On p-adic haromonic Maass functions

TL;DR

This paper constructs $p$-adic analogs of harmonic Maass forms of weight 0 and 1/2 on square-free level, extending classical theory to the full supersingular locus and connecting to modular forms and theta lifts.

Contribution

It provides an analytic construction of $p$-adic harmonic Maass forms that coincide with existing theories where they intersect, and extends their domain to the entire supersingular locus.

Findings

Constructed $p$-adic harmonic Maass forms on full supersingular locus.

Connected $p$-adic forms to weight 2 cusp forms and their derivatives.

Interpolated coefficients of half-integer weight modular forms via CM values.

Abstract

Modular and mock modular forms possess many striking $p$-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 $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ 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 $p$-adic analogs are connected to weight $2$ cusp forms and their modular derivatives are weight $2$ 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 $p$-adic analogs of two families of theta lifts for these forms.