Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics

Number Theory Seminar


Wednesday, August 29th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Richard Gottesman (Queen's University)

Title: VECTOR VALUED MODULAR FORMS ON GAMMA_0 ( 2 )

Abstract: I will give an introduction to vector-valued modular forms and describe my research on the arithmetic of vector-valued modular forms with respect to a representation of Gamma_0(2). The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, we can use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the project line minus three points. We then are able to use Gaussian hypergeometric series to explicitly solve this differential equation. Finally, we make use of these ideas together with some algebraic number theory to study the prime numbers that divide the denominators of the Fourier coefficients of the component functions of vector-valued modular forms.