Carleton University School of Mathematics and Statistics University of Ottawa Department of Mathematics and Statistics

NUMBER THEORY SEMINAR
Fall 2005/ Winter 2006

Time and Place: Friday, Sept. 23, 11am-noon, room KED B-03 Ottawa

Speaker: Nathan Ng (University of Ottawa)

Title: Primes and Sieves

Abstract: In this talk I will discuss some basic concepts concerning prime numbers and sieve methods.
In particular, I would like to show the importance of Selberg's sieve in prime number theory. My goal is
to expose the recent groundbreaking work of Goldston, Pintz, and Yildirim concerning small gaps between
primes. In the second talk, I will present an outline of their proof that there are infinitely many consecutive
primes with very small gaps between them. In fact,their approach presents a promising method for attacking
the twin prime conjecture.

Time and Place: Friday, Oct. 7, 11am-noon, room KED B-03 Ottawa

Speaker: Nathan Ng (University of Ottawa)

Title: Primes and Sieves II- Small gaps between primes

Abstract: In this talk I will sketch a proof of the following groundbreaking theorem of Goldston, Pintz, Yildirim:
Let p_n be the n-the prime number. We show that liminf (p_(n+1)-p_n)/log(p_n) =0. I will discuss how this
proof uses an upper bound sieve and the Bombieri-Vinogradov theorem. Moreover, we discuss how the method
applies to finding bounded gaps between primes assuming certain distribution properties of primes in arithmetic progressions.

Time and Place: Friday, Oct. 14, 11am-noon, room 4325 HP Carleton

Speaker: Gonzalo Tornaria (CRM)

Title: Modular forms of weight 3/2 and the central values of twisted L-series

Abstract: Let f be an eigenform of weight 2 and prime level. A method of Gross constructs, provided L(f,1)
does not vanish, a modular form of weight 3/2 whose Fourier coefficients relate to the central values
of the L-series of the imaginary quadratic twists of f, giving an explicit formula for such central values.

We will present joint work with Z. Mao and F. Rodriguez-Villegas which generalize Gross's algorithm and
formula to include the case L(f,1)=0, and to the case of the real quadratic twists of f.

Time and Place: Friday, Oct. 28, 11am-noon, room KED B-03 Ottawa

Speaker: Gary Walsh (University of Ottawa)

Title: Integer points on prime power congruent number curves

Abstract: Using a host of new diophantine results, we provide sharp estimates for the number of integer
points on elliptic curves of the form $y^2=x^3 \pm nx$, with $n$ a prime power.

Time and Place: Friday, Nov.25, 11am-noon, room KED B-03 Ottawa

Speaker: Damien Roy (University of Ottawa)

Title: Simultaneous rational approximation to a real number, its square and its cube (part I)

Abstract: Abstract: Let xi be a real number which is either transcendental over Q or algebraic over Q of degree >3.
Dirichlet box principle shows that the uniform exponent of simultaneous approximation of xi, xi^2 and xi^3
by rational numbers with the same denominator is at least 1/3 ~ 0.3333. In this talk, we show that this
exponent is at most the smallest positive root lambda ~ 0.4245 of the polynomial X^2-(2*gamma+1)X+gamma
where gamma denotes the golden ratio.

Time and Place: Friday, Feb.3, 2006, 11am-noon, room KED B-03 Ottawa

Speaker: Nathan Jones (CRM)

Title: Almost all elliptic curves are "Serre curves"

Abstract: A Serre curve is an elliptic curve defined over the rational numbers whose
torsion subgroup, roughly speaking, has as much Galois symmetry as is
possible. I will discuss a theorem that, when counted according to
height, almost all elliptic curves are Serre curves. If time permits,
I'll discuss an application of this theorem to a question of averages of
the constants in the Lang-Trotter conjecture.

Time and Place: Friday, Feb.24, 2006, 11am-noon, room KED B-03 Ottawa

Speaker: Florian Luca (UNAM, Mexico)

Title: On a conjecture of Ma

Abstract: In 1992, investigating so-called reversible difference sets in abelian
groups, S. L. Ma proposed the following conjecture: Ma's conjecture.
Let p be an odd prime and b, m, r be positive integers. Then (1): x^2
= 2^(2b+2)p^(2m) ? 2^(b+1)p^(m+r) + 1 holds with some positive integer
x if and only if p = 5, b = 3, m = 1 and r = 2 (when x = 49). While we
cannot prove Ma's conjecture, in my talk I will show that if p is
fixed, then the diophantine equation (1) has at most 2^30,000 positive
integer solutions (x, b, m, r). The proof uses the Subspace Theorem
and results on S-unit equations. This is joint work with Pantelimon Stanica.

Time and Place: Friday, March 10, 2006, 11am-noon, room 4325 Herzberg Carleton

Speaker: Emmanuel Knafo (Toronto)

Title: Variance of Distribution of Almost Primes in Arithmetic Progressions

Abstract: We sketch the proof for a lower bound for the variance of distribution of almost primes in arithmetic progressions. The emphasis is placed on the key ideas and techniques involved.

Time and Place: Friday, March 31, 2006, 11am-noon, room KED B-03 Ottawa

Speaker: Abdellah Sebbar (Ottawa)

Title: TBA

Abstract: