Factorizations of Eisenstein Series of Level up to 4

TL;DR

This paper reveals new factorizations of classical Eisenstein series and theta functions, expressing them as products of conjugate Eisenstein series of half their weight and double their level, with identities across levels 2, 3, and 4.

Contribution

It introduces novel factorizations of Eisenstein series and theta functions into products of conjugate series, extending understanding of their structure across multiple levels.

Findings

E_4( au) factors into conjugate Eisenstein series

2E_2(2τ)-E_2(τ) factors similarly

Identities for E_6 and levels 2, 3, 4

Abstract

We show that the most standard Eisenstein series such as $E_4(\tau)$ or $2E_2(2\tau)-E_2(\tau)$, and also the function $\theta^2(\tau)$, are in a natural way the product of two conjugate Eisenstein series of half their weight and double their level, as well as a number of similar elementary identities for $E_6$ and Eisenstein series of levels $2$, $3$, and $4$.