Vol. 5, No. 1, 2012

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Elliptic curves, eta-quotients and hypergeometric functions

David Pathakjee, Zef RosnBrick and Eugene Yoong

Vol. 5 (2012), No. 1, 1–8

The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a nice representation of the modular form associated to each elliptic curve. Here we provide explicit representations of the modular forms associated to certain Legendre form elliptic curves 2E1(λ) as linear combinations of quotients of Dedekind’s eta-function. We also give congruences for some of the modular forms’ coefficients in terms of Gaussian hypergeometric functions.

number theory, elliptic curves, eta quotients, hypergeometric functions
Mathematical Subject Classification 2000
Primary: 11F11, 11F20, 11G05
Secondary: 11T24, 33C99
Received: 3 May 2010
Revised: 22 April 2011
Accepted: 14 September 2011
Published: 28 April 2012

Communicated by Kenneth S. Berenhaut
David Pathakjee
Department of Mathematics
University of Wisconin - Madison
480 Lincoln Drive
Madison, WI 53706-1388
United States
Zef RosnBrick
Department of Mathematics
University of Wisconsin - Madison
480 Lincoln Drive
Madison, WI 53706-1388
United States
Eugene Yoong
Department of Pure Mathematics
University of Waterloo
200 University Avenue West
Waterloo, ON
N2L 3G1