Multiplicative 𝜂-quotients

Martin, Yves

doi:10.1090/S0002-9947-96-01743-6

Multiplicative $\eta$-quotients
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by Yves Martin

Trans. Amer. Math. Soc. 348 (1996), 4825-4856

DOI: https://doi.org/10.1090/S0002-9947-96-01743-6

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Abstract:

Let $\eta (z)$ be the Dedekind $\eta$-function. In this work we exhibit all modular forms of integral weight $f(z) = \eta (t_1z)^{r_1}\eta (t_2z)^{r_2}\dots \eta (t_sz)^{r_s}$, for positive integers $s$ and $t_j$ and arbitrary integers $r_j$, such that both $f(z)$ and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group $2 \mathrm {Co}_1$ via a generalized McKay-Thompson series.

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