Modular parametrizations of certain elliptic curves

TL;DR

This paper explores modular parametrizations of elliptic curves over Q, generalizing previous observations about eta-quotients and differential equations, and providing new insights into their properties.

Contribution

It introduces a generalized framework for modular parametrizations of elliptic curves, extending prior work on eta-quotients and differential equations.

Findings

Characterization of elliptic curves via differential equations

Generalization of eta-quotient related properties

Enhanced understanding of modular parametrizations

Abstract

Kaneko and Sakai recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of modular parametrizations associated to the elliptic curves over Q, and as a consequence we generalize and explain some of their findings.