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# Number Theory Seminar

Tuesday, February 5, 2019 — 12:30 PM EST

Richard Gottesman, Queen's University

"Vector-Valued Modular Forms on $\Gamma_0(2)$"

The collection of vector-valued modular forms form a graded module over the 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 projective line minus three points. We then use the 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.

MC 5417

### March 2021

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