The Ramanujan-Serre differential operators and certain elliptic curves

TL;DR

This paper explores differential equations satisfied by Eisenstein series of weight 4 for certain congruence subgroups and links them to elliptic curves and specific new forms of weight 2, enriching the understanding of their interrelations.

Contribution

It establishes explicit differential equations for Eisenstein series and connects them to elliptic curves and new forms classified by Martin and Ono, revealing new structural insights.

Findings

Differential equations for Eisenstein series derived for specific subgroups

Connection established between these equations and elliptic curves

Identification of related new forms of weight 2 from eta-products

Abstract

For several congruence subgroups of low levels and their conjugates, we derive differential equations satisfied by the Eisenstein series of weight 4 and relate them to elliptic curves, whose associated new forms of weight 2 constitute the list of Martin and Ono of new forms given by eta-products/quotients.