## On $p$-adic harmonic Maass functions

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- by Michael J. Griffin PDF
- Trans. Amer. Math. Soc.
**373**(2020), 7019-7066 Request permission

## 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 $0$ 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.

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## Additional Information

**Michael J. Griffin**- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 943260
- ORCID: 0000-0002-9014-3210
- Email: mjgriffin@math.byu.edu
- Received by editor(s): April 8, 2019
- Received by editor(s) in revised form: December 31, 2019
- Published electronically: July 28, 2020
- Additional Notes: This research was supported by the National Science Foundation grant DMS-1502390 and by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 7019-7066 - MSC (2010): Primary 11F33, 11F37
- DOI: https://doi.org/10.1090/tran/8105
- MathSciNet review: 4155199