Schedule for: 25w2030 - Alberta Number Theory Days XVI

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

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. Funding for breakfasts at Vistas for participants and observers provided by PIMS, NTF, and JNT. You will need to show your name tag to the staff at the entrance. Guests must pay for their own meals.

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

In this talk we discuss joint work with Fatma Çiçek on establishing asymptotic formulae with power saving error term for first and second moments of Rankin-Selberg $L$-functions $L(1/2+i\mu,f\otimes g)$ where $f$ varies over newforms of conductor $p^n$ with $p$ being a fixed prime and $n\to\infty$

Number theory is often seen as the science of integer numbers. Yet, arithmetical similarities between the ring of integers, and that of polynomials over a finite field, have led to establishing a "number theory" in positive characteristic that strikingly resembles its classical analogue. It is, however, easier to work in positive characteristic: function fields come from algebraic varieties, which give new tools for their study. For example, in positive characteristic, the Riemann hypothesis as well as some cases of the Langlands program are actually theorems. The lack of geometrical interpretation is a fundamental obstruction to our understanding of the classical case. In this presentation, we will talk about computations related to Drinfeld modules—the positive characteristic counterpart of elliptic curves, fundamental tools in classical number theory. Although their definition seems rather abstract, they encode many finite linear structures, that make algorithms very efficient. We will explore this through through the study of well-chosen computational problems, and algorithms that solve them.

The divisor class group (which is isomorphic to the Jacobian) of hyperelliptic curves is of great interest in both the contexts of mathematical significance and cryptographic applications. Regarding the theoretical aspect, the divisor class group stands at the heart of many conjectures and open problems in algebraic geometry and number theory. Significant examples include the Birch and Swinnerton-Dyer conjecture and the Sato-Tate conjecture, which were initially developed for elliptic curves, yet they have generalized statements in the case of hyperelliptic curves and other algebraic varieties. For experimental approaches to investigate these and other related problems, efficient implementation of the divisor addition in the divisor class group plays a crucial role. On the other hand, hyperelliptic curves have applications in lightweight and public-key cryptography where they can provide the same level of security as other schemes such as RSA cryptosystem and elliptic curve cryptography but using shorter key sizes. However, to keep them competitive, one remaining challenge is being able to do the divisor class arithmetic fast. In this study, we introduce a software framework that automates the generation of “explicit formulas”. The latter approach has shown significant promise in literature by achieving the current state-of-the-art results. However, these formulas have only been obtained for curves of small genera through manual efforts, which are cumbersome to extend as the genus of the curve grows. The main contribution of this project is developing a comprehensive software package extensible by design, which automates the process of generating and verifying improved formulas by adapting the prior computational techniques to a generic setting and taking advantage of highly reliable testing mechanisms.

The group law on elliptic curves is an important subject in number theory. When we generalize this group law to algebraic curves that are not necessarily elliptic, we get the degree zero divisor class group (which is isomorphic to the Jacobian or Picard variety). In this talk, I will discuss two algorithms for efficient hyperelliptic Jacobian arithmetic. The first is due to Paulus and Rück, and the second is due to Galbraith, Harrison, and Mireles. I will then compare these algorithms to an algorithm by Hess that performs Jacobian arithmetic on general algebraic curves. We will also discuss the importance of unique representatives of equivalence classes and how each of these algorithms finds unique representatives.

The original Ramanujan conjecture, which concerns the behavior of the Fourier coefficients of a certain a modular form, sits at the very heart of number theory. The proof of the conjecture depends on the Weil conjectures, and its consequences reach as far as the theory of expander graphs. In an effort to generalize this powerful result, one is lead to the study of generic and tempered representations of p-adic groups. In this talk, we will first introduce Vogan's geometric version of the local Langlands correspondence. Taking this perspective will then allow us to discuss recent results which articulate the geometric features of tempered and generic representations.

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. Funding for lunches at Vistas for participants and observers provided by PIMS, NTF, and JNT. You will need to show your name tag to the staff at the entrance. Guests must pay for their own meals.

(Joint work with Punya Satpathy) For a reductive group $G$, Borel and Serre introduced a compactification of a large class of arithmetic quotients of the symmetric space attached to $G$. After reviewing some aspects of their construction, we explain how to generalize it to the case when $G$ is replaced by an infinite-dimensional analogue $LG$, the loop group of $G$. Along the way, we describe a partition of an arithmetic quotient of $LG$, inspired by the work of P.-H. Chaudouard for $\operatorname{GL}_n$ and related to earlier constructions of Harder-Narasimhan and Behrend.

Semi-stable lattices are, heuristically, lattices in which the successive minima are not far from each other. They were first studied by Stuhler in 1976-1977, when he realized that he could use some work of Harder and Narasimhan on stable vector bundles to gain new information about lattices in Euclidean spaces. Later on, Grayson assigned these lattices a graphical content, called a "canonical plot," which is useful in partitioning the spaces of lattices. In this presentation, I will discuss the notion of semi-stable lattices in dimension 2 and above, as well as the equivalence between two different ways of defining semi-stable lattices.

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!

We computed the first Whittaker coefficients of a metaplectic Eisenstein series induced from Borel and showed that they are certain Weyl group multiple Dirichlet series associated to the metaplectic root datum.

The study of integers and primes represented by binary quadratic forms is a classical problem, going back to Fermat. We will discuss a Fourier analysis approach to this problem, based on joint work with Andrés Chirre. For a given form and integer $\ell \geq 2$, this approach gives us strong estimates for the average number of representations of integers that are multiples of $\ell$. This leads to unconditional upper bounds on the number of primes in short intervals represented by a given form, and, conditionally on the generalized Riemann hypothesis, an upper bound on the maximum gap between such consecutive primes. The latter extends a method of Carneiro, Milinovich, and Soundararajan.

This talk provides a Lucas-Lehmer type of test to prove the primality of numbers of the form $$10^{2n+m+1}+d10^{n+1}+1$$ where $m$ is odd and $d$ is an $m$ digit palindromic number. For example, numbers of the form $$10^{2n+4}+666\cdot10^{n+1}+1$$ belong to this class of numbers; also, when they are prime for some value of $n$ these have been called both "beastly palindromic primes" and, perhaps more commonly, "Belphegor primes." These Belphegor primes have garnered some interest from recreational number theory enthusiasts. Some related computational results are also included.

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. Other options exist at the Banff Center and in the town. Note that BIRS does not pay for meals for 2-day workshops.

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. Funding for breakfasts at Vistas for participants and observers provided by PIMS, NTF, and JNT. You will need to show your name tag to the staff at the entrance. Guests must pay for their own meals.

An order $O$ in a number field is called monogenic if it can be generated by one element over the integers, that is $O = \mathbb{Z}[\alpha]$. In this case we call $\alpha$ a monogenizer of $O$. Since $\mathbb{Z}[\alpha] = \mathbb{Z}[\pm \alpha+ c]$, for any integer $c$, we call two algebraic integers $\alpha$ and $\alpha'$ equivalent if $\alpha + \alpha'$ or $\alpha - \alpha'$ is a rational integer. By a monogenization of $O$, we mean an equivalence class of monogenizers of $O$. Győry has shown that there are finitely many monogenizations for a given order. An interesting (and open) problem is to count the number of monogenizations of a given monogenic order. First, we will observe, for a given order $O$, that $O= \mathbb{Z}[\alpha]$ in $\alpha$, is indeed a Diophantine equation, namely an index form equation. Then we will modify some algorithmic approaches, due to Gaál, Pethő and Pohst for finding solutions of index form equations in quartic number fields to obtain new and improved upper bounds for the number of monogenizations of a quartic order. If time permits, we will discuss how the resolution of an index form equation can give precise information on the number of distinct monogenized orders of a given index.

Let $F/k[x]$ be a function field of degree $n$, where $k$ is a field and $x$ is a transcendental element. The divisor group $\operatorname{Cl}(F)$ is the quotient of the group of all divisors of $F$ modulo principal divisors. In this talk, we discuss explicit algorithms for performing arithmetic in $\operatorname{Cl}(F)$, and we compare our approach and algorithms compare to the existing methods.

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.

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 determine an upper bound for the number of non-trivial zeros, denoted as $N(\sigma, T)$, within a specific rectangular region defined by $\sigma < \operatorname{Re}[s] < 1$ and $0 < \operatorname{Im}[s] < T$ . 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 improves their upper bound and also update the result to use better bounds on $\zeta$ on the half line among other improvements.

This talk will first provide a survey of explicit results on zero-free regions and zero densities of the Riemann zeta function and their relationship to error terms in the prime number theorem. We will extend this discussion to Dirichlet L-functions and Dedekind zeta functions, exploring estimates for error terms in prime counting functions across various contexts, with specific attention to number fields. Chebotarev's density theorem establishes that prime ideals are equidistributed among conjugacy classes of the Galois group in any normal extension of number fields. While Lagarias and Odlyzko first established an effective version of this theorem in 1977, we present an explicit refinement of their result. This research represents joint work with Sourabh Das and Nathan Ng.

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. Funding for lunches at Vistas for participants and observers provided by PIMS, NTF, and JNT. You will need to show your name tag to the staff at the entrance. Guests must pay for their own meals.