Relations and Derivatives of Multiple Eisenstein Series
Henrik Bachmann, Hayato Kanno
TL;DR
This work develops a comprehensive framework connecting multiple zeta values and modular forms through multiple Eisenstein series, proving a broad lattice of relations by comparing harmonic and shuffle regularizations and proposing an explicit derivative formula in terms of the Drop1 operator. It introduces a formal space of formal MES with an sl2-algebra structure, and proves analogous sl2-structure for an alternate formal space, enabling an organized approach to relations and derivatives. A key conjecture asserts that all linear relations among MES are generated by the identified relations and their derivatives, supported by explicit weights (e.g., weight 6 and 7 examples) and a Fibonacci-based generator-count pattern; the paper also defines a diamond-regularized version G^{\diamondsuit} to provide an alternative regularization with a corresponding kernel given by Drop1- type relations. Together, these results advance a unified algebraic and analytic treatment of MES, set the stage for proving or refining conjectures about their structure, and introduce multiple algebraic formalisms that capture the intricate interplay between regularization, derivations, and modular-analytic phenomena.
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.