Norton's replicability conjecture for rational formal q-series

Let f=q1+n=1anqnf=q^{-1}+\sum_{n=1}^\infty a_nq^n be a formal qq-series with rational coefficients that is replicable. Norton's conjecture. The series ff is either a Hauptmodul for a genus-zero group of moonshine type, or one of q1q^{-1} and q1±qq^{-1}\pm q. This conjecture proposes a converse to the result of Conway and Norton that Hauptmoduln with rational qq-coefficients for moonshine-type groups are replicable; the source gives no resolution.

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Primary source

Kazuki Tomiyama, “Generalized modular equations and the CM values of Hauptmoduln”, arXiv:2505.05135 (2025).

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