Formal multiple Eisenstein series and their derivations

Abstract: We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.

• The paper formalizes multiple Eisenstein series via derivations, establishing an algebraic structure akin to classical modular forms.
• It introduces derivation operators (D, W, δ) forming an sl₂-triple that mirrors quasimodular forms in algebraic manipulations.
• The framework connects formal Eisenstein series with multiple zeta values through symbolic and product structure analogues.

Formal Multiple Eisenstein Series and Their Derivations

Introduction

The paper introduces the algebra of formal multiple Eisenstein series, which explores their algebraic properties and their connections to multiple zeta values (MZVs) via derivations. Formalizing these concepts allows us to extend the classical notions of Eisenstein series into a purely algebraic framework that connects to the classical MZVs through a quotient algebra isomorphism. This algebraic setting is crucial for understanding the relationships among various types of zeta values and modular forms.

Multiple Zeta Values and Product Structures

Multiple Zeta Values (MZVs): The multiple zeta values are defined for positive integers such that $k_1 \geq 2$ and $k_i \geq 1$ for $i > 1$. They can be expressed as:

$$\zeta(k_1, \ldots, k_r) = \sum_{m_1 > \cdots > m_r > 0} \frac{1}{m_1^{k_1} \cdots m_r^{k_r}}$$

These values are known to satisfy relational structures such as the shuffle and stuffle products, providing an algebraic structure necessary to perform symbolic manipulations.

Product Structures on Words: Two primary product structures are used in the manipulation of word series:

• Index Shuffle Product denoted as $\shuffle$.
• Stuffle Product denoted as $\ast$.

These product operations are crucial in defining the relationships among the series involved and are central to the formal algebra of Eisenstein series.

Formal Multiple Eisenstein Series

The formal multiple Eisenstein series analogize the classical Eisenstein series. This formal space contains symbols that mimic the behavior of multiple zeta values but are structured according to generalized relations satisfied by modular forms.

Definition and Properties:

1. Algebraic Setup: The algebra of formal multiple Eisenstein series is constructed over a free monoid and equipped with a product defined via the stuffle product, imbued with swap invariance.
2. Symbolic Manipulation: The formal series use symbols $(k_1, \ldots, k_r)$ which satisfy similar relations to the classical Eisenstein series moduli lower weight terms.

Derivations and Algebraic Properties

The derivations play a crucial role in establishing the algebraic structure of formal Eisenstein series:

Derivation Operators:

• $D$ (Depth Derivation): Mimics the derivation of quasimodular forms and acts by increasing each exhibit's depth.
• $W$ (Weight Derivation): Multiplies elements by their weight, crucial for maintaining homogeneity in expansions and manipulations.
• $\delta$ (Formal Modular Derivation): Acts to reflect the behavior of classical modular transformations algebraically.

An $sl$-Algebra Structure

The algebra of formal multiple Eisenstein series is shown to be an $sl$-algebra:

• Commutator Relations: The derivations $D$, $W$, and $\delta$ form an $sl_2$-triple and satisfy the commutator relations typical for $sl_2$ algebras.
• Implications: The $sl$ algebra structure provides a grounding framework for interpreting the algebraic symbol manipulations analogously to how they operate on classical quasimodular forms.

Connections and Applications

Formal Modularity and Cusp Forms:

• The formal setup provides a means to simulate modular and quasimodular forms, involving experiences in formal modular spaces.
• The definition of formal cusp forms mimics the vanishing at the boundary points in classical modular theory, reflecting the depth-structural properties specific to derivations acting on zero terms.

Conclusion

This formalization approach not only advances the understanding of multiple zeta values and Eisenstein series in an algebraic framework but also links deeper algebraic properties like those derived from $sl_2$ actions. These developments facilitate further exploration and utilization of formal multiple Eisenstein series in theoretical and applied contexts, providing links between number theory, algebra, and elliptic modular forms.