Greater Vancouver Number Theory Day

Abstract: Shimura varieties of PEL type are moduli spaces of abelian varieties with extra structure. Rapoport-Zink spaces (moduli spaces of p-divisible groups with extra structure) are key tools in studying the supersingular loci of Shimura varieties of PEL type. In this talk, we'll discuss the geometry of the GL(4) Rapoport-Zink space. As an application, we'll also discuss the geometry of the supersingular locus of a particular Shimura variety.

Abstract: This is joint work with G. Glebov and R. Goenka. We develop a conceptual and systematic method to derive Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we use it to determine Ramanujan type series for $\frac{1}{\pi}$ whose coefficients are a product of a hypergeometric coefficient and a linear function of the summation index of the series.

Abstract: I will discuss a new result stating that many equations of the shape $F(x , y) = 1$ have no solutions in integers $x, y$, where $F(x , y)$ is a quartic form with integer coefficients. In this recent work, in order to construct a dense subset of forms that do not represent 1, the quartic forms are ordered by the two generators of their rings of invariants. Previously, in a joint work with Manjul Bhargava, we showed a similar result, but we ordered forms by their naive heights.

1. Pedro Mendoza Roca (SFU) -- Arithmetic progressions of integral points on congruent elliptic curves of rank two
2. Simone Coccia (UBC) -- S-integral points on rules surfaces with elliptic base
3. Aven Bross (SFU) -- Transcendental Brauer-Manin obstructions on genus 1 fibered surfaces
4. Sharon Robins (SFU) -- Algebraic hyperbolicity
5. Nils Bruin (SFU) -- Genus 0 curves on the "perfect cuboid" surface

Abstract: In this talk, I will talk about Darmon's program and the resolution of the generalized Fermat equation of signature $(p,p,5)$ using Frey hyperelliptic curves. This is joint work with Imin Chen (Simon Fraser University).