Multiple Eisenstein Series in Modular Forms

• Multiple Eisenstein series are advanced generalizations of classical Eisenstein series defined by iterated lattice summations that exhibit modular transformation properties.
• Their Fourier expansions decompose the series into constant terms and q-series components that establish a natural q-analogue framework linked to multiple zeta values.
• Employing regularization techniques like shuffle and stuffle, these series satisfy double shuffle relations, enriching studies in number theory, algebraic combinatorics, and mathematical physics.

The concept of multiple Eisenstein series represents a significant generalization of classical Eisenstein series, extending their reach into fields such as modular forms and number theory. Multiple Eisenstein series create a bridge between the theory of modular forms, particularly Eisenstein series, and multiple zeta values (MZVs), enriching the understanding of both domains through their Fourier expansions, algebraic structures, and the various regularization techniques developed to manage their convergence properties.

1. Definition and Key Properties

Multiple Eisenstein series extend classical Eisenstein series by considering series over multiple lattice points in the complex upper-half plane. For classical Eisenstein series, the function is defined for an even integer $k$ by:

$$E_k(\tau) = \sum_{(m,n) \neq (0,0)} \frac{1}{(m\tau + n)^k},$$

where $\tau$ is in the upper half-plane. In contrast, multiple Eisenstein series incorporate iterated sums over several indices, such as:

$$G_{n_1, n_2, \ldots, n_r}(\tau) = \sum_{\lambda_1 \succ \lambda_2 \succ \ldots \succ \lambda_r \succ 0} \frac{1}{\lambda_1^{n_1} \ldots \lambda_r^{n_r}},$$

with each $\lambda_i = m_i \tau + n_i$ and $\succ$ denoting a predefined ordering.

These series converge when all indices $n_j$ satisfy a minimal admissibility condition, typically $n_j \geq 2$. They exhibit modular transformation properties, serving as modular forms of specified weights or as components in the expansion of modular forms.

2. Fourier Expansion and Connection to Multiple Zeta Values

The Fourier expansion of multiple Eisenstein series relies heavily on the decomposition of series into constant terms and $q$-series, which are linked to multiple zeta values (MZVs). For example, a typical multiple Eisenstein series can be expanded as:

$$G_{n_1, \ldots, n_r}(\tau) = \zeta(n_1, \ldots, n_r) + \sum \text{MZV terms} \cdot g(\tau),$$

where $\zeta(n_1, \ldots, n_r)$ is a multiple zeta value and the $g(\tau)$ functions encode the $q$-series component.

This Fourier expansion demonstrates how MES act as $q$-analogues of MZVs, bridging modular forms with MZVs by setting $q \to 1$, a process that recovers the zeta values. The inclusion of q-analogues provides a natural way to explore relations between MZVs and modular forms.

3. Regularization Techniques

The definition and paper of MES require careful handling of convergence issues, especially when indices include 1. Regularization techniques such as "stuffle" (quasi-shuffle) and "shuffle" regularization have been developed to handle divergent series parts. The stuffle regularization, for instance, modifies the multiple zeta and Eisenstein values based on algebraic operations in quasi-shuffle algebras, ensuring consistent algebraic properties across the entire MES.

A formal algebraic structure, often leveraging Hopf algebras, helps manage these regularizations and maintain the algebraic consistency of the MES, akin to structures known from multiple zeta value studies.

4. Algebraic Structures and Double Shuffle Relations

The algebraic structure of multiple Eisenstein series displays a fascinating parallel to the double shuffle relations observed in MZVs. These are additional relations that stem from the iterated sum (stuffle) and iterated integral (shuffle) representations of the series. MES have been shown to satisfy similar kinds of relations, significantly enriching the algebraic landscape.

This underpins a robust connection between MES and MZVs, as MES can be seen as formal algebras where duality and symmetry operations (like the "swap" operation) contribute to forming a consistent mathematical framework that embraces both the iterative nature of the values and their modularity properties.

5. Applications and Implications

Multiple Eisenstein series extend their utility into various fields:

• Number Theory: MES provide insights into higher-dimensional aspects of modular forms and their L-values, contributing to understanding periods and special values of L-functions.
• Algebraic Combinatorics: They suggest new combinatorial identities related to partitions and divisor sum functions, as seen in works refining classical results of Ramanujan.
• Mathematical Physics: In contexts like string theory, MES relate to modular graph functions and can be connected to expansion coefficients of modular forms, impacting the understanding of effective string actions and other physical models.

In conclusion, the paper of multiple Eisenstein series is a multifaceted exploration, drawing tools from and contributing to a broad range of mathematical disciplines, including the deep algebraic structures governing MZVs, the analysis of modular forms, and nuanced aspects of number theory and theoretical physics.