Number Theory and Combinatorics Seminar
Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Thursday, January10, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  Dragos Ghioca (University of Lethbridge)

Title:
The Skolem-Mahler-Lech theorem

Abstract
:  We present an old result for linear recurrence sequences whose proof relies on p-adic analysis.
Based on it we derive an interesting result for automorphisms of the affine space.

Thursday, January 24, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title:  Lower bounds for moments of L-functions

Abstract:
Hardy and Littlewood initiated the study of high moments of the Riemann
zeta function.  They were initially interested in this problem because of its
connection to the Lindelof hypothesis. In recent years, the moments of central
values of families of L-functions have attracted much attention.
In this talk I will explain a recent method of Rudnick and Soundararajan
which establishes lower bounds for moments of L-functions of the
correct order or magnitude.

Thursday, January 31, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker: Amir Akbary (University of Lethbridge)

Title:  On the reduction mod p of a rational point of an elliptic curve

Abstract:  Let E be an elliptic curve over Q.  For any prime p of good reduction, let E_p
be the elliptic curve over the finite field F_p obtained by reducing E modulo p.  We investigate
that for a point of infinite order in the Mordell group E(Q), how the order of the reduction of
this point mod p varies as p goes to infinity. We study this problem by comparison with reduction
of integers mod p.

Thursday,  February 7, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker: Nathan Ng (University of Lethbridge)

Title:  Upper bounds for moments of the Riemann zeta function

Abstract
:  I will explain a recent method of Soundararajan that  proves an upper
bound for moments of the Riemann zeta function which is nearly
as sharp as the expected order of magnitude. This technique assumes
the Riemann hypothesis and relies on obtaining an upper bound for
the frequency of values for which the zeta function is large on the
critical line.

Thursday,  February 14, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  John Irving (Saint Mary's University)

Title:
Lattice Paths Under a Shifting Boundary

Abstract
:  The generalized ballot theorem gives a well-known formula for
the number of lattice paths in the first quadrant lying weakly under the
line x=ay, where a is an arbitrary positive integer.  While there is
almost certainly no simple formula for the number of paths under an
arbitrary piecewise linear boundary, we show that nice enumerative
results are available if we allow for cyclic shifts of such a general
boundaries. A refinement of this result allows for the counting of paths
with a specified number of corners, and we also examine paths dominated
by periodic boundaries.  This is joint work with A. Rattan.

Thursday,  February 28, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  Nathan Ng (University of Lethbridge)

Title:  Upper bounds of the Riemann zeta function

Abstract
:  Hadamard and Weierstrass proved a product formula for entire
functions of order one. This factorization theorem leads to a formula for the
logarithmic derivative of the zeta function in terms of the zeros of the zeta function.
I will explain how such formulae can be applied to obtain upper bounds for the
Riemann zeta function on the critical line. This talk is based on recent work of Soundararajan.

Thursday,  March 13, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker: Matthew Greenberg (University of Calgary)

Title:  Modular forms, Stark units and Heegner points

Abstract
:  In this talk, we will discuss constructions and conjectures concerning
units in rings of algebraic integers and algebraic points on elliptic
curves over number fields.  Modular forms, in various guises, are the
key ingredients in all known systematic constructions of such objects.

Thursday,  March 20, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  Brandon Fodden (PIMS/University of Lethbridge)

Title:  Fractional moments of the Riemann zeta function

Abstract
:  We discuss the Heath-Brown method to find upper and lower bounds
for the fra
ctional moment of the Riemann zeta function for s with real
part 1/2.

Thursday,  March 27, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  Hadi Kharaghani (University of Lethbridge)

Title:  The energy of matrices

Abstract:  Let  M be an m by n matrix , m<= n. Let  \lambda_i, i=1, 2, ..., m
be the eigenvalues of the matrix MM^t.  Motivated by an application in theoretical chemistry,
Gutman defined the energy of M to be the sum of the square roots of  \lambda_i's.
The (0,1)-matrices attaining maximum energy will be discussed.

Thursday,  April 3, 2008
Room: C756
Time: 3:05-3:55 PM

Speaker:  Habiba Kadiri (University of Lethbridge)

Title:
A bound for the least prime ideal in the Chebotarev density problem.

Abstract
:  A classical theorem due to Linnik gives a bound for the least prime number
in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization
of this result to any number field. Their proof relies on some results about the distribution
of the zeros of the Dedekind Zeta function (zero free regions, Deuring Heilbronn phenomenon).
In this talk, I will present some new results about these zeros.  As a consequence, we are able to
prove an effective version of the theorem of Lagarias et al.

Thursday,  April  17, 2008
Room: B650
Time:  12:00-12:50 PM

Speaker: Harald Helfgott  (University of Bristol)

Title:
Growth in SL_3

Abstract
:  Let K be R, C or a Z/pZ. Let G = SL_2(K). Not long ago, I proved the  following theorem:
for every subset A of G that is not contained in a
proper subgroup, the set A A A is much larger than A.
A generalization to groups of higher rank was desired by many, but seemed hard to obtain.
I have obtained a generalization to SL_3(K). The role of both  linearity and the group structure of G should
now be clearer than they
were at first. Bourgain, Gamburd and Sarnak derived various consequences on
expander graphs from my SL_2 result; analogous
consequences should follow in the case of SL_3.

Friday,  May 9, 2008
Room: E575
Time:  11:00-11:50 AM

Speaker: Nathan Jones (CRM, Montreal)

Title:
A refined version of the Lang-Trotter conjecture

Abstract
:  Let E be an elliptic curve defined over the rational numbers and r a fixed integer.  Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r.  However, when one sums the main term in their asymptotic over r in a fixed residue class modulo q, one does not recover the main term in the Chebotarev theorem for the q-th division field, but rather 8/\pi times the main term.
In this talk, I will state a refinement of the Lang-Trotter conjecture and demonstrate consistency of this refinement with the Chebotarev Theorem for a fixed division field.  This is based on joint work with S. Baier.