Maryam Khaqan: Elliptic Curves and Moonshine

Abstract

Moonshine began as a series of numerical coincidences connecting finite groups to modular forms and 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 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.

We can in fact classify all such Thompson-modules where the graded dimension is a specific weakly-holomorphic modular form as well as 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.

Zoom Notes: The meeting ID is 657 9019 8929 and the passcode is 3517257.