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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eta-quotients and elliptic curves


Authors: Yves Martin and Ken Ono
Journal: Proc. Amer. Math. Soc. 125 (1997), 3169-3176
MSC (1991): Primary 11F20, 11GXX
MathSciNet review: 1401749
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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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Additional Information

Yves Martin
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: ymartin@math.berkeley.edu

Ken Ono
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: ono@math.ias.edu, ono@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03928-2
PII: S 0002-9939(97)03928-2
Keywords: Eta-quotient, elliptic curves
Received by editor(s): March 2, 1996
Received by editor(s) in revised form: May 17, 1996
Additional Notes: The second author is supported by NSF grants DMS-9508976 and DMS-9304580.
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society