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September 1, 2020

Elliptic Curves and Moonshine

Moonshine began as a series of numerical coincidences connecting finite groups to modular forms. It has since evolved into a rich theory that sheds light on the underlying structures that these coincidences reflect. We prove the existence of one such structure, a module for the Thompson group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then proceed to use this module to study the ranks of certain families of elliptic curves. This serves as an example of moonshine being used to answer questions in number theory. This talk is based on arXiv: 2008.01607, where we classify all such Thompson-modules where the graded dimension is a specific weakly-holomorphic modular form and prove more subtle results concerning geometric invariants of certain families of elliptic curves. Time permitting, we will talk about some of these results as well.