Schedule for: 16w2687 - Alberta Number Theory Days VIII

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. Your welcome package will include meal tickets for breakfasts and lunches.

Improving a result of Montgomery and Weinberger, we establish the existence of infinitely many real quadratic fields for which the class numbers are as large as possible. These values are achieved using a special family of fields, first studied by Chowla. In a subsequent work, joint with A. Dahl, we investigate the distribution of class numbers in Chowla’s family, and show a strong similarity between this distribution and that of class numbers of imaginary quadratic fields. As an application of our results, we determine the average order of the number of real quadratic fields in Chowla’s family with class number \(h\).

In this talk I will discuss how to extend the extend the argument of Erdos-Pomerance which gave a lower bound for the average number of Fermat pseudoprimes to obtain lower bounds for the average number of Quadratic Frobenius pseudoprimes. This is joint work with Andrew Shallue.

Compact representations are explicit representations of algebraic numbers or functions, with size polynomial in the logarithm of their height or, respectively, degree. These representations enable much more efficient manipulations of large algebraic numbers or functions than would be possible using a standard representation, and have proved to be useful in a variety of applications. In this talk, we will describe two such applications - how compact representations are essential for short certificates of the unit group and ideal class group of a number field, and how they can be used to speed the resolution of certain Diophantine equations. We will also present recent improvements that reduce the size of compact representations, efforts to generalize these to hyperelliptic function fields, and applications of the latter to speeding the computation of bilinear pairings.

In this talk, we show that the central values of the Rankin-Selberg convolutions, \(L(g\otimes f, s)\), uniquely determine an adelic Hilbert modular form \(g\); here \(f\) varies in a carefully chosen infinite family of adelic Hilbert modular forms. We prove our results in both the level and weight aspects. This is joint work with Naomi Tanabe.

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. Your welcome package will include meal tickets for breakfasts and lunches.

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!

Let \( \pi(x;q,a) \) denote the number of primes up to \(x\) that are congruent to \(a( \text{mod }q) \). A "prime number race", for fixed modulus \(q\) and residue classes \(a_1, \ldots,a_r\), investigates the system of inequalities \[ \pi(x;q,a_1)> \pi(x;q,a_2)> \cdots > \pi(x;q,a_r). \] We expect that this system should have arbitrarily large solutions x, and moreover we expect the same to be true no matter how we permute the residue classes \(a_j\); if this is the case, the prime number race is called "inclusive". Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet L-functions. We show that the same conclusion can be reached with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.

Rogers-Ramanujan identities, their analogues and generalizations are important from various points of view. In particular, they hold interesting number-theoretic and representation-theoretic information. In this talk, I will present some new conjectural partition identities which were discovered experimentally in a joint work with Matthew C. Russell. All of these conjectures have been verified up to partitions of 1500 and are still open.

One approach to studying the p-adic behavior of L-functions relies on understanding p-adic properties of certain modular forms. In this talk, I will discuss an approach to studying these properties, and I will introduce a construction of certain p-adic families of modular forms. As part of the talk, I will explain how this construction allows one to p-adically interpolate certain values of both holomorphic and non-holomorphic Eisenstein series. I will also mention some applications to number theory and beyond.

The irreducible smooth representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, which form a subset of the usual Langlands parameters. The set of irreducible representations associated with a single Arthur parameter is called an Arthur packet. Following Arthur's classification theory of automorphic representations of symplectic and orthogonal groups, the Arthur packets are now known in these cases. On the other hand, Moeglin independently constructed these packets in the p-adic case by using very different methods. In this talk, I would like to describe a combinatorial procedure to study the structure of the Arthur packets following the works of Moeglin. As an application, we show the size of Arthur packets in these cases can be given by counting integral (or half-integral) points in certain polytopes.

In this talk we extend results on Kneser-Hecke-operators for codes over finite fields, to the setting of codes over finite chain rings. In particular, we consider chain rings of the form \( \mathbb{Z}/p^2\mathbb{Z} \) for \( p \) prime. On the set of self-dual codes of length \(N\), we define a linear operator, \(T\), and characterize its associated eigenspaces.

Any vertex operator algebra that is \(C_2\)-cofinite and non-rational has both a finite number of simple modules and a non-semisimple representation theory. Such VOAs have some of the simpler non-obvious structures one can think of. However, only one such family of examples is known. With a mid term aim of exposing new examples, we come across modular objects by looking at certain characters. Consider an affine VOA at rational admissible negative level. Then some extension of a coset sub-VOA is believed to provide new families of examples. In the case of the affine \(sl_2\) simple VOA, we show that the linear span of the characters of the lifted simple coset-modules give rise to a vector valued modular form. This non-obvious result is to be expected if one has faith in the conjectured properties of the extended coset VOA in question.

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.

Hyperelliptic curves of low genus are good candidates for curve-based cryptography. Hyperelliptic curves comes in two models: imaginary and real. The existence of two points at infinity in real models makes them more complicated than their imaginary counterparts. However, real models are more general than the other model, every imaginary hyperelliptic curve can be transformed into a real curve over the same base field $\mathbb{F}_q$, while the reverse process requires a larger base field. Real hyperelliptic curves have not received as much attention by the cryptographic community as imaginary models, but more recent research has shown them to be suitable for cryptography. Real models admit two structures, the Jacobian (a finite abelian group) and the infrastructure (almost group just fails associativity). In this talk, we explain these two structures and compare their arithmetic based on some recent research. We show that the Jacobian makes a better performance in the real model. We also confirm our claim with some numerical evidence for genus 2 and 3 hyperelliptic curves.

Don't Try to Solve these Problems!