Polar harmonic Maaß forms and holomorphic projection
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Bibliographic record
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 [Formula: see text] and prove that these generate the holomorphic part of polar harmonic Maaß forms of weight [Formula: see text]. 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 [Formula: see text] and Hurwitz class numbers, and ask for more such identities. Moreover, we prove [Formula: see text]-adic congruences of our small divisor functions when [Formula: see text] is an odd prime. If [Formula: see text] is non-trivial we rewrite the generating function of [Formula: see text] 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 [Formula: see text] and false theta functions.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Science and technology studies | 0.000 | 0.000 |
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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