  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.
