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

NUMBER THEORY SEMINAR
Fall 2006/ Winter 2007

Time and Place: Friday, Sept. 29, 11am-noon, room KED B-015 Ottawa

Speaker: Nathan Ng (University of Ottawa)

Title: Extreme values of L-functions

Abstract:

In this talk I will discuss the resonance method of Soundararajan which is used in detecting
large values of L-functions and character sums. In particular, I will discuss an application of
this method to finding extreme values of the derivative of the zeta function evaluated at the
zeros of zeta.

Time and Place: Friday, Oct. 13, 11am-noon, room KED B-015 Ottawa

Speaker: Hugo Chapdelaine (McGill)

Title: Elliptic units and modular symbols

Abstract:

Let $K$ be a real quadratic number field and let $p$ be a prime number inert in $K$. We denote
by $K_p$ the completion of $K$ at $p$. Using periods of modular functions we construct a family
of $\ZZ$-valued measures on $\PP^1(\QQ_p)$. We then use those measures to do $p$-adic integration
and construct $p$-adic invariants in $K_p^{\times}$. Those $p$-adic invariants are conjectured to be
global $p$-units in abelian extensions of $K$. The truth of this conjecture would entail an explicit
class field theory for $K$. We also construct $p$-adic $L$-functions for which we relate their first
derivative at $s=0$ with our $p$-adic invariants. This is an analogue of the classical Kronecker limit formula.

Time and Place: Friday, Nov.3, 10am-11am, room TBA Ottawa

Speaker: Michael Rubinstein (University of Waterloo)

Title: Hide and Seek - a naive factoring algorithm

Abstract:

I present a factoring algorithm that factors N=UV, where U < V, provably in O(N^(1/3+epsilon))
time. I also discuss the potential for improving this to a sub-exponential algorithm. Along the way,
I consider the distribution of solutions (x,y) to xy=N modulo a, using estimates for Kloosterman sums.

Time and Place: Friday, Nov.3, 11am-noon, room KED B-015 Ottawa

Speaker: Micah Milinovich (University of Rochester)

Title: Moments of the Riemann Zeta-function and its Derivatives.

Abstract:

Mean-value theorems (or moments) for the Riemann zeta-function and other families of L-functions are a
central problem in analytic number theory, but relatively few have been proved. The recent introduction of
random matrix theory into the field led to a greater understanding of the structure of such moments conjecturally,
and this in turn has led to a very general but detailed estimate for various moments called the Ratios Conjecture.
Here we prove some mean value formulas which confirm predictions of the Ratios Conjecture in a number of cases.

Time and Place: Friday, Nov.17, 11am-noon, room 4325 HP Carleton

Speaker: Kenneth S. Williams (Carleton)

Title: Hypergeometric functions and representations of integers by quaternary quadratic forms

Abstract: It will be shown how hypergeometric functions can be used todetermine the number of representations of an
integer by certain quaternary quadratic forms. This is joint work with A. Alaca, S. Alaca and M. Lemire.

Time and Place: Friday, Jan.19, 11:30am-12:30pm, room KED B-015 Ottawa

Speaker: Adam Logan (Waterloo)

Title: Geometry and arithmetic of Kummer surfaces and their twists
Abstract: I will start by reviewing $2$-descent on elliptic curves, and
then I will explain the basics of $2$-descent on curves of genus $2$. The
principal homogeneous spaces of the Jacobian of the curve that arise are
often awkward to analyze---in particular, it is quite difficult to prove
that they have no rational points. I will describe some interesting curves
on the quotient of the homogeneous space by its natural involution and show
how to use them in some special cases to prove that the quotient has no
rational points. From this I obtain an infinite family of twists of a
curve of genus $2$ all with nontrivial Tate-Shafarevich group. This is
joint work with Ronald van Luijk.

Time and Place: Friday, Feb.2, 2007, 11:30am-12:30pm, room KED B-015 Ottawa

Speaker: Gary Walsh (Ottawa)

Title: Sharp bounds for the number of integer points on families of elliptic curves
Abstract: Abstract: Over the course of more than 30 years during the last century,
Wilhelm Ljunggren proved many remarkable theorems on the number
of integer solutions to certain families of quartic Diophantine equations.
However, many fundamental problems remained unsettled after his
investigations. In joint work with Shabnam Akhtari and Alain Togbe,
some of the outstanding problems from Ljunggren's work have been
completely solved. For instance, we prove the best possible result
that for any positive integers $a,b$ the quartic equation $aX^4-bY^2=1$
has at most two solutions in positive integers (X,Y).
Other results of this type will also be discussed.

Time and Place: Friday, Feb.16, 2007, 11:30am-12:30pm, room KED B-015 Ottawa

Speaker: Nathan Ng (Ottawa)

Title: Non-vanishing of L-functions and an application to Fermat equations
Abstract: In recent years, a popular research topic in analytic number
theory has been the non-vanishing of L-functions. Many of the
central ideas for proving the non-vanishing of an L-function date
back to Selberg's proof that a positive proportion of the zeros of the
Riemann zeta function lie on the half-line. In this talk I will discuss
some non-vanishing results which imply that the Fermat equation
A^4+B^2=C^p for p >7 a prime has no non-trivial solutions. This is
joint work with Michael Bennett and Jordan Ellenberg.

Time and Place: Friday, March 9, 2007, 11:30am-12:30pm, room KED B-015 Ottawa

Speaker: Eric Villani (Ottawa)

Title: Transcendence measures for algebraic points of Siegel modular functions.
Abstract: We give an effective version of a result of Cohen, Shiga and Wolfart,
which is a generalisation to the case of Siegel spaces of arbitrary
degree, of the classical theorem of Schneider on the modular invariant
j(\tau). Given a point \tau of the Siegel space parametrizing a
principally polarised abelian variety A defined over
the algebraic closure of the rationals, we obtain a lower bound for the distance between
\tau and algebraic points \beta of the Siegel space, in terms of the
geometrical data of the problem. To achieve this,
we establish a simultaneous measure of linear independance for periods of
abelian integrals, using Baker's method.

Time and Place: Friday, March 16, 2007, 11:30am-12:30pm, room KED B-015 Ottawa

Speaker: Damien Roy (Ottawa)

Title: Small value estimates for the multiplicative group
Abstract: We generalize Gel'fond's criterion of algebraic independence
to the context of a sequence of polynomials taking small values on the
initial terms of a fixed geometric sequence of complex numbers.

Time and Place: Friday, March 16, 2007, 2pm-3pm, room KED B-04 Ottawa

Speaker: Sinnou David (IAS Princeton and Universite de Paris VI)

Title: Points of small height on tori
Abstract: We shall present a few conjectures on small points of subvarieties
of multiplicative group varieties and describe how they are natural
generalizations of the classical Lehmer problem. Applications to other
arithmetic geometry problems will be discussed as well as the latest results in
the direction of theses conjectures. If time permits, we shall devote a few
words to the abelian variety situation.

For further information on the seminar please contact:
Nathan Ng
nng362 at science.uottawa.ca
Tel. 613-562-5800 ext 3515
or
Damien Roy
droy at uottawa.ca
Tel. 613-562-5800 ext 3504