arXiv 2108.010272021-08math.NTmath.AGmath.SG

椭圆展开式月光猜想的证明

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

Letong Hong, Michael H. Mertens, Ken Ono, Shengtong Zhang

备注：: 10 pages. Revised based on the referee's comments. Added reference to work of Shen and Zhou pertaining to Conjecture (1)
发表信息：: Proc. Am. Math. Soc., 150 (2022), 5047-5056
DOI：: 10.1090/proc/16032

摘要：

利用镜像对称的预测，Czeldkiraru，他和黄最近制定了一个“月光猜想在朗道-金斯堡点”克莱因的模块$j$ -功能在$j=0$和$j=1728.$的猜想断言，$j$ -功能，当专门在特定的平面坐标上的模空间的相应CM椭圆曲线的变形，产生简单的有理函数。我们证明了这个猜想，并表明，这些有理函数产生于经典的$ _2F_1$ -超几何反演公式的$j$ -功能。

英文摘要：

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 $ _2F_1$-hypergeometric inversion formulae for the $j$-function.

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