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

DOI:
https://doi.org/10.1090/S0002-9939-97-03928-2

MathSciNet review:
1401749

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Abstract | References | Similar Articles | Additional Information

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.

- J. E. Cremona,
*Algorithms for modular elliptic curves*, Cambridge University Press, Cambridge, 1992. MR**1201151** - D. Dummit, H. Kisilevsky, and J. McKay,
*Multiplicative properties of $\eta$-functions*, Contemp. Math.**45, Amer. Math. Soc.**(1985), 89-98. - Basil Gordon and Dale Sinor,
*Multiplicative properties of $\eta $-products*, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp. 173–200. MR**1019331**, DOI https://doi.org/10.1007/BFb0086404 - Basil Gordon and Sinai Robins,
*Lacunarity of Dedekind $\eta $-products*, Glasgow Math. J.**37**(1995), no. 1, 1–14. MR**1316958**, DOI https://doi.org/10.1017/S0017089500030329 - Basil Gordon and Kim Hughes,
*Multiplicative properties of $\eta $-products. II*, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 415–430. MR**1210529**, DOI https://doi.org/10.1090/conm/143/01008 - Neal Koblitz,
*Introduction to elliptic curves and modular forms*, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR**766911** - G. Köhler,
*Theta series on the theta group*, Abh. Math. Sem. Univ. Hamburg**58**(1988), 15–45. MR**1027431**, DOI https://doi.org/10.1007/BF02941367 - Günter Köhler,
*Theta series on the Hecke groups $G(\sqrt 2)$ and $G(\sqrt 3)$*, Math. Z.**197**(1988), no. 1, 69–96. MR**917851**, DOI https://doi.org/10.1007/BF01161631 - I. G. Macdonald,
*Affine root systems and Dedekind’s $\eta $-function*, Invent. Math.**15**(1972), 91–143. MR**357528**, DOI https://doi.org/10.1007/BF01418931 - Yves Martin,
*Multiplicative $\eta $-quotients*, Trans. Amer. Math. Soc.**348**(1996), no. 12, 4825–4856. MR**1376550**, DOI https://doi.org/10.1090/S0002-9947-96-01743-6 - Yves Martin,
*On Hecke operators and products of the Dedekind $\eta $-function*, C. R. Acad. Sci. Paris Sér. I Math.**322**(1996), no. 4, 307–312 (English, with English and French summaries). MR**1378504** - Geoffrey Mason,
*$M_{24}$ and certain automorphic forms*, Finite groups—coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 223–244. MR**822240**, DOI https://doi.org/10.1090/conm/045/822240 - Geoffrey Mason,
*On a system of elliptic modular forms attached to the large Mathieu group*, Nagoya Math. J.**118**(1990), 177–193. MR**1060709**, DOI https://doi.org/10.1017/S0027763000003068 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210** - Joseph H. Silverman,
*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368**

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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

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