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On p-adic harmonic Maass functions.
- Source :
-
Transactions of the American Mathematical Society . Oct2020, Vol. 373 Issue 10, p7019-7066. 48p. - Publication Year :
- 2020
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HARMONIC functions
*MODULAR forms
*CUSP forms (Mathematics)
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 373
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 146167605
- Full Text :
- https://doi.org/10.1090/tran/8105