  The Thompson sporadic group admits special relationships to modular forms of
two kinds. On the one hand, last century's generalized moonshine for the
monster equipped the Thompson group with a module for which the associated
McKay-Thompson series are distinguished weight zero modular functions. On the
other hand, Griffin and Mertens verified the existence of a module for which
the McKay-Thompson series are distinguished modular forms of weight one-half,
that were assigned to the Thompson group in this century by the last two
authors of this work. In this paper we round out this picture by proving the
existence of two new avatars of Thompson moonshine: a new module giving rise to
weight zero modular functions, and a new module giving rise to forms of weight
one-half. We explain how the newer modules are related to the older ones by
Borcherds products and traces of singular moduli. In so doing we clarify the
relationship between the previously known modules, and expose a new arithmetic
aspect to moonshine for the Thompson group. We also present evidence that this
phenomenon extends to a correspondence between other cases of generalized
monstrous moonshine and penumbral moonshine, and thereby enriches these
phenomena with counterparts in weight one-half and weight zero, respectively.
