## $$M_{24}$$ 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

### MSC:

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

### Citations:

Zbl 0578.10028; Zbl 0578.10030; Zbl 0565.00006; Zbl 0424.20010