Elliptic Curves, eta-quotients, and hypergeometric functions

Details Elliptic Curves, eta-quotients, and hypergeometric functions Elliptic Curves, eta-quotients, and hypergeometric functions

2012-02-02

arxiv.org/abs/1202.0337v1

Mathematics

Number Theory

Accepted for publication by journal Involve

Scientific paper

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({\lambda}) 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.

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