  Motivated by a conjecture of Lian and Yau concerning the mirror map in string
theory, we determine when the mirror map q-series of certain elliptic curve and
K3 surface families are Hauptmoduln (genus zero modular functions). Our
geometric criterion for modularity characterizes orbifold uniformization
properties of their Picard-Fuchs equations, effectively demystifying the
mirror-moonshine phenomenon. A longer, more comprehensive treatment of these
results will appear shortly.
