Multiplicative products of \(\eta\)-functions. (English) Zbl 0578.10028

Finite groups - coming of age, Proc. CMS Conf., Montreal/Que. 1982, Contemp. Math. 45, 89-98 (1985).
[For the entire collection see Zbl 0565.00006.]
For positive integers \(n_ 1\leq n_ 2\leq...\leq n_ t\), consider the products \(f(z)=\eta (n_ 1 z)\cdot...\cdot \eta (n_ t z)\) where \(\eta\) is the Dedekind eta-function. These products are cusp forms of weight \(k=t/2\) on a congruence group \(\Gamma_ 0(N)\) of some level N. A necessary condition for f to have multiplicative Fourier coefficients is that \(n_ 1+...+n_ t=24.\)
The authors find that exactly 30 of the partitions of 24 yield products f with multiplicative coefficients. Among them are \(\eta\) (24z) and \(\eta (8z)^ 3\). The remaining 28 products f have integral weight k and belong to partitions which are balanced in the sense that the sequence \(n_ 1,...,n_ t\) is invariant with respect to \(n_ j\mapsto N/n_ j\) where \(N=n_ 1n_ t\) is the level of f. Remarkably, among these partitions occur all the 21 partitions which are cycle shapes of elements of the Mathieu group \(M_{24}\) (see the following reviews of papers by G. Mason [ibid., 223-244 (1985)] and M. Koike [Nagoya Math. J. 99, 147-157 (1985; Zbl 0578.10030)]).
Two methods are used to show that the 28 products under consideration have in fact multiplicative coefficients: For some of them, including all those of weight 1, the associated Dirichlet series is an L-function corresponding to a grössencharacter on some imaginary quadratic number field. All those of weight \(k\geq 2\) belong to one-dimensional spaces of cusp forms and hence are eigenforms of the Hecke operators.
Reviewer: G.Köhler


11F11 Holomorphic modular forms of integral weight
20C15 Ordinary representations and characters