Proof of the elliptic expansion Moonshine Conjecture of Căldăraru, He, and Huang

Abstract

Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a "Moonshine Conjecture at Landau-Ginzburg points" for Klein's 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 ${}_2{F}_1$-hypergeometric inversion formulae for the $j$-function.

10 pages. Revised based on the referee's comments. Added reference to work of Shen and Zhou pertaining to Conjecture (1)

Cited by in corpus (1)

• Inversion Formulas for the $j$-function Around Elliptic Points