Modular equations and the genus zero property of moonshine functions
In the original Moonshine paper [Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)], \textit{J. H. Conway} and \textit{S. P. Norton} conjectured that there exist characters \(a_m\) of the Monster finite simple group such that for each element \(g\) of the Monster, the \(q\)-series \(j_g(q)=q^{-1} + \sum_{m=1}^\infty a_m(g) q^m\) (taking \(q=e^{2\pi i z}\)) is a Hauptmodul for a genus zero subgroup of \(\text{PSL}(2,R)\) that contains \(\Gamma_0(N)\), where \(N=ho(g)\) for some \(h\mid \gcd(24,o(g))\) (\(o(g)=\) order of \(g\)). \textit{R. E. Borcherds} subsequently proved the Moonshine conjectures [Invent. Math. 109, 405-444 (1992; Zbl 0799.17014)] as part of his Fields medal-winning work. However, the last part of Borcherds' proof that the \(j_g(q)\) are Hauptmoduls requires a (short) case-by-case verification. The main result of this paper is to give a more conceptual explanation as to why the \(j_g(q)\) are Hauptmoduls. Now, as part of his proof of Moonshine, Borcherds showed that the \(j_g(q)\) satisfied a recurrence property known as complete replicability. (See the paper for definitions of replicability and complete replicability.) \textit{D. N. Kozlov} [Can. J. Math. 51, 1020-1034 (1999; Zbl 0969.11016)] later showed that if a formal \(q\)-series \(f(q)\) is completely replicable, then \(f(q)\) (taking \(q=e^{2\pi i z}\)) actually gives an analytic function on the upper half plane that satisfies modular equations of order \(n\) for all \(n \equiv 1\pmod K\) for some integer \(K>0\). (See Shimura's book on automorphic forms for more on modular equations.) The main theorem in this paper (Theorem 1.3) is that if \(f(q)\) satisfies modular equations of order \(n\) for all \(n\equiv 1\pmod K\) for some integer \(K>0\), then \(f(q)\) is a Hauptmodul of the desired type. Combined with the above results of Borcherds and Kozlov, this theorem implies that \(j_g(q)\) are Hauptmoduls, without a case-by-case verification. The authors also obtain a converse to their main theorem (Theorem 1.4).
scientific article; zbMATH DE number 1054504