Relations and Derivatives of Multiple Eisenstein Series

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.

💡 Research Summary

The paper investigates the algebraic and analytic structure of multiple Eisenstein series (MES), objects that sit at the crossroads of multiple zeta values (MZV) and classical modular forms. After recalling the harmonic product () and shuffle product (⧢) that generate the well‑known double‑shuffle relations for MZV, the authors introduce two regularizations of MES: the harmonic regularization (G^{}) (following Bachmann) and the shuffle regularization (G^{\shuffle}) (following Brown–Tsuchiya). Both are constructed via generating functions and admit Fourier expansions whose constant term is a (regularized) multiple zeta value.

The first major result, Main Theorem A, shows that on the subspace (H_{\ge2,\mathrm{alm}}) – consisting of words in the non‑commutative alphabet ({z_k}) where all but possibly one letter are (z_{k\ge2}) – the two regularizations coincide: (G^{\shuffle}(w)=G^{*}(w)). This extends the previously known restricted double‑shuffle relations (Theorem 1.4) and allows one to treat words containing a single (z_1) as well.

Using this coincidence the authors define a linear operator (\varphi(w)=w* z_2-w\shuffle z_2) and a bilinear map
(R(u,v)=\varphi(u*v)-\varphi(u)v-u\varphi(v)).
The operator (\varphi) measures the failure of (z_2) to be a derivation for the harmonic product, while (R) measures the failure of (\varphi) itself to be a derivation. Main Theorem B proves that for any (u,v\in H_{\ge2}) the element (R(u,v)) lies in the kernel of the natural map (G:H_{\ge2}\to\mathcal{O}(\mathbb{H})). Consequently, (R(u,v)) yields new linear relations among MES. The authors verify that these relations reproduce all known relations up to weight 16 (but not weight 17), suggesting that additional structure is needed for higher weight.

The paper then turns to an (\mathfrak{sl}_2)-algebra structure on the space (E) spanned by all MES. Building on the classical (\mathfrak{sl}_2)-action on the algebra of quasi‑modular forms (\mathbb{Q}