Schedule for: 20w2254 - Alberta Number Theory Days XII (Online)

You are responsible for your own beverage supply

Is it possible to arrange the integers 1, 2, ... , n in a circle such that any two adjacent entries sum to a square? Cube? Fibonacci number? What if the word sum is changed to difference? Or the square, cube, Fibonacci number restriction is replaced by some other permissible finite set of values? If such arrangements are possible, how many are there? Are there infinitely many? Richard Guy loved these types of questions which live at the interface between number theory and combinatorics and sound like simple brain teasers, but for which coming up with proofs is frequently extremely difficult if not outright impossible. Our protagonist in this talk is an (a,b)-difference necklace, which is a circular arrangement of the first n non-negative integers such that the absolute difference of any two neighbours takes on one of two possible values (a,b). We prove that such arrangements almost always exist for sufficiently large n, and provide explicit recurrence relations for the cases (a,b) = (1,2), (1,3), (2,4) and (1,4). Using the transfer matrix method from graph theory, we then prove that for any pair (a,b) -- and in fact for any finite set of two or more difference values -- the number of such arrangements satisfies a linear recurrence relation with fixed integer coefficients. This work began in summer 2017 as an NSERC USRA project undertaken by U of C undergraduate student Ethan White, now a PhD candidate at UBC, and jointly supervised by Richard Guy and me.

The current state of the art algorithm for computing a system of fundamental units in a number field without relying on any unproven assumptions or heuristics is due to Buchmann and has an expected run time O(\Delta_K^{1/4 + \epsilon}). If one is willing to assume the GRH, then one can use the index-calculus method, which computes the logarithm lattice corresponding to the unit group. This method has subexponential complexity with respect to the logarithm of the discriminant, though both complexity and correctness of this method depend on the GRH. We discuss a hybrid algorithm which computes the basis of the logarithm lattice for a number field given a full rank sublattice as input. The algorithm is capable of certifying the output of the index calculus algorithm in asymptotically fewer operations than Buchmann's method.

You are responsible for having your own coffee and cookies available. This coffee break is dedicated to the memory of Richard Guy. We encourage everyone to share stories about him.

Let $f_1$ and $f_2$ be (holomorphic) newforms of same weight and with same nebentypus, and let $a_{f_1}(n)$ and $a_{f_2}(n)$ denote the normalised Fourier coefficients of $f_1$ and $f_2$, respectively. If $a_{f_1}(p)=a_{f_2}(p)$ for almost all primes $p$, then it follows from the strong multiplicity one theorem that $f_1$ and $f_2$ are equivalent. Furthermore, a result of Ramakrishnan states that if $a_{f_1}(p)^2=a_{f_2}(p)^2$ outside a set of primes $p$ of density less than $\frac{1}{18}$, then $f_1$ and $f_2$ are twist-equivalent. In this talk, we will discuss some refinements of the strong multiplicity one theorem and Ramakrishnan's result for general $\rm{GL}(2)$-forms. In particular, we will analyse the set of primes $p$ for which $|a_{f_1}(p)| \neq |a_{f_2}(p)|$ when $f_1$ and $f_2$ are not twist-equivalent.

We study Hecke operators on vector-valued modular forms (vvmfs) for the Weil representation of a lattice $L$. In particular, we construct Hecke operators that map vector-valued modular forms (vvmfs) of a lattice $L$ to those of the same lattice $L$, and also Hecke operators that map vvmfs to those of a rescaled lattice. The components of these Hecke operators have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators and compare our operators with the alternative construction of Bruinier-Stein. This is joint work done with V. Bouchard and T. Creutzig.

Let $q\equiv 1 \pmod 4$ be a prime power. Consider $D$ to be a square-free monic polynomial over $\mathbb{F}_q[T]$ and $\chi_D$ the Kronecker symbol associated to $D$. In this talk we will discuss the distribution of large values for $L(\sigma,\chi_D)$ for $1/2< \sigma \le 1$. We will note the expected similarities to the situation over quadratic extensions of $\mathbb{Q}$ and the surprising differences.

This is a 1.5 hour lunch break.

In this talk, I will survey a variety of arithmetic problems related to the sequence of differences between perfect powers, highlighting what is known, what is expected to be true, and what is (possibly) within range of current technology.

Chebotarev’s density theorem asserts that the prime ideals are equidistributed over the conjugacy classes of Galois group of any given normal extensions of number fields. An effective version of this theorem was first proved by Lagarias and Odlyzko in 1977. In this talk, I will discuss a new explicit version of Lagarias and Odlyzko’s result on Chebotarev’s density theorem. Some of the main ideas are deriving an explicit formula for the smooth version of a certain prime ideal counting function, and proving new bounds for the number of low-lying zeros for Hecke L-functions to estimate certain sums over the non-trivial zeros of the Dedekind ζ-function. I will give an overview of how these ideas lead to a new explicit version.

You are responsible for your own supply of coffee and cookies.

Benford's Law, a notorious gem of mathematics folklore, asserts that significant digits of numerical data are not usually equidistributed, as might be expected, but rather follow one particular logarithmic distribution. Since first recorded by S. Newcomb in 1881, this apparently counter-intuitive phenomenon has attracted much interest from scientists, statisticians, and mathematicians. This talk will discuss several intriguing aspects of the phenomenon, and relate them to problems in dynamics and number theory.

Let $F$ be a $p$-adic field. Arthur recently completed endoscopic classification of irreducible admissible representations of $SO_n(F)$ and $Sp_{2n}(F)$, and constructed Arthur packets for those representations in the process. These results were extended to other classical groups by several authors. The next natural question is: what can we say about exceptional groups. In this talk, I will report our recent work on the construction of Arthur packets for unipotent representations of $G_2(F)$ arising from sub-regular unipotent orbit of $\widehat{G}_2$. This is a joint work with Clifton Cunningham and Andrew Fiori.

You are responsible for your own "Coffee" supply. We will be experimenting with zoom options to try to encourage social interaction.

The properties of a pair of integer valued sequences, similar to those of Lucas, are used to produce a sufficiency test for the primality of numbers $N$ such that $N^2 + N + 1$ is divisible by a large power of a prime p.

One of the main difficulties in extending Macdonald’s theory of spherical functions from p-adic Chevalley groups to p-adic Kac-Moody groups is the absence of Haar measure in the infinite dimensional case. Related to this problem is the question of how to generalize the integral defining Harish-Chandra’s c-function to the p-adic Kac-Moody setting. These questions are answered by proving the Approximation Theorem, which is a formal analogue of Harish-Chanda's limit relating spherical and c-fucntion, and two certain finiteness’. We call these finiteness results the Gindikin-Karpelevich Finiteness and the Spherical Finiteness. In this talk, we will present the proofs of these results for infinite dimensional local Kac-Moody groups. This work was completed under the supervision of Manish M. Patnaik.

We will investigate the fourth moment of quadratic Dirichlet $L$-functions. Under the generalized Riemann hypothesis, we show an asymptotic formula for the fourth moment. Unconditionally, we establish a precise lower bound. This work is mainly based on the Soundararajan and Young's paper in 2010 on the second moment of quadratic twists of modular $L$-functions.

Let $s(n) = \sigma(n) - n$ denote the proper sum of divisors function. In his 1976 M.Sc. thesis, Stan Devitt (supervised by Richard Guy) presented theoretical and numerical evidence, using a ``new method of factoring called POLLARD-RHO'', that the average order of $s(n)/n$ in successive iterations of $s(n)$ (Aliquot sequences) is greater than 1. These results seemingly lent support to the Guy/Selfridge Conjecture that there exist unbounded Aliquot sequences. In this talk, we describe the results of a project suggested by Richard in his efforts to provide more evidence in support of the Guy/Selfridge conjecture. In particular, we expand and update Devitt's computations by considering the more-appropriate geometric mean of $s(n)/n$ as opposed to the arithmetic mean considered by Devitt, and greatly extending Devitt's computations using modern factoring algorithms. This is joint work with K. Chum, R. Guy, and A. Mosunov.

Please turn on your camera to participate in the "group photo" -- a screenshot of Zoom with Gallery View enabled. You don't have to wear pants, but a shirt is recommended!

All the traditional activities... including trying to select next years organizing committe.