On harmonic weak Maass forms of half integral weight

Abstract:

Since Zwegers found a connection between mock theta functions and harmonic weak Maass forms, this subject has been of vast research interest. In this paper, we obtain isomorphisms among the space $H_{k + \frac {1}{2}}^{+}(\Gamma _0(4m))$ of (scalar valued) harmonic weak Maass forms of half integral weight whose Fourier coefficients are supported on suitable progressions, the space $H_{k + \frac {1}{2x1}, \bar {\rho }_L}$ of vector valued ones, and the space x1$\mathbb {\widehat {J}}_{k+1,m}^{cusp}$ of certain harmonic Maass-Jacobi forms of integral weight: \[ H_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)) \simeq H_{k + \frac {1}{2}, \bar {\rho }_L} \simeq \mathbb {\widehat { J}}_{k+1,m}^{cusp}\] for $k$ odd and $m = 1$ or a prime. This is an extension of a result developed by Eichler and Zagier, which shows that \[ M_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)) \simeq M_{k + \frac {1}{2}, \bar {\rho }_L} \simeq J_{k+1,m}.\] Here $M_{k + \frac {1}{2}}^{+}(\Gamma _0(4m)), M_{k + \frac {1}{2}, \bar {\rho }_L}$ and $J_{k+1,m}$ are the Kohnen plus space of (scalar valued) modular forms of half integral weight, the space of vector valued ones, and the space of Jacobi forms of integral weight, respectively. To extend the result, another approach is necessary because the argument by Eichler and Zagier depends on the dimension formulas for the spaces of holomorphic modular forms, but the dimensions for the spaces of harmonic weak Maass forms are not finite. Our proof relies on some nontrivial properties of the Weil representation.