  We describe a relationship between the representation theory of the Thompson
sporadic group and a weakly holomorphic modular form of weight one-half that
appears in work of Borcherds and Zagier on Borcherds products and traces of
singular moduli. We conjecture the existence of an infinite dimensional graded
module for the Thompson group and provide evidence for our conjecture by
constructing McKay--Thompson series for each conjugacy class of the Thompson
group that coincide with weight one-half modular forms of higher level. We also
observe a discriminant property in this moonshine for the Thompson group that
is closely related to the discriminant property conjectured to exist in Umbral
Moonshine.
