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
Abstract

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 ${}_{2}{E}_{1}\left(\lambda \right)$ 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.

Keywords
number theory, elliptic curves, eta quotients, hypergeometric functions
Mathematical Subject Classification 2000
Primary: 11F11, 11F20, 11G05
Secondary: 11T24, 33C99