Events

Yifan Yang, National Taiwan University

Shimura curves are generalizations of classical modular  curves. Because of the lack of cusps on Shimura curves, there are very few explicit methods for Shimura curves. In this talk, we will introduce Borcherds forms and use them to determine the equations of Shimura curves. The construction of Borcherds forms is done by solving certain integer programming problems. This is a joint work with Jia-Wei Guo.

MC 5403

Karl Dilcher, Dalhousie University

Neha Prabhu, Queen's University

Melissa Emory, University of Toronto

I will define the general spin groups, and discuss the Gan-Gross-Prasad conjecture including the global GGP conjecture for GSpin groups. I will show that the conjecture is verified in several low-rank cases.

MC 5417

Jerry Wang, Department of Pure Mathematics, University of Waterloo

Ananth Shankar, MIT

Let A denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that A does not have real multiplication by any real quadratic field with discriminant a multiple of p. Then we prove that there are infinitely many places modulo which A is isogenous to the product of two elliptic curves. This is joint work with Davesh Maulik and Yunqing Tang.

MC 5417

Daniel Smertnig, Department of Pure Mathematics, University of Waterloo

Quaternion orders can possess many different ring-theoretical properties, such as being maximal, hereditary, Eichler, Bass, or Gorenstein. I will recall these properties and their relations to each other, summarizing a 'taxonomy' of quaternion orders. A quaternion order O over a domain R is basic if it contains an integrally closed quadratic R-order. I will show that a quaternion order is Bass if and only if it is basic, in the local and global settings.

Richard Gottesman, Queen's University

Brad Rogers, Queen's University

Vandita Patel, University of Toronto

Let $k$ be an even integer such that $k$ is at least $2$. We give a (natural) density result to show that for almost all $d$ at least $2$, the equation $(x+1)^k + (x+2)^k + ... + (x+d)^k = y^n$ with $n$ at least $2$, has no integer solutions $(x,y,n)$. The proof relies upon some Galois theory and group theory, whereby we deduce some interesting properties of the Bernoulli polynomials. This is joint work with Samir Siksek (University of Warwick).

MC 5417