Elliptic curves and Thompson's sporadic simple group

TL;DR

This paper constructs special modules for Thompson's sporadic simple group with modular trace functions and uses them to study important arithmetic groups related to elliptic curves.

Contribution

It characterizes infinite-dimensional modules for Thompson's group with modular trace properties and applies these to elliptic curve arithmetic.

Findings

Modules have trace functions as weight 3/2 modular forms

Modules detect non-triviality of Mordell--Weil groups

Modules relate to Selmer and Tate--Shafarevich groups

Abstract

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group, whose graded traces are weight 3/2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell--Weil, Selmer, and Tate--Shafarevich groups of quadratic twists of certain elliptic curves.