Involve: A Journal of Mathematics
- Volume 5, Number 1 (2012), 1-8.
Elliptic curves, eta-quotients and hypergeometric functions
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 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.
Involve, Volume 5, Number 1 (2012), 1-8.
Received: 3 May 2010
Revised: 22 April 2011
Accepted: 14 September 2011
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F11: Holomorphic modular forms of integral weight 11F20: Dedekind eta function, Dedekind sums 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11T24: Other character sums and Gauss sums 33C99: None of the above, but in this section
Pathakjee, David; RosnBrick, Zef; Yoong, Eugene. Elliptic curves, eta-quotients and hypergeometric functions. Involve 5 (2012), no. 1, 1--8. doi:10.2140/involve.2012.5.1. https://projecteuclid.org/euclid.involve/1513733443