Polar harmonic Maa{\ss} forms and holomorphic projection

TL;DR

This paper constructs and analyzes small divisor functions related to polar harmonic Maa{ ss} forms of weight 3/2, establishing identities, congruences, and connections to classical and mock modular objects.

Contribution

It introduces new small divisor functions generating holomorphic parts of polar harmonic Maa{ ss} forms and explores their properties, identities, and relations to other modular objects.

Findings

Construction of small divisor functions $\sigma^{ ext{sm}}_{2, ext{Id}}$ and $\sigma^{ ext{sm}}_{2, ext{chi}}$

Identity between $\sigma^{ ext{sm}}_{2, ext{Id}}$ and Hurwitz class numbers

Proven $p$-adic congruences for the small divisor functions

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 $\sigma^{\text{sm}}_{2,\chi}$ and prove that these generate the holomorphic part of polar harmonic (weak) Maa{\ss} forms of weight $\frac{3}{2}$. To this end, we essentially compute the holomorphic projection of mixed harmonic Maa{\ss} 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 identitiy between $\sigma^{\text{sm}}_{2,\ \text{Id}}$ 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 $\sigma^{\text{sm}}_{2,\chi}$ 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.