April 23, 2014

June 8-10, 2013
Workshop on Number Theory
with a view towards Transcendence and
Diophantine Approximation

Damien Roy, University of Ottawa
Cameron L. Stewart, University of Waterloo

Speaker Abstracts

Henri Darmon (McGill University)
Stark-Heegner points

Stark-Heegner points are analogues for elliptic curves of Stark units arising from leading terms of Artin L-functions at s=0. Defined analytically in terms of certain ostensibly transcendental (complex, or p-adic) periods, the main challenge is to prove the predicted algebraicity of these local points. I will give a brief overview of some of a few of the known constructions, of the experimental evidence that has been gathered in their support, and of possible approaches to relating Stark-Heegner points to invariants of a more global nature.

Edward Dobrowolski (University of Northern British Columbia)
On a question of Schinzel about the length and Mahler's measure of polynomials that have zero on the unit circle

We prove the inequality L(P) =2M(P) for any polynomial P in C[x] having a zero on the unit circle. Here L(P) and M(P) are the length and Mahler's measure of P, respectively. The problem was stated by A. Schinzel and applies to the problem of so called reduced length of a polynomial.


John Friedlander (University of Toronto)
Primes plus other things

In this case, the "other things" are "the intervals between them". The lecture is inspired by the recent work of Yitang Zhang: "Bounded gaps between primes".


Jing-Jing Huang (University of Toronto)
Metric Diophantine approximation on planar curves

Since Kleinbock and Margulis established the fundamental Baker-Sprindzuk conjecture concerning homogeneous Diophantine approximation on manifolds in 1998, some tremendous progress has been made toward the much stronger Khintchine-Jarnik type theorem for non-degenerate manifolds in the last decade or so. In particular, the metric theory of planar curves has been well understood now, thanks to Vaughan and Velani for the convergence theory and Beresnevich, Dickinson and Velani for the divergence theory. Both of these two results rely on estimates on the number of rational points with small denominators which are near the curve (actually the convergence theory concerns the upper bound and the divergence theory concerns the lower bound). However the two proofs differ quite significantly in nature. In this talk, I will prove that the above mentioned counting function actually satisfies an asymptotic formula as the height of the denominator approaches infinity and the distance approaches 0, and hence this yields a unified proof of both results.


Matilde Lalín (Université de Montréal)
Mahler measure and special values of L-functions of elliptic curves

The Mahler measure of a Laurent polynomial P is defined as the integral of log|P| over the unit torus with respect to the Haar measure. For multivariate polynomials, it often yields special values of L-functions. In this talk I will discuss some of these relationships and the meaning behind them and present some results involving L(E,3) for E an elliptic curve.

Claude Levesque (Université Laval)
Système fondamental d'unités de certaines familles de corps de nombres de degré 12

Le but de cet exposé est d'exhiber un système fondamental d'unités pour une famille infinie de corps de nombres algébriques de degré 12 sur Q, dont le rang du groupe des unités est 7. Ceci est un travail conjoint avec Hans-Joachim Stender.


Jan Minac (Western University)
Small quotients of absolute Galois groups, Massey triple products, and memories from Budapest --Dedicated to Professor Michel Waldschmidt

The structure of maximal pro-p-quotients of absolute Galois groups seems to be governed by deep laws which include the Bloch-Kato conjecture recently proved by Rost and Voevodsky, and the vanishing of Massey products. How all of this works is very fascinating and amazing, but also a bit mysterious.

And what happened in Budapest, and how is this related to absolute Galois groups?

This is a report on my joint work with Sunil Chebolu, Ido Efrat, Nguyen Duy Tan, and related mathematics.


Kumar Murty (University of Toronto)
Special values of Class Group L-functions

We review recent work on the transcendence of special values of L functions associated to ray class characters.

Ram Murty (Queen's University)
Li's criterion and transcendence

Li's criterion for the Riemann hypothesis can be rephrased as a statement involving special values of the Riemann zeta function at positive integer arguments. We will discuss some research in progress with Sanoli Gun and Purusottam Rath in this context.

Damien Roy (University of Ottawa)
On the algebraic independence of values of the exponential function

Since C. Hermite opened the way by proving the transcendence of e in 1873, much progress has been done in our understanding of the algebraic independence of values of the exponential function. However, much remains to be done as several important conjectures remain widely open. We present a survey of the question.


Cameron L. Stewart (University of Waterloo)
On the distribution of small denominators in the Farey series of order N.

Alain Togbé
(Purdue University North Central)
On the number of solutions of a family of Diophantine inequalities

Recently, using linear forms in logarithms, C. Levesque and M. Waldschmidt studied the effective bounds for the solutions of families of cubic Thue equations. In this talk, we will recall their result and give the effective bound of a family of quartic Diophantine equations.


Michel Waldschmidt (University Paris VI)
Families of Diophantine equations with finitely many solutions

During the spring 2010 in Rio de Janeiro, Claude Levesque suggested a question on the finiteness of the set of solutions of families of Diophantine equations. In this lecture we report on the progress which has been done during the last three years, and on the program which we plan to work on.

Gary Walsh
(University of Ottawa)
Common terms in Lucas-type sequences

In joint with with Alain Togbé, we discuss the problem of determining all terms which are common to two sequences of Lucas' type, and some of the ideas underlying the methods. We end by discussing some related open problems.