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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On $p$-adic harmonic Maass functions

Author: Michael J. Griffin
Journal: Trans. Amer. Math. Soc. 373 (2020), 7019-7066
MSC (2010): Primary 11F33, 11F37
Published electronically: July 28, 2020
MathSciNet review: 4155199
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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

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
Article copyright: © Copyright 2020 American Mathematical Society