Elliptic Curves, eta-quotients, and hypergeometric functions

2012/02/02 by Eugene Yoong, Yoong, Eugene, David Pathakjee +3
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1202.0337

Accepted for publication by journal Involve

arxiv created 2012/02/02 · openalex publication_date 2012/02/02 · arxiv updated 2012/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_E1(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.

Related

• Modular parametrizations of certain elliptic curves
• The Explicit Hypergeometric-Modularity Method I
• Quasi-modular forms attached to elliptic curves, I
• Hypergeometric functions over F_p and relations to elliptic curves and modular forms
• Elliptic curves and Fourier coefficients of meromorphic modular forms