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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we list all the weight newforms that are products and quotients of the Dedekind eta-function

where There are twelve such and we give a model for the strong Weil curve whose Hasse-Weil function is the Mellin transform for each of them. Five of the 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 series infinite product identities.

**1.**J. E. Cremona,*Algorithms for modular elliptic curves*, Cambridge Univ. Press, Cambridge, 1992. MR**93m:11053****2.**D. Dummit, H. Kisilevsky, and J. McKay,*Multiplicative properties of -functions*, Contemp. Math.**45, Amer. Math. Soc.**(1985), 89-98.**3.**B. Gordon and D. Sinor,*Multiplicative properties of products*, Springer Lect. Notes Math.**1395, Number Theory, Madras**(1987), 173-200. MR**90k:11050****4.**B. Gordon and S. Robins,*Lacunarity of Dedekind products*, Glasgow Math. J. (1995), 1-14. MR**96d:11044****5.**B. Gordon and K. Hughers,*Multiplicative properties of products II*, Contemp. Math.**143, Amer. Math. Soc.**(1993), 415-430. MR**94a:11058****6.**N. Koblitz,*Introduction to elliptic curves and modular forms*, Springer-Verlag, New York, 1984. MR**86c:11040****7.**G. Köhler,*Theta series on the theta group*, Abh. Math. Sem. Univ., Hamburg**58**(1988), 15-45. MR**90m:11067****8.**G. Köhler,*Theta series on the Hecke groups and*, Math. Z.**197**(1988), 69-96. MR**89b:11040****9.**I. G. Macdonald,*Affine root systems of Dedekind's -function*, Invent. Math.**15**(1972), 91-143. MR**50:9996****10.**Y. Martin,*Multiplicative eta-quotients*, Trans. Amer. Math. Soc.**348**(1996), 4825-4856. MR**97d:11070****11.**Y. Martin,*On Hecke operators and products of the Dedekind function*, C.R. Acad. Paris,**322**(1996), 307-312. MR**97a:11068****12.**G. Mason,*and certain automorphic forms*, Contemp. Math.**45, Amer. Math. Soc.**(1985), 223-244. MR**87c:11041****13.**G. Mason,*On a system of elliptic modular forms attached to the large Mathieu group*, Nagoya Math. J.**118**(1990), 177-193. MR**91h:11038****14.**J. Silverman,*The arithmetic of elliptic curves*, Springer-Verlag, New York, 1986. MR**87g:11070**; MR**95m:11054****15.**J. Silverman,*Advanced topics in the arithemtic of elliptic curves*, Springer-Verlag, New York, 1994. MR**96b:11074**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11F20,
11GXX

Retrieve articles in all journals with MSC (1991): 11F20, 11GXX

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

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