## Number Theory Seminar

### Wednesday, July 10th, 2019

**Time:** 3:30-4:30 p.m. **Place:** Jeffery Hall 319

**Speaker:** Richard Gottesman (Queen's University)

**Title:** Vector-Valued Modular Forms and Modular Linear Differential Equations

**Abstract:** The sequence of denominators of the Fourier coefficients of a modular form on a congruence subgroup is always bounded. It has been conjectured that the converse is also true. We will consider this problem in the context of vector-valued modular forms and explain a strategy for proving such an unbounded denominator result. In this talk, we will also explain how the module structure of vector-valued modular forms leads one to study modular linear differential equations. A key point is the importance of understanding the solutions of the modular linear differential equation at all of the cusps. No previous background on vector-valued modular forms will be assumed.