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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.