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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
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Abstract |
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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. |
Keywords
number theory, elliptic curves, eta quotients,
hypergeometric functions
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Mathematical Subject Classification 2000
Primary: 11F11, 11F20, 11G05
Secondary: 11T24, 33C99
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Milestones
Received: 3 May 2010
Revised: 22 April 2011
Accepted: 14 September 2011
Published: 28 April 2012
Communicated by Kenneth S. Berenhaut
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Authors |
| David Pathakjee | |
| Department of Mathematics University of Wisconin - Madison 480 Lincoln Drive Madison, WI 53706-1388 United States |
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| Zef RosnBrick | |
| Department of Mathematics University of Wisconsin - Madison 480 Lincoln Drive Madison, WI 53706-1388 United States |
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| Eugene Yoong | |
| Department of Pure Mathematics University of Waterloo 200 University Avenue West Waterloo, ON N2L 3G1 Canada |
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