  For any square-free integer $N$ such that the "moonshine group"
$\Gamma_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the
Hauptmoduli of $\Gamma_0(N)^+$ to certain McKay-Thompson series associated to
the representation theory of the Fischer-Griess monster group. In particular,
the Hauptmoduli admits a $q$-expansion which has integer coefficients. In this
article, we study the holomorphic function theory associated to higher genus
moonshine groups $\Gamma_0(N)^+$. For all moonshine groups of genus up to and
including three, we prove that the corresponding function field admits two
generators whose $q$-expansions have integer coefficients, has lead coefficient
equal to one, and has minimal order of pole at infinity. As corollary, we
derive a polynomial relation which defines the underlying projective curve, and
we deduce whether $i\infty$ is a Weierstrass point. Our method of proof is
based on modular forms and includes extensive computer assistance, which, at
times, applied Gauss elimination to matrices with thousands of entries, each
one of which was a rational number whose numerator and denominator were
thousands of digits in length.
