Elliptic Curves, eta-quotients, and hypergeometric functions

TL;DR

This paper provides explicit eta-quotient representations of modular forms for certain elliptic curves and explores their coefficients' congruences using hypergeometric functions, advancing understanding of elliptic curve modularity.

Contribution

It introduces explicit eta-quotient formulas for modular forms of specific elliptic curves and links their coefficients to hypergeometric functions, offering new explicit representations.

Findings

Explicit eta-quotient representations for elliptic curve modular forms

Congruences for modular form coefficients in terms of hypergeometric functions

Enhanced understanding of the structure of elliptic curve modular forms

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_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.