Quasi-modular forms attached to elliptic curves: Hecke operators

TL;DR

This paper introduces Hecke operators on quasi-modular forms and constructs vector fields that encode relations between Eisenstein series and modular curves of degree d isogenies.

Contribution

It develops a new framework for Hecke operators on quasi-modular forms and explicitly constructs vector fields related to modular curves of elliptic curves.

Findings

Constructed vector fields for each natural number d in six dimensions.

Determined polynomial relations between Eisenstein series and their transformations.

Characterized modular curves of degree d isogenies via these vector fields.

Abstract

In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2,4 and 6 and their transformation under multiplication of the argument by $d$, and in particular, it determines uniquely the modular curve of degree $d$ isogenies between elliptic curves.