Analytic number theory and Diophantine equations
Org:
Michael Bennett (UBC) and
Patrick Ingram (Colorado State University)
[
PDF]
 AMIR AKBARY, University of Lethbridge
LangTrotter conjecture for two elliptic curves [PDF]

We propose explicit Euler product representation for the universal conjectural constant in the LangTrotter conjecture for two elliptic curves. This is joint work with James Parks (KTH Royal Institute of TechnologySweden).
 SHABNAM AKHTARI, University of Oregon
Many Thue equations have no solutions [PDF]

I will talk about my recent joint work with Manjul Bhargava. We show that a positive proportion of integral binary cubic forms that locally everywhere represent 1 do not globally represent 1. We order all
classes of binary cubic forms by their absolute discriminants. We prove
the same result for binary forms of any degree greater than 2, provided that
these forms are ordered by the maximum of the absolute values of their
coefficients.
 NILS BRUIN, Simon Fraser University
Obstructions for primitive solutions to $Ax^3+By^3=Cz^2$ [PDF]

It is classical that the only obstructions for homogeneous quadratic
equations to have primitive solutions $(x,y,z)$ are local; the Hasse
principle holds. The picture for $Ax^p+By^q=Cz^r$ is much more
complicated. A result by Beukers shows that if $1/p+1/q+1/r > 1$ then
the primitive solutions correspond to rational points on finitely many
genus 0 curves (subject to certain local conditions). However, as for
instance $x^2+31y^2=5z^3$ shows, obstructions to primitivity are not
just local anymore: there is also a class group that can provide
obstructions.
For other exponents, obstructions are no longer directly explained by
class groups. We will explore some statistics in the case $(p,q,r)=(3,3,2)$
This is joint work with Patrick McMahon.
 KARL DILCHER, Dalhousie University
Derivatives and fast evaluation of the Witten zeta function [PDF]

We study analytic properties of the Witten zeta function ${\mathcal W}(r,s,t)$, which is also named after Mordell and Tornheim. In particular, we evaluate the function ${\mathcal W}(s,s,\tau s)$ ($\tau>0$) at $s=0$ and, as our main result, find the derivative of this function at $s=0$, which turns out to be surprisingly simple. These results were first conjectured using highprecision calculations based on an identity due to Crandall that involves a free parameter and provides an analytic continuation. This identity was also the main tool in the eventual proofs of our results. Finally, we derive special values of a permutation sum and study an alternating analogue of ${\mathcal W}(r,s,t)$. (Joint work with Jon Borwein).
 AMY FEAVER, The King's University
 ALIA HAMIEH, University of Lethbridge
Valuedistribution of logarithmic derivatives of $L$functions [PDF]

In this talk, we describe a method for studying the valuedistribution of $L$functions based on the JessenWintner theory. This method has been explored recently by Ihara and Matsumoto for the case of logarithmic derivatives of Dirichlet $L$functions of prime conductor and by Mourtada and V. K. Murty for the case of logarithmic derivatives of Dirichlet $L$functions associated with quadratic characters. We show how one may extend such results to the case of cubic characters. This is a work in progress joint with Amir Akbary.
 DIMITRIS KOUKOULOPOULOS, Université de Montréal
Sieve weights and their smoothings [PDF]

I will discuss moments of partially smoothed, truncated divisor sums of the M\"obius function. Such divisor sums appear naturally in the theory of the Selberg sieve and they play a key role in the GPY sieve and its recent improvements due to Maynard and Tao. It turns out that if the truncation is smooth enough, the main contribution to the moments comes from almost primes. However, for rougher truncations, the dominant contribution comes from integers with many prime factors.
Analogous questions can be asked for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, a rare exception. As we will see, a plausible explanation for this phenomenon is given by studying the analogous sums for Dirichlet characters and obtaining different answers depending on whether or not the character is ``exceptional''.
This is joint work with Andrew Granville and James Maynard.
 MATILDE LALIN, Université de Montréal
Polylogarithms and multizeta values in Mahler measure [PDF]

The Mahler measure of a multivariable polynomial or rational function P is given by the integral of $\logP$ where each of the variables moves on the unit circle and with respect to the Haar measure. We will discuss some results involving formulas of Mahler measure of polynomials and rational functions yielding polylogarithms and multizeta values. We will focus specially in families of an arbitrary number of variables.
 PETER CHO HO LAM, Simon Fraser University
 ALLYSA LUMLEY, York University
A Zero Density Result for the Riemann Zeta Function [PDF]

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a zero density result. In 1940, Ingham showed the following asymptotic result
\[N(\sigma,T)=O(T^{\frac{3(1\sigma)}{2\sigma}}\log^5T).\]
Ramaré recently proved an explicit version of this estimate:
\[N(\sigma,T)\le 4.9(3T)^{\frac83(1\sigma)}\log^{52\sigma}(T)+51.5\log^2T,\]
for $\sigma\ge 0.52$ and $T\ge 3.061\cdot10^{10}$.
We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants. Furthermore, we present the effect of these improvements on explicit estimates for the prime counting function $\psi(x)$. This is joint work with Habiba Kadiri and Nathan Ng.
 GREG MARTIN, UBC
 MYRTO MAVRAKI, University of British Columbia
Simultaneous torsion points in a Weierstrass family of elliptic curves [PDF]

In 2010, Masser and Zannier proved that there are at most finitely many
complex numbers $t$, not equaling $0$ or $1$, such that the two points on the
Legendre elliptic curve $y^2=x(x1)(xt)$ with xcoordinates $2$ and $3$ are
simultaneously torsion. Recently, Stoll proved that there is in fact no
such $t$, and it is his result that inspires our work. In this talk we will
focus on the Weierstrass family of elliptic curves $E_t: y^2=x^3+t$, and show
that in many instances there is no parameter $t$ such that the points
$(a,*)$ and $(b,*)$ are simultaneously torsion in $E_t$. In contrast to the
original approach of Masser and Zannier, our approach is dynamical. We
focus on studying whether a and b are simultaneously preperiodic for a
Latt\`es map.
 JAMES PARKS, KTH Royal Institute of Technology
Lowlying zeros of quadratic Dirichlet Lfunctions [PDF]

In this talk we study the $1$level density of lowlying zeros of Dirichlet $L$functions attached to real primitive characters of conductor at most $X$. We obtain an asymptotic expansion of this quantity with lower order terms in descending powers of $\log X$. We show that this is valid under GRH when the support of the Fourier Transform of the implied even test function $\phi$ is contained in $(2,2)$. We also uncover a phase transition when the supremum of the support of $\widehat \phi $ reaches $1$, where a new lower order term appears. This is joint work with Daniel Fiorilli and Anders Södergren.
 KATHERINE STANGE, University of Colorado Boulder
Arithmetic properties of the Frobenius traces defined by a rational abelian variety [PDF]

Let $A$ be an abelian variety over the rationals. Under suitable hypotheses, we formulate a conjecture about the asymptotic behaviour of the Frobenius traces $a_{1,p}$ of $A$ reduced modulo varying primes $p$. This generalizes a wellknown conjecture of S. Lang and H. Trotter from 1976 about elliptic curves. We prove upper bounds for the counting function $\#\{p \leq x: a_{1,p} = t\}$ and we investigate the normal order of the number of prime factors of $a_{1,p}$. This is joint work with Alina Carmen Cojocaru, Rachel Davis and Alice Silverberg.
 GARY WALSH, Universite d'Ottawa
Solving systems of simultaneous Pell equations [PDF]

We survey recent quantitative results on the subject matter, and then
focus on the problem of completely solving certain specific families,
generalizing a recent result of Jian Hua Chen. This is joint work with Paul Voutier.
© Canadian Mathematical Society