\(M_{24}\) and certain automorphic forms. (English) Zbl 0578.10029

Finite groups - coming of age, Proc. CMS Conf., Montreal/Que. 1982, Contemp. Math. 45, 223-244 (1985).
[For the entire collection see Zbl 0565.00006.]
There is a connection between the sporadic simple groups and certain modular forms which has come to be known as moonshine [J. H. Conway and S. P. Norton, Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)]. The author studies moonshine for the Mathieu group \(M_{24}\). It is a subgroup of the symmetric group \(S_{24}\), hence (the conjugacy class of) any g in \(M_{24}\) has a certain cycle shape \(n_ 1,...,n_ t\) which means that the permutation g is a product of disjoint cycles of length \(n_ j\) where \(n_ 1\leq n_ 2\leq...\leq n_ t\) and \(n_ 1+...+n_ t=24\). With g we associate the function \(\eta_ g(z)=\eta (n_ 1 z)\cdot...\cdot \eta (n_ t z)\) where \(\eta\) is the Dedekind eta-function. Then \(\eta_ g\) is a cusp form of weight \(k=k(g)=t/2\) and some character \(\epsilon\) on the congruence group \(\Gamma_ 0(N)\) of level \(N=n_ 1n_ t\). Remarkably, all the functions \(\eta_ g\) have multiplicative Fourier coefficients. They include, with but two exceptions, all those eta products of even weights for which the corresponding space of cusp forms has dimension 1 [see the preceding review of the paper by D. Dummit, H. Kisilevsky and J. McKay, ibid., 89-98 (1985)].
One of the exceptions is related to a Conway group.
If we write \(\eta_ g(z)=\sum^{\infty}_{n=1}\theta_ n(g) \exp (2\pi inz),\) then \(\theta_ n(g)\) is a multiplicative function of n for every fixed g. Moreover, as was noticed by Conway and Norton, \(\theta_ n(g)\) is a generalized character of \(M_{24}\) as a function of g, for every fixed n. The author shows that \(\psi_ p=\theta^ 2_ p- \theta_{p^ 2}\) yields, for every prime \(p\neq 3\), a character of \(M_{24}\) with values \(\psi_ p(g)=\epsilon_ g(p) p^{k(g)-1}\) where \(\epsilon_ g\) is a real Dirichlet character modulo N(g) which is trivial if and only if k(g) is even.
Reviewer: G.Köhler


11F11 Holomorphic modular forms of integral weight
20C15 Ordinary representations and characters
20D08 Simple groups: sporadic groups