  In this paper we investigate the Fourier coefficients of harmonic Maass forms
of negative half-integral weight. We relate the algebraicity of these
coefficients to the algebraicity of the coefficients of certain canonical
meromorphic modular forms of positive even weight with poles at Heegner
divisors. Moreover, we give an explicit formula for the coefficients of
harmonic Maass forms in terms of periods of certain meromorphic modular forms
with algebraic coefficients.
