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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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