Lian–Yau mirror-moonshine conjecture for algebraic K3 surfaces

Let $z(q)$ be the mirror map for a one-parameter deformation of an algebraic K3 surface arising from an orbifold construction whose Picard–Fuchs equation is third order. A Lian–Yau conjecture. For some $c\in\mathbb{Z}$, the $q$-series

$$\frac{1}{z(q)}+c$$

is a Thompson series $T_g(q)$ for some element $g$ of the Monster. This conjecture expresses the observed mirror-moonshine phenomenon, in which reciprocals of mirror maps agree, up to an additive constant, with Thompson series; the statement concerns the modularity of mirror maps for the specified class of K3 families.