Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang

Hong, Letong; Mertens, Michael; Ono, Ken; Zhang, Shengtong

doi:10.1090/proc/16032

Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang
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by Letong Hong, Michael H. Mertens, Ken Ono and Shengtong Zhang PDF

Proc. Amer. Math. Soc. 150 (2022), 5047-5056 Request permission

Abstract:

Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] 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.

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