Monday, March 9th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: David Wehlau

Title: Planes in Finite Fields, Lehmer Numbers and the Multiplicative Order of Elements mod $p$.

Abstract: Let $\mathbb{F}_p$ denote the finite field of order $p$ and $\mathbb{F}$ its algebraic closure. Classifying the $\mathbb{F}$-representations of $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ leads to a simply stated geometric problem involving $\mathbb{F}_p$-planes in $\mathbb{F}$. Solving this leads in turn to an infinite family of polynomials in $\mathbb{F}[t]$. These polynomials have a number of surprising algebraic and combinatorial properties and satisfy a recursion relation related to that studied by D.H. Lehmer in his thesis. This is joint work with H. E. A. Campbell.

This presentation will be accessible to graduate students and senior undergraduates.

Monday, February 24th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Ertan Elma (University of Waterloo)

Title: Discrete Mean Values of Dirichlet L-functions.

Abstract: Let $\chi$ be a Dirichlet character modulo a prime number $p \geq 3$ and let $\mathfrak{a}_{\chi}:=(1-\chi(-1))/2$. Define the mean value $$ \mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})$$ for a complex number $s$ such that $s\neq 1$ if $\mathfrak{a} _{\chi}=1.$ Mean values of the form above have been considered by several authors when $\chi$ is the principal character modulo $p$ and $\Re(s)>0$ where one can make use of the series representations of the Dirichlet $L$-functions being considered. In this talk, we will investigate the behaviour of the mean value $\mathcal{M}_{p}(-s,\chi)$ where $\chi$ is a non-principal Dirichlet character modulo $p$ and $\Re(s)>0$. Our main result is an exact formula for $\mathcal{M}_{p}(-s,\chi)$ which, in particular, shows that $$ \mathcal{M}_{p}(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi 2p^sL(1,\chi)\zeta(-s)+o(1), \quad (p\rightarrow \infty)$$ for fixed $0<\sigma:=\Re(s)<\frac{1}{2}$ and $|\Im s|=o\left(p^{\frac{1-2\sigma}{3+2\sigma}}\right)$.

Monday, January 27th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Seoyoung Kim (Queen's University)

Title: On the density of irreducible polynomials which generate $k$-free polynomials over function fields.

Abstract: Let $M \in \FF_{q}[t]$ be a polynomial, and let $k \geq 2$ be an integer. In this talk, we will compute the asymptotic density of irreducible monic polynomials $P\in\FF_{q}[t]$ for which $P+M$ is not divisible by the $k$-th power of any irreducible polynomial.

Monday, January 13th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: On Picard-type theorems involving $L$-functions.

Abstract: The little Picard's theorem states that any non-constant entire function takes all complex values or all complex values except one point. In a similar flavour, suppose $f$ is an entire function such that for complex values $a$ and $b$, the set of zeros of $f$ is same as the set where $f'$ takes values $a$ and $b$, then it is possible to show that $f$ is a constant function. Such results are called Picard-type theorems. In this talk, we will discuss similar questions for $L$-functions, where it is possible to prove much stronger results.

Thursday, November 21st, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Richard Leyland

Title: Isogenies of Elliptic Curves with Complex Multiplication.

Abstract: In my thesis work I seek to answer Mazur's Question which asks if there exists any isomorphisms of mod $N$ Galois representations attached to elliptic curves that are not induced by isogenies. The first step in answering this question is determining which isogenies of elliptic curves are defined over a field $F$. In this talk, I will show how to construct isogenies between CM elliptic curves by using ideals of the endomorphism rings. In particular, we will see that if the field of definition $F$ does not contain the CM field, then we can reduce the problem to finding cyclic isogenies.