Schedule for: 17w2672 - Alberta Number Theory Days (ANTD IX)

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, additionally, Amy may bring some more wine and snacks.

A buffet breakfast is served daily between 7:00am and 8.45am 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.

Integral Apollonian circle packings are certain fractal packings of the plane with circles of disjoint interior, and integer curvatures. The set of curvatures which appears has been of recent interest as a challenging problem in the study of orbits of thin groups. Work of Bourgain, Fuchs and Kontorovich culminated in the demonstration that density one of the integers appear as curvatures, up to a congruence restriction. In this talk, we'll rediscover Apollonian circle packings as part of the essential nature of the Gaussian integers and their Diophantine approximation, generalize to other quadratic fields to discover new circle packings, and discuss the extension of results on curvatures to these and other Kleinian packings.

The size function $h^0$ for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. In this talk, we introduce this function and discuss the conjecture of Schoof and Van der Geer on the maximality of $h^0$ at the trivial divisor.

Metaplectic groups have had a rich interplay with number theory, generally via the theory of theta functions and their Fourier-Whittaker coefficients. We describe a recent construction of metaplectic covers of infinite-dimensional groups (joint with Anna Puskas) and explain its conjectural link to some concrete questions in analytic number theory.

In 1962, Rosser and Schoenfeld gave a method to estimate the error term in the approximation of the prime counting function $\psi(x)$. Since then, progress on the numerical verification of the Riemann Hypothesis and widening the zero-free region of the Riemann zeta function have allowed numerical improvements of these bounds. It is only recently that explicit zero density estimates have been used in this context. We will present some of these results as well as consequences to the distribution of primes.

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. Please use the coupons provided to you for the lunch. Note that BIRS does not pay for meals for 2-day workshops.

As the modularity theorem shows, classical modular forms are connected to Tate modules of elliptic curves over $\mathbb{Q}$ through their $L$-functions. This connection is built through the automorphic representations of GL(2) and its subgroups. This talk concerns a generalization of this story to abelian varieties. The Langlands program predicts that for abelian varieties $A$ over $\mathbb{Q}$, there should be an automorphic representation of GSpin over $\mathbb{Q}$ such that the $L$-function of the automorphic representation coincides with the $L$-function coming from the Tate module of the abelian variety $A$. Recently, Gross has refined this prediction for certain abelian varieties $A$, showing exactly how to describe the weight and level of a type-$B$ modular form $f_A$ whose $L$-function matches the $L$-function of the Tate module of $A$. In this talk, I will review some of this story and will describe my own work on the group scheme of the level of the GSpin modular forms that arise in Gross' conjecture.

In 1909, Landau showed that \[\limsup \tfrac{n}{\phi(n) \log\log{n}} = e^\gamma,\] where $\phi(n)$ is Euler's function. Later, Rosser and Schoenfeld asked whether there were infinitely many $n$ for which ${n}/{\phi(n)} > e^\gamma \log\log{n}$. This question was answered in the affirmative in 1983 by Jean-Louis Nicolas, who showed that there are infinitely many such $n$ both in the case that the Riemann Hypothesis is true, and in the case that the Riemann Hypothesis is false. One can prove a generalization of Landau's theorem where we restrict our attention to integers whose prime divisors all fall in a fixed arithmetic progression. In this talk, I will discuss the methods of Nicolas as they relate to the classical result, and also provide evidence that his methods could be generalized in the same vein to provide answers to similar questions related to the generalization of Landau's theorem.

Let $K/\mathbb{Q}$ be a number field with ring of integers $\mathbb{Z}_K$. If $K$ has class number one, the set of irreducible elements of $\mathbb{Z}_K$ coincides with the set of prime elements; in general, this need not be the case. One is led to wonder: Do statements about primes in $\mathbb{Z}$ have analogues for irreducibles in $\mathbb{Z}_K$, for a general choice of $K$? This talk concerns two instances where the answer is yes. We will discuss the maximal order of the number of irreducible divisors of an element of $\mathbb{Z}_K$, and we will provide an asymptotic formula for the number of irreducible elements of norm up to $x$ belonging to a given arithmetic progression.

We prove that if two Calabi-Yau invertible pencils in projective space have the same dual weights, then they share a common polynomial factor in their zeta functions related to a hypergeometric Picard-Fuchs differential equation. The polynomial factor is defined over the rational numbers and has degree greater than or equal to the order of the Picard-Fuchs equation. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight.

Recent work has related the equivariant elliptic genera of sigma models with K3 surface target space to a vertex operator superalgebra that realizes moonshine for Conway’s group. Motivated by this we consider conditions under which a self-dual vertex operator superalgebra may be identified with the bulk Hilbert space of a superconformal field theory. After presenting a classification result for self-dual vertex operator superalgebras with central charge up to 12, several examples of close relationships with bulk superconformal field theories are described, including those arising from sigma models for tori and K3 surfaces.

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 the workshop will not pay for the dinner.

A conjectured generalization of Fermat's Last Theorem states that the equation $x^p + y^q = z^r$ has no solutions in non-zero mutually coprime integers $x, y, z$ whenever the integer exponents $p, q, r \geq 3$. Since the proof of Fermat's Last Theorem, it was natural to attempt to study this generalization using a similar approach by Galois representations and modular forms. In this talk, I will survey some of the successes of applying this method, current ongoing approaches, and fundamental challenges in carrying out a complete resolution.

An odd prime $p$ is called a \emph{Wieferich prime} (in base $2$), if $$2^{p-1} \equiv 1 \pmod {p^2}.$$ These primes first were considered by A. Wieferich in $1909$, while he was working on a proof of Fermat's last theorem. This notion can be generalized to any integer base $a>1.$ In this talk, we discuss the work that has been done regarding the size of the set of non-Wieferich primes and show that, under certain conjectures, there are infinitely many non-Wieferich primes in certain arithmetic progressions. Also we consider the congruence $$a^{\varphi(m)} \equiv 1 \pmod{m^2},$$ for an integer $m$ with $(a,m)=1,$ where $\varphi$ is Euler's totient function. The solutions of this congruence lead to Wieferich numbers in base $a$. In this talk we present a way to find the largest known Wieferich number for a given base. In another direction, we explain the extensions of these concepts to other number fields such as quadratic fields of class number one.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. 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.

Note that the check out time is Noon. Check out is in Front Desk – Professional Development Centre.

Algebraic curves with many points are useful in coding theory, but are also of number theoretic and geometric interest in their own right. Their symmetries are described by their automorphism group. Other information, such as the number of rational points on the curve and on the associated Jacobian variety over any field, is encoded in their zeta function. Unfortunately, all these objects are generally notoriously difficult to compute. In this talk, we describe a class of Artin-Schreier curves whose unusually big automorphism group can be explicitly described. The automorphism group contains a large extraspecial subgroup, precise knowledge of which makes it possible to compute the zeta functions of these curves after extending the base field to contain the appropriate field of definition. We find that over fields of square cardinality, these curves are either maximal or minimal, and we classify which curves fall into which category. This is joint work with Irene Bouw, Wei Ho, Beth Malmskog, Padmavathi Srinivasan and Christelle Vincent.

This talk will give a brief review of basic pseudoprime history. It will also give the principles of a few generalizations of the Lucas functions and how these yield a 'new' type of pseudoprime. Finally, we will conjecture how these new pseudoprimes fit into the more general theory.