Symmetric Multiple Eisenstein Series in Modular Forms

• Symmetric multiple Eisenstein series are holomorphic q-series that generalize classic Eisenstein series and symmetric multiple zeta values.
• Their construction employs shuffle-regularization and Hopf algebra techniques to establish essential algebra homomorphism properties.
• They reveal deep connections with modular forms, period polynomials, and elliptic zeta values, offering a modular framework for multiple zeta relations.

Symmetric multiple Eisenstein series are a class of holomorphic q-series on the modular group, motivated by the theory of symmetric multiple zeta values and possessing deep connections to modular forms, period polynomials, and elliptic zeta values. They generalize both classical Eisenstein series and the symmetric multiple zeta values introduced by Kaneko and Zagier, and are constructed so as to satisfy distinguished shuffle-type relations within a noncommutative algebraic framework. Recent work by Hara, Sakugawa, and Tasaka details their definition, algebraic properties, structural results, and interplay with modular forms and elliptic zeta relations (Hara et al., 20 Jan 2026).

1. Construction and Regularization of Symmetric Multiple Eisenstein Series

Let $\tau$ lie in the complex upper half-plane, and consider the lattice $\Lambda = \mathbb{Z}\tau + \mathbb{Z}$. For strictly positive integers $k_1, \dots, k_d \geq 2$, the (depth-$d$) multiple Eisenstein series is defined as the iterated, conditionally convergent sum

$$G_{k_1,\ldots,k_d}(\tau) = \lim_{M\to\infty} \lim_{N\to\infty} \sum_{\substack{0<\lambda_1<\cdots<\lambda_d \ \lambda_i\in(-M\tau-N,\dotsc, M\tau+N)} } \lambda_1^{-k_1}\cdots \lambda_d^{-k_d}$$

with respect to a total order $<$ on $\Lambda$ determined by a positive cone $P$. This generalizes the classical Eisenstein series ($d=1$).

Multiple Eisenstein series $G_{k_1,\ldots,k_d}(\tau)$ do not a priori satisfy the shuffle product relations characteristic of multiple zeta values. To remedy this, a shuffle-regularized version $G^{\text{shuffle}}_{k_1, \ldots, k_d}(\tau)$ is defined, providing an algebra homomorphism from the shuffle algebra on noncommutative generators $e_k$. This construction utilizes generating functions $H^{(d)}$ satisfying (quasi-)shuffle or stuffle relations, and their Taylor expansions give rise to $g^{\text{shuffle}}_{k_1, \ldots, k_d}(\tau)$, which do satisfy shuffle product rules. The regularization is made precise using Hopf algebra techniques, notably Goncharov's coproduct.

Motivated by the notion of symmetric multiple zeta values, the symmetric shuffle-regularized multiple Eisenstein series is defined as

$$G^{\text{shuffle},S}_{k_1,\ldots,k_d}(\tau) = \sum_{j=0}^d (-1)^{k_{j+1}+\cdots+k_d} G^{\text{shuffle}}_{k_1,\ldots,k_j}(\tau)\cdot G^{\text{shuffle}}_{k_d,\ldots,k_{j+1}}(\tau)$$

Alternatively, this can be expressed as a sum over strictly ordered indices with respect to the total Kontsevich order on $\Lambda\setminus\{0\}$, which extends the lattice order to a cyclic, infinite total order identifying $+\infty$ with $-\infty$.

2. Linear Shuffle Relations

Symmetric multiple Eisenstein series satisfy a fundamental class of algebraic identities: linear shuffle relations. For a decomposition $d = j + (d-j)$ and the shuffle product $*$ in the noncommutative algebra $\mathbb{Q}\langle e_k \rangle$, the following holds: $$G^{\text{shuffle},S}( e_{k_1, \ldots, k_j} * e_{k_{j+1}, \ldots, k_d}) = (-1)^{k_{j+1}+\cdots+k_d} G^{\text{shuffle},S}( e_{k_1, \ldots, k_j, k_d, \ldots, k_{j+1}})$$ This relation is most naturally phrased via generating series in commuting variables $x_1, \dotsc, x_d$: $$F^{(d)}(x_1, \ldots, x_d) = \sum_{k_1, \ldots, k_d \geq 1} G^{\text{shuffle},S}_{k_1, \ldots, k_d}(\tau) x_1^{k_1-1} \cdots x_d^{k_d-1}$$ with corresponding identities involving application of shuffle operators $sh_j'$. The only obstruction to $G^{\text{shuffle},S}$ being an algebra homomorphism arises from these specific “twisted” linear shuffle relations, which mirror the algebraic constraints found for symmetric multiple zeta values.

3. Structure and Dimensions of Spaces Spanned by Symmetric Double Eisenstein Series

Define, for weight $k\geq 3$,

$$\mathrm{DE}_k^S = \mathrm{span}_{\mathbb{Q}} \{\, \widetilde G^{\text{shuffle},S}_{r,s}(\tau)\ :\ r+s = k,\ r,s\geq1 \,\}, \quad\text{with}\ \widetilde G^{\text{shuffle},S}_{r,s} = G^{\text{shuffle},S}_{r,s}/(2\pi i)^k$$

The structure of $\mathrm{DE}_k^S$ depends on the parity of $k$:

• Even weight $k\geq 6$:

  $$\mathrm{DE}_k^S = M_k(\mathrm{SL}_2(\mathbb{Z})) \oplus q\, \frac{d}{dq} G^{\text{shuffle}}_{k-2}$$

  with explicit dimension

  $$\dim\, \mathrm{DE}_k^S = \left\lfloor \frac{k+4}{4} \right\rfloor - \left\lfloor \frac{k-2}{6} \right\rfloor$$

  The proof relies on shuffle and stuffle arguments, explicit formulas for $G^{\text{shuffle},S}_{r,s}(\tau)$, and known decompositions for classical modular forms.

• Odd weight $k\geq 3$:

  $$\left\{\, \widetilde G^{\text{shuffle},S}_{j,k-j} : 1 \leq j \leq \left\lfloor \frac{k}{3} \right\rfloor\, \right\}$$

  is a $\mathbb{Q}$-basis, so

  $$\dim\, \mathrm{DE}_k^S = \left\lfloor \frac{k}{3} \right\rfloor$$

  The argument uses the linear shuffle relations, an $\mathfrak{S}_3$-representation argument for dimension, and linear independence established via a rank argument on a matrix of binomial coefficients extracted from $q$-expansions.

4. Fay–Shuffle Correspondence and Elliptic Double Zeta Values

Elliptic double zeta values, as introduced by Matthes, satisfy two classes of linear relations: classical shuffle, and Fay–shuffle stemming from Fay’s identity for elliptic integrals. The space of polynomial Fay–shuffle relations in weight $w$,

$$\mathrm{FSh}_w^{\mathrm{pol}} = \{ Q(X,Y) \in \mathbb{Q}[X,Y],\ \deg Q = w \mid Q|_{1-\epsilon'}=0,\ Q|_{1+\gamma'+(\gamma')^2}=0 \}$$

has dimension $\left\lfloor (w+2)/3 \right\rfloor$. There is an explicit correspondence between these Fay–shuffle spaces and the linear shuffle space $L\mathrm{Sh}_w^{(2)}$ satisfied by the symmetric Eisenstein series:

• For $w$ even, the odd period polynomial subspace $W_w^{\text{od}}\subseteq L\mathrm{Sh}_w^{(2)}$.
• For $w$ odd, there is a $\mathbb{Q}$-linear bijection:

  $$\mathrm{FSh}_w^{\mathrm{pol}} \simeq L\mathrm{Sh}_w^{(2)}$$

achieved by transformations involving generators $\epsilon,\gamma,\delta\in GL_2(\mathbb{Q})$ of $S_3$-type symmetries, with linear conditions checked via explicit substitution.

This correspondence elucidates the parallel between the algebraic relations among symmetric double Eisenstein series and the structure of the elliptic double zeta value relations.

5. Modular Forms as Sums of Symmetric Triple Eisenstein Series

Every modular form for $\mathrm{SL}_2(\mathbb{Z})$ with rational coefficients and weight $k\geq 4$ can be written as a finite $\mathbb{Q}$-linear combination of symmetric triple Eisenstein series: $$f = \sum_{r + s + t = k} c_{r,s,t}\, \widetilde{G}^{\text{shuffle},S}_{r,s,t}(\tau)$$ Defining $\mathrm{TE}_k^S$ as the span of all depth-3 symmetric Eisenstein series in weight $k$, the argument leverages the fact that products $G_{2j} G_{k-2j}$, $G_k$, and $q \frac{d}{dq} G_{k-2}$ belong to $\mathrm{TE}_k^S$ and that these are known to span all modular forms. Thus, symmetric triple Eisenstein series suffice to generate the entire space of modular forms of a given weight.

Taking the limit as $\tau\to i\infty$, the constant terms give rise to linear relations among symmetric multiple zeta values $\zeta^{\text{shuffle},S}(r,s,t)$, in accordance with the modular relations originally posited by Kaneko–Zagier.

6. Implications and Connections

The introduction and systematic study of symmetric multiple Eisenstein series reveals several significant structures:

• Linear shuffle relations satisfied by these series directly parallel those found in the theory of symmetric multiple zeta values.
• Their expansion space in depth two and three aligns with the structure and dimension formulas of modular forms and the corresponding period polynomial spaces.
• They establish the first concrete modular framework to systematically derive Kaneko–Zagier type relationships for multiple zeta values, particularly the “modular phenomena” conjectured in higher weight and depth (Hara et al., 20 Jan 2026).
• The correspondence with Fay–shuffle relations for elliptic zeta values indicates a unity between the algebraic structures of modular forms, period polynomials, and elliptic and classical multiple zeta values.

A plausible implication is that symmetric multiple Eisenstein series act as a bridge between classical modular form theory and the arithmetic of multiple zeta values, providing a unifying perspective for further study of their special values, $q$-expansions, and motivic or cohomological interpretations.

7. Summary Table

Structure	Object/Invariants	Dimension/Decomposition
Symmetric double Eisenstein	$\mathrm{DE}_k^S$	Even $k$: $$\dim = \lfloor\frac{k+4}{4}\rfloor-\lfloor\frac{k-2}{6}\rfloor$$; Odd $k$: $\dim = \lfloor k/3\rfloor$
Fay–shuffle polynomials	$\mathrm{FSh}_w^{\mathrm{pol}}$	$\dim = \lfloor(w+2)/3\rfloor$
Modular forms in triple series	$\mathrm{TE}_k^S$	All cusp forms admit decomposition by symmetric triple series

Further technical details, Fourier expansions, and coproduct computations are elaborated in Hara–Sakugawa–Tasaka’s paper (Hara et al., 20 Jan 2026).