Relations and Derivatives of Multiple Eisenstein Series

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 give an explicit formula for their derivatives. This formula is expressed using the double shuffle structure and the Drop1 operator introduced by Hirose, Maesaka, Seki, and Watanabe. In particular, the space of multiple Eisenstein series is closed under the derivative. Further we construct bi-multiple Eisenstein series, which give a realization of the formal multiple Eisenstein series as holomorphic functions on the upper half-plane, and we prove a conjecture of Okounkov on derivatives of $q$-analogues of multiple zeta values. Based on the derivative formula, 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.

42 pages. Please Drop 1 comment