Cubic theta functions and modular forms of level six

TL;DR

This paper derives differential equations for modular forms of level six, extends the field of cubic theta functions, and uncovers new relations among Eisenstein series, enhancing understanding of modular forms and their applications in number theory.

Contribution

It introduces new differential equations and extends the differential field of cubic theta functions for level six modular forms, including previously unstudied forms.

Findings

Derived Riccati equations for level six modular forms

Constructed extensions of the cubic theta function field

Discovered new relations among Eisenstein series

Abstract

The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by modular forms of level six and to construct extensions of the differential field of the cubic theta functions, generalizing the classical Ramanujan and Halphen fields. We treat both modular forms that appear in the literature and others that do not. We find Riccati equations satisfied by level six modular forms, explore applications to number theory, and find new relations among the Eisenstein series.