Eta-quotients and elliptic curves

Martin, Yves; Ono, Ken

doi:10.1090/S0002-9939-97-03928-2

Eta-quotients and elliptic curves
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by Yves Martin and Ken Ono PDF

Proc. Amer. Math. Soc. 125 (1997), 3169-3176 Request permission

Abstract:

In this paper we list all the weight $2$ newforms $f(\tau )$ that are products and quotients of the Dedekind eta-function \begin{equation*} \eta (\tau ):=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}), \end{equation*} where $q:=e^{2\pi i \tau }.$ There are twelve such $f(\tau ),$ and we give a model for the strong Weil curve $E$ whose Hasse-Weil $L-$function is the Mellin transform for each of them. Five of the $f(\tau )$ have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Grössencharacter values. These formulae follow easily from well known $q-$series infinite product identities.

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