Schedule for: 24w2020 - Alberta Number Theory Days XV

Note: the Lecture rooms are available after 16:00.

Dinner is available at Vistas Dining Room between 5:30pm-7:30pm.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A well-rounded (WR) ideal lattice or a WR ideal is an ideal of a number field for which the associated lattice is well-rounded. WR ideal lattices can be used to investigate various problems such as kissing numbers, sphere packing problems, and Minkowski’s conjecture. They also have a variety of applications to coding theory. In this talk, we show criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We prove that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded.

The log-unit lattice of a number field is the image of the units of the ring of integers under Minkowski embedding in $\mathbb{R}^n$. Computing the log-unit lattice (or a fundamental unit) of a number field is a hard problem and is linked to the problem of computing class numbers which is one of the main tasks of computational algebraic number theory. Knowing the geometry of these lattices may help us to find better ways to compute them. In this talk, we will discuss the geometry of these lattices. Among different properties, orthogonality and well-roundedness of these lattices are two properties that are more interesting to us. As an example, we will discuss the geometry of log unit lattices of totally real bi-quadratic fields.

In 2020, in the article "Parity bias in Partitions", Kim, Kim, and Lovejoy presented a curious result regarding the parity of partitions which stated that the number of partitions of n with more odd parts (than even parts), denoted by $p_o(n)$ is greater than the number of partitions with more even parts (than odd parts), denoted by $p_e(n)$ whenever $n \geq 2$. Their paper majorly used q-series analysis to show this result. Moreover, they conjectured that if the partitions had an added condition that the parts were distinct, an equivalent inequality holds for $n>19$. This talk is based on a joint work with Kaustav Banerjee, Manosij Ghosh Dastidar, Pankaj Jyoti Mahanta and Manjil P. Saikia and I will be proving the first result combinatorially, using injective mappings from the smaller set to the larger set . The conjecture will be proved analogously and I will also show that if we add another condition, that the smallest part of the partition equals 2, the parity bias is reversed for all $n>7$.

Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erdős, Granville, Pomerance, and Spiro conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then the preimage set $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when $\mathcal{A}$ is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set $s^{-1}(\mathcal{A})$. The talk is based on the joint work with Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky and Lola Thompson.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk we will give an overview of recent advances in the theory of Galois representations attached to elliptic curves. In particular, we will discuss recent results of the speaker and collaborators (Enrique González-Jiménez, Asimina Hamakiotes, Benjamin York) in the special case of Galois representations attached to elliptic curves with complex multiplication, where we can give a complete classification of the $\ell$-adic representations that occur, together with the arithmetic applications of such classification.

Given a rank-$n$ lattice, $\Lambda$, we define the shape to be its equivalence class under scaling, rotation, and reflection. The shape lies in the following double coset space, which we refer to as the space of shapes of rank $n$ lattices: \[ \mathcal{S}_n = \text{GL}_n(\mathbb{Z}) \setminus \text{GL}_n(\mathbb{R}) /\text{GO}_n(\mathbb{R}). \] Now, given a family of lattices we can ask about where they lie in this space—maybe they lie on an explicit subspace of the space of shapes, or some collection of subspaces—and once we know where they lie we can ask about how they are distributed. This question is studied in different areas of mathematics but in this talk we will focus on lattices coming from both the additive and multiplicative structures of number fields. Specifically, we will talk about the shape of the integral lattice and the unit lattice in families of number fields with prescribed Galois conditions. We will approach this from the purview of arithmetic statistics and some of the open questions in this area of number theory.

The Riemann zeta function is a fundamental function in number theory. The study of zeros of the zeta function has important applications in studying the distribution of the prime numbers. Riemann hypothesis conjectures that all non-trivial zeros lie on the critical line, while the trivial zeros occur at negative even integers. A less ambitious goal than proving there are no zeros is to deter- mine an upper bound for the number of non-trivial zeros, denoted as $N(\sigma, T)$, within a specific rectangular region defined by $ \sigma < Rs < 1$ and $0 < Im s < T$ . Previous works by various authors like Ingham and Ramare have provided bounds for $N(\sigma, T)$. In 2018, Habiba Kadiri, Allysa Lumley, and Nathan Ng presented a result that provides a better estimate for $N(\sigma, T)$. In this talk I will give an overview of the method they provide to deduce an upper bound for $N(\sigma, T)$. My thesis will improve their upper bound. I will do this by revisiting the argument of Habiba Kadiri, Allysa Lumley, and Nathan Ng but using some improvements in the estimation of the zeta function.

The degree of a dominant rational map \(f:\mathbb{P}^n\to \mathbb{P}^n\) is the common degree of its homogeneous components. By considering iterates of \(f\), one can form a sequence \({\rm deg}(f^n)\), which is submultiplicative and hence has the property that there is some $\lambda\ge 1$ such that $({\rm deg}(f^n))^{1/n}\to \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We’ll give an overview of the significance of the dynamical degree in complex dynamics and describe an example of a birational self-map of $\mathbb{P}^3$ in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller, Mattias Jonsson, and Holly Krieger.

In this talk, inspired by a theorem of Ishida, I show that if a pure cubic number field $K=\mathbb{Q}(\sqrt[3]{m})$, with $m \neq 1$ a cube-free integer, has at least three ramified primes then its class number is divisible by three. The proof is based on the Galois cohomology which can be used also to get an alternative proof for Ishida's result.

The local Langlands correspondence posits finite-to-one map between the set of equivalence classes of smooth irreducible $\mathbb{C}$-representations of a connected reductive algebraic group $G$, over a non-archimedean field $F$, and the so-called Langlands parameters for $G$, which may be thought of as generalisations of the Galois representations associated with the Langlands dual group $\widehat{G}$. The conjecture seeks to uncover a deep relationship between the disciplines of harmonic analysis and number theory through the language of $L$-functions. In his work from the 90’s, David Vogan reformulated the local Langlands correspondence as a bijection between those smooth irreducible representations $G$ and certain irreducible equivariant perverse sheaves on moduli spaces built from the corresponding Langlands parameters. In this talk, we recall how the $L$-functions arise in the Langlands Programme and describe how Vogan’s conception of the theory hints at a possible categorical local Langlands correspondence.

Products of the form $\alpha_1^{c_1}\cdots\alpha_n^{c_n}$ where the $\alpha_i$ are algebraic are of interest across much of number theory, especially since Baker's results on linear forms in logarithms are widely applicable. In this talk, we explore the scenario where $\alpha_1,\dots,\alpha_n$ consist only of algebraic integer conjugates, though the $\alpha_i$ need not comprise a full set of algebraic integer conjugates. In particular, for some integers $d \geq 2$ and $1 \leq k \leq d-1$ we describe the set $E_{k,d}$ of all tuples $(c_2,\dots,c_{k+1}) \in (\mathbb{R}_{\geq 0})^k$ for which $|\alpha_1||\alpha_2|^{c_2}\cdots|\alpha_{k+1}|^{c_{k+1}} \geq 1$ for every tuple of degree $d$ algebraic integer conjugates $\alpha_1,\dots,\alpha_d$ which are written in descending order of absolute value. Furthermore, for any fixed tuple $(c_2,\dots,c_{k+1}) \in E_{k,d}$, we ask whether or not there exists a tuple of degree $d$ algebraic integer conjugates $\alpha_1,\dots,\alpha_d$ (written in descending order of absolute value) so that $|\alpha_1||\alpha_2|^{c_2}\cdots|\alpha_{k+1}|^{c_{k+1}} = 1$. This talk features joint work with Seda Albayrak, Samprit Ghosh, and Khoa Nguyen.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Gauss sums are fundamental objects in number theory. Quadratic Gauss sums were studied by Gauss who gave a simple formula depending only on the argument of the Gauss sums modulo 4. Higher degree Gauss sums behave differently. It was conjectured by Kummer that the cosines of the angles of (normalized) cubic Gauss at prime arguments are not equidistributed, and exhibit a bias towards positive values. This was disproved by Heath-Brown and Patterson in 1979, and they showed that (normalized) cubic Gauss at prime arguments are equidistributed. This was later generalized by Patterson to general \(n\)th-order Gauss sums. We explain in this talk what is involved in proving those results, and how to improve the results of Patterson for the distribution of quartic Gauss sums at prime arguments. Joint work with A. Dunn, A. Hamieh and H. Lin.

I will explain how the Geometric Lemma allows us to classify parabolically induced representations of the p-adic group $G_2$ distinguished by $SO_4$. In particular, I will describe a new approach, in progress, where we use the structure of the p-adic octonions and their quaternionic subalgebras to describe the double coset space $P \backslash G_2 / SO_4$, where $P$ stands for the maximal parabolic subgroups of $G_2$.

2-day workshop participants are welcome to use BIRS facilities (TCPL) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 11 M. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

In this talk, I will discuss recent work on how to reduce computational problems to subfield calculations via the framework of norm relations. For example, this enables efficient class group and $S$-unit group computations in large degree number fields. This also enables the search for small generators of principle ideals in fields of large degree, which has applications to the computation of approximate short vectors in ideal lattices (a topic with applications to cryptography).

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.