Proof of the elliptic expansion Moonshine Conjecture of C\u{a}ld\u{a}raru, He, and Huang

TL;DR

This paper proves a conjecture linking the Klein modular j-function at special points to rational functions derived from hypergeometric formulas, confirming predictions from mirror symmetry.

Contribution

It provides a proof of the elliptic expansion Moonshine Conjecture for the Klein j-function at specific points, connecting it to classical hypergeometric inversion formulas.

Findings

The conjecture is rigorously proven.

Rational functions at special points are derived from hypergeometric formulas.

The proof confirms mirror symmetry predictions.

Abstract

Using predictions in mirror symmetry, C\u{a}ld\u{a}raru, 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 $ _2F_1$-hypergeometric inversion formulae for the $j$-function.