Proof of the elliptic expansion Moonshine Conjecture of Cu{a}ldu{a}raru, He, and Huang

Using predictions in mirror symmetry, Cu{a}ldu{a}raru, He, and Huang recently formulated a Moonshine Conjecture at Landau-Ginzburg points for Kleins modular $j$-function at $j=0$ and $j=1728.$ The conjecture asserts that the $j$-function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical $ _2F_1$-hypergeometric inversion formulae for the $j$-function.