Greater Vancouver Number Theory Day

Abstract: A rational distance set is a subset of the real plane in which every two points have rational distance. A famous question of Ulam, based on work of Anning and Erdös, asks whether any such set can be dense for the Euclidean topology. Tao and Shaffaf showed that Lang's Conjecture implies that a rational distance set can never be dense. We show, following works of Solymosi, de Zeeuw, Makhul and Shaffaf that one can in fact derive from Lang’s Conjecture bounds for the size of rational distance sets in general position. This is based on joint work with K. Ascher and L. Braune.

Abstract: In 2010, Zagier defined a quantum modular form to describe certain examples of complex-valued functions defined on the rationals, which exhibit something close to modular behavior. These functions lie at the boundary of the complex upper half plane, and can be shown, in some cases, to extend to modular objects that naturally occur in the upper half plane. We will discuss examples of quantum modular forms and their connection to mock modular forms, in particular.

Abstract: The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding $\ell$-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point $\alpha$ under iterates of a rational map $f$ (points that eventually map to $\alpha$ as we apply $f$ repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation. As in Serre's open image theorem, we expect the image of this representation to have finite index in the automorphism group except in certain cases.

• Peter Lam -- Simultaneous Prime Values of Two Binary Forms
• Nicholas Jian Hao Lai -- On the Zero-Free Region of Certain Families of Dedekind Zeta-Functions
• Amir Hossein Parvardi -- On Subproducts of Residue Classes Modulo a Prime
• Adela Gherga -- Implementing Algorithms to Compute Elliptic Curves Over $\mathbb{Q}$
• Sepehr Yadegarzadeh -- On the local density of parametrizations of primitive solutions to $Ax^2+By^3+Cz^3=0$
• Angelos Koutsianas -- Solving a family of Lebesgue-Nagell equations
• Alexandre Zotine -- Global Generation of Vector Bundles over Elliptic Curves

Abstract: We study the Galois cohomology of Selmer groups of Galois representations with good reduction at a prime $p>3$ and investigate their behaviour over infinite Galois extensions. We also study the behaviour of Iwasawa theoretic invariants for two such representations that are residually isomorphic.