Elliptic curves and Thompson's sporadic simple group

Abstract

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 32 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.