Involve: A Journal of Mathematics

  • Involve
  • Volume 5, Number 1 (2012), 1-8.

Elliptic curves, eta-quotients and hypergeometric functions

David Pathakjee, Zef RosnBrick, and Eugene Yoong

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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 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.

Article information

Source
Involve, Volume 5, Number 1 (2012), 1-8.

Dates
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
https://projecteuclid.org/euclid.involve/1513733443

Digital Object Identifier
doi:10.2140/involve.2012.5.1

Mathematical Reviews number (MathSciNet)
MR2924308

Zentralblatt MATH identifier
1284.11072

Subjects
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

Keywords
number theory elliptic curves eta quotients hypergeometric functions

Citation

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


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References

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