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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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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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
  • MR Author ID: 342109
  • Email: ono@math.ias.edu, ono@math.psu.edu
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3169-3176
  • MSC (1991): Primary 11F20, 11GXX
  • DOI: https://doi.org/10.1090/S0002-9939-97-03928-2
  • MathSciNet review: 1401749