconditional, not the running time) for computing the regulator of a
real quadratic field is Lenstra's O(D^{1/5+e}) Las Vegas
Algorithm. In this talk we describe a technique for rigorously
verifying the regulator produced by the subexponential algorithm.
This technique is of complexity O(D^{1/6+e}).
Furthermore, these methods can be extended to the problem of
determining rigorously for real quadratic fields of large discriminant
whether a given ideal is principal. This is of great importance in
solving certain Diophantine equations.
Joint work with Robbert de Haan and Mike Jacobson.
 GUANGWU XU, University of Toronto, Toronto, Ontario M5S 3G3
Nonadjacent Radixt Expansions of Integers in Euclidean
Imaginary Quadratic Number Fields

In his seminal papers, Koblitz proposed families of elliptic curves
for cryptographic use. For the fast operations on the curves these
papers also initiated a study of radixt expansion of integers in
the number fields Q(Ö{3}) and Q(Ö{7}). The (window)
nonadjacent form of texpansion of integers in Q(Ö{7})
was first investigated by Solinas. For integers in Q(Ö{3}),
the window nonadjacent form of texpansion were studied by the
authors. These forms are used for efficient point multiplications on
Koblitz curves. In this work, we complete the picture by producing
the (window) nonadjacent radixt expansions for integers in all
Euclidean imaginary quadratic number fields.
 LIANGYI ZHAO, Dept. Math., Univ. of Toronto, 100 Saint George Street,
Toronto, ON M5S 3G3
Large Sieve Inequality for Special Characters

The groups of Dirichlet characters modulo prime powers decompose
nicely via a formula of A. G. Postnikov. This fact leads to various
results concerning the corresponding Lfunctions. I will talk about
a large sieve type inequality restricted to characters resulting from
the Postnikov decomposition: what is known already and what further
progress can and has been made.

