Lethbridge Number Theory and Combinatorics Seminar
Abstract:
Julie Desjardins (University of Toronto, Canada)
The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X.
A natural question arises when studying the configuration of those curves: If a point of X is contained in “many” exceptional curves, is it torsion on its fiber on E?
In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.
Lethbridge Number Theory and Combinatorics Seminar
Abstract:
Elchin Hasanalizade (University of Lethbridge, Canada)
The Fibonacci sequence \(F(n) : (n\geq 0) is the binary recurrence sequence defined by
F(0)=F(1)=1andF(n+2)=F(n+1)+F(n)∀n≥0.
There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality
|F(n)+F(m)−2a|<2a/2
in positive integers n,m and a with n≥m. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Petho version of the Baker-Davenport reduction method in Diophantine approximation.
The behavior of quadratic twists of modular L-functions at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving
an optimal large sieve type bound.
In the past century, the studies of moments of L-functions have been important in number theory and are well-motivated by a variety of arithmetic applications. This talk will begin with two problems in elementary number theory, followed by a survey of techniques in the past and the present. We will slowly move towards the perspectives of period integrals which will be used to illustrate the interesting structures behind moments. In particular, we shall focus on the “Motohashi phenomena”.
We compute extreme values of the Riemann Zeta function at the critical points of the zeta function in the critical strip. i.e. the points where ζ′(s)=0 and Rs<1.. We show that the values taken by the zeta function at these points are very similar to the extreme values taken without any restrictions. We will show geometric significance of such points.
We also compute extreme values of Dirichlet L-functions at the critical points of the zeta function, to the right of Rs=1. It shows statistical independence of L-functions and zet function in a certain way as these values are very similar to the values taken by L-functions without any restriction.
Lethbridge Number Theory and Combinatorics Seminar
Abstract:
Dave Morris (University of Lethbridge, Canada)
We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.
Lethbridge Number Theory and Combinatorics Seminar
Abstract:
Khoa D. Nguyen (University of Calgary, Canada)
A power series f(x1,…,xm)∈C[[x1,…,xm]] is said to be D-finite if all the partial derivatives of f span a finite dimensional vector space over the field C(x1,…,xm). For the univariate series f(x)=∑anxn, this is equivalent to the condition that the sequence (an) is P-recursive meaning a non-trivial linear recurrence relation of the form: Pd(n)an+d+⋯+P0(n)an=0 where the Pi's are polynomials. In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients. This is from a joint work with Jason Bell and Umberto Zannier in 2019 and a more recent work in June 2022.
In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as
μk(n),ϕ(n)n,nϕ(n),μ2(n)ϕ(n)n,σα(n),(−1)#{p:pk|n}
and many others and it provides various new results and improvements to the previous result
in the literature. This is a joint work with Mithun Kumar Das.
It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.
The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters 0<α⩽. We consider the integral moments of the Hurwitz zeta function on the critical line \Re(s)=\frac12. We focus on rational \alpha. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet L-functions, which leads us into investigating moments of products of L-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters \alpha, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.