On 𝑝-adic harmonic Maass functions

Abstract

Modular and mock modular forms possess many striking p p -adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent p p -adic modular forms, Candelori and Castella showed this leads to p p -adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct p p -adic analogs of harmonic Maass forms of weight 0 0 with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super-singular locus as well as on the ordinary locus. As with classical harmonic Maass forms, these p p -adic analogs are connected to weight 2 2 cusp forms and their modular derivatives are weight 2 2 weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half-integer weight modular and mock modular forms. We demonstrate this through the construction of p p -adic analogs of two families of theta lifts for these forms.