Relations and Derivatives of Multiple Eisenstein Series

TL;DR

This paper explores the structure and relations of multiple Eisenstein series, connecting them to multiple zeta values and modular forms, and proposes conjectures on their derivatives and linear relations.

Contribution

It introduces a conjectural formula for derivatives of multiple Eisenstein series and a formal space exhibiting an $rak{sl}_2$-algebra structure, advancing understanding of their relations.

Findings

Proved a large family of relations among multiple Eisenstein series.

Proposed an explicit conjectural formula for their derivatives.

Introduced a formal space of multiple Eisenstein series with an $rak{sl}_2$-algebra structure.

Abstract

In this paper, we study multiple Eisenstein series, which build a natural bridge between the theory of multiple zeta values and modular forms. We prove a large family of relations among these series and propose an explicit conjectural formula for their derivatives. This formula is expressed using the double shuffle structure and the Drop1 operator introduced by Hirose, Maesaka, Seki, and Watanabe. Based on this, we propose a family of linear relations that is conjectured to generate all linear relations among multiple Eisenstein series. Motivated by this conjecture, we introduce a space of formal multiple Eisenstein series and show that it is an $\mathfrak{sl}_2$-algebra.