Realizations of the formal double Eisenstein space

TL;DR

This paper introduces the formal double Eisenstein space, generalizing the formal double zeta space, and constructs realizations linking it to quasimodular forms, providing new insights into Eisenstein series and Ramanujan's differential equations.

Contribution

It defines the formal double Eisenstein space, proves key identities, and constructs realizations connecting it to quasimodular forms and classical Eisenstein series.

Findings

Established analogues of sum formula and parity result for formal double Eisenstein series.

Constructed the Kronecker realization linking the space to quasimodular forms.

Provided a combinatorial proof of Ramanujan's differential equations.

Abstract

We introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double Eisenstein series. We show that $\mathbb Q$-linear maps $\mathcal{E}_k\rightarrow A$, for some $\mathbb Q$-algebra $A$, can be constructed from formal Laurent series (with coefficients in $A$) that satisfy the Fay identity. As the prototypical example, we define the Kronecker realization $\rho^{\mathfrak{K}}: \mathcal{E}_k\rightarrow \mathbb Q[[q]]$, which lifts Gangl-Kaneko-Zagier's Bernoulli realization $\rho^B: \mathcal{D}_k\rightarrow \mathbb Q$, and whose image consists of quasimodular forms for the full modular group. As an application to the theory of modular forms, we obtain a purely combinatorial proof of Ramanujan's differential equations for classical Eisenstein series.