Universität zu Köln

Males, Joshua, Mono, Andreas and Rolen, Larry (2022). Polar harmonic Maass forms and holomorphic projection. Int. J. Number Theory, 18 (9). S. 1975 - 2005. SINGAPORE: WORLD SCIENTIFIC PUBL CO PTE LTD. ISSN 1793-7310

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Abstract

Recently, Mertens, Ono and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a class of small divisor functions sigma(sm)(2,chi) and prove that these generate the holomorphic part of polar harmonic Maass forms of weight 3/2. To this end, we essentially compute the holomorphic projection of mixed harmonic Maass forms in terms of Jacobi polynomials, but without assuming the structure of such forms. Instead, we impose translation invariance and suitable growth conditions on the Fourier coefficients. Specializing to a certain choice of characters, we obtain an identity between sigma(sm)(2,1) and Hurwitz class numbers, and ask for more such identities. Moreover, we prove p-adic congruences of our small divisor functions when p is an odd prime. If chi is non-trivial we rewrite the generating function of sigma(sm)(2,chi) as a linear combination of Appell-Lerch sums and their first two normalized derivatives. Lastly, we offer a connection of our construction to meromorphic Jacobi forms of index - 1 and false theta functions.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Males, Joshua UNSPECIFIED UNSPECIFIED UNSPECIFIED Mono, Andreas UNSPECIFIED UNSPECIFIED UNSPECIFIED Rolen, Larry UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-673362
DOI:	10.1142/S1793042122501019
Journal or Publication Title:	Int. J. Number Theory
Volume:	18
Number:	9
Page Range:	S. 1975 - 2005
Date:	2022
Publisher:	WORLD SCIENTIFIC PUBL CO PTE LTD
Place of Publication:	SINGAPORE
ISSN:	1793-7310
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MODULAR-FORMS Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/67336