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On McKay’s conjecture. (English) Zbl 0548.10018

Let \(\eta(z)\) be the Dedekind \(\eta\)-function. For any set of integers \(g=(k_ 1,...,k_ s)\), \(k_ 1\geq k_ 2\geq...\geq k_ s\geq 1\), we associate the \(\eta\)-product defined by \(\eta_ g(z)=\eta (k_ 1z)\eta (k_ 2z)...\eta (k_ sz).\) J. McKay, D. S. Dummit and H. Kisilevsky [Multiplicative products of \(\eta\)-functions, preprint] considered \(\eta\)-products associated to all sets \(g\) such that \(\sum^{s}_{i=1}k_ i=24\). These functions have Fourier expansion of the form \(\eta_ g(z)=\sum^{\infty}_{n=1}a(n)q^ n, q=\exp (2\pi iz)\). By computing \(a(n)\) for small \(n\), they found all sets \(g\) whose \(\eta\)-products have multiplicative Fourier coefficients. They also found a certain combinatorial condition on \(k_ i\) which just all the above sets satisfy.
In this paper, we use the theory of new forms to give an intrinsic proof of their results and also give a real meaning of their combinatorial condition.

MSC:

11F11 Holomorphic modular forms of integral weight
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[1] DOI: 10.2969/jmsj/02620362 · Zbl 0273.14007
[2] DOI: 10.2969/jmsj/02610056 · Zbl 0266.12009
[3] M24 and certain automorphic forms, preprint
[4] Multiplicative products of {\(\eta\)}-functions, preprint
[5] DOI: 10.1007/BF01390245 · Zbl 0369.10016
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