Symmetric Double Eisenstein Series

• Symmetric double Eisenstein series are q-series that generalize classical Eisenstein series from one to two variables, defined through shuffle regularization and lattice sums.
• They satisfy both shuffle and stuffle relations, establishing deep algebraic connections with modular forms, multiple zeta values, and period polynomials.
• Their q-expansions interpolate between rational double-shuffle solutions and classical multiple zeta values, offering insights into the structure of modular forms and related vector spaces.

Symmetric double Eisenstein series are algebraic and analytic objects in the theory of modular forms and multiple zeta values, constituting a family of $q$-series constructed to satisfy shuffle-type and stuffle-type relations, with an explicit symmetric structure in their indices. They generalize classical Eisenstein series from one to two variables and serve as $q$-analogues interpolating between double-shuffle solutions at $q\to0$ and multiple zeta values as $q\to1$, providing deep connections to the structure of modular forms, double zeta values, and period polynomials. Their definition and properties have direct implications for the structure of spaces of modular forms and the algebraic relations among multiple zeta and elliptic zeta values (Bachmann et al., 2022, Hara et al., 20 Jan 2026, Kaneko et al., 2011).

1. Definition and Construction

Let $\tau$ be in the upper half-plane and $q = e^{2\pi i\tau}$. For integers $r,s \geq 1$, the symmetric double Eisenstein series $G^{\shuffle,S}_{r,s}(\tau)$ are defined via the shuffle-regularized multiple Eisenstein series $G^\shuffle_{k_1,\dots,k_d}(\tau)$, whose constant term is the shuffle-regularized multiple zeta value and whose higher-order terms arise via the Goncharov coproduct–Ihara-law construction:

$G^{\shuffle,S}_{r,s}(\tau) = G^\shuffle_{r,s}(\tau) + (-1)^s\, G^\shuffle_r(\tau) G^\shuffle_s(\tau) + G^\shuffle_{s,r}(\tau).$

For $r,s\geq2$ this series converges conditionally and admits a lattice sum representation:

$G^{\shuffle,S}_{r,s}(\tau) = \lim_{M,N\rightarrow\infty} \sum_{\substack{\lambda_1\prec\lambda_2\ \lambda_i\in(\tau\mathbb{Z}+\mathbb{Z})\cap[-M,N]}} \frac{1}{\lambda_1^r\lambda_2^s},$

where $\prec$ denotes Kontsevich's total order on the lattice outside the origin (Hara et al., 20 Jan 2026).

These series can be interpreted as symmetrized variants of double Eisenstein series, specifically constructed to enforce symmetry in the indices $r$ and $s$, and they can accommodate all $r,s\geq1$ through shuffle-regularization.

2. Algebraic Relations: Shuffle and Stuffle Structures

Symmetric double Eisenstein series satisfy characteristic algebraic identities that generalize the double-shuffle relations known from multiple zeta values:

• Linear Shuffle Relation. For $r,s\geq1$,

$G^{\shuffle,S}_{r,s}(\tau) + G^{\shuffle,S}_{s,r}(\tau) = (-1)^s\, G^{\shuffle,S}_{r+s}(\tau).$

Since $G^{\shuffle,S}_k=2G_k^\shuffle$ for even $k$, and vanishes for $k$ odd, this specializes to

$G^{\shuffle,S}_{r,s} + G^{\shuffle,S}_{s,r} = \begin{cases} 2(-1)^s G^\shuffle_{r+s} & r+s \text{ even},\ 0 & r+s \text{ odd}. \end{cases}$

• Stuffle-Type and Swap Symmetry. In the context of combinatorial Eisenstein series, suitable generating series satisfy both stuffle (harmonic) and shuffle (symmetrility) products, which enforce

$$E_{n_1,n_2}(q) = E_{n_2,n_1}(q)\quad\forall\,n_1,n_2,$$

as well as a swap-functional relation among the generating series:

$G_2(X_1, X_2) = G_2(X_2, X_1-X_2)$

(Bachmann et al., 2022, Hara et al., 20 Jan 2026).

These identities codify the double-symmetry and ensure the analytic objects match the formal double-zeta structure.

3. Vector Spaces and Modular Structures

The vector space spanned by symmetric double Eisenstein series of fixed weight $k$ exhibits a distinct structural dichotomy between even and odd weights:

• For even $k \geq 6$:

$\mathcal{DE}^S_k = M_k(\mathrm{SL}_2(\mathbb{Z})) \oplus \bigl\langle q\,\frac{d}{dq}\,\widetilde{G}^\shuffle_{k-2} \bigr\rangle_{\mathbb{Q}},$

where $M_k$ is the space of weight $k$ modular forms, and the extra generator is a $q$-derivative of the Eisenstein series of weight $k-2$ (Hara et al., 20 Jan 2026).

The dimension is explicitly:

$$\dim \mathcal{DE}^S_k = \dim M_k + 1 = \left\lfloor \frac{k+4}{4}\right\rfloor - \left\lfloor \frac{k-2}{6}\right\rfloor.$$

• For odd $k\geq3$:

$$\dim_{\mathbb{Q}} \mathcal{DE}^S_k = \left\lfloor \frac{k}{3} \right\rfloor,$$

and an explicit basis is given by:

$\left\{\,\widetilde G^{\shuffle,S}_{j, k-j}(\tau)\mid j=1,\dots,\lfloor k/3\rfloor\,\right\}.$

This construction provides a concrete link between automorphic forms and the algebraic properties of multiple Eisenstein series (Hara et al., 20 Jan 2026).

4. Relation to Modular Forms, Level 2 Identities, and Period Polynomials

Analogs of symmetric double Eisenstein series for modular forms of level $2$ have been studied, notably by Kaneko and Tasaka. Definitions involve parity-twisted double series:

$$G_{r,s}^{\mathrm{ev,ev}}(\tau),\quad G_{r,s}^{\mathrm{ev,od}}(\tau),\quad G_{r,s}^{\mathrm{od,od}}(\tau),$$

with explicit lattice sums over $\lambda, \mu$ in discrete sublattices. These series satisfy stuffle and shuffle-type relations, which are explicitly symmetric under the exchange $r\leftrightarrow s$, and their function equations encode double-shuffle structures at the modular level.

For even $k\geq 4$, the span of $G_{r,k-r}$ is of dimension $k/2-1$ over $\mathbb{Q}$ and contains both the Eisenstein and cusp spaces for $\Gamma_0(2)$. The imaginary parts assemble period-polynomial matrices with right-kernel matching the space of even period-polynomials for $\Gamma_0(2)$, paralleling the Eichler–Shimura isomorphism (Kaneko et al., 2011).

5. Fay–Shuffle Relations and Elliptic Double Zeta Values

The linear shuffle space $LSh^{(2)}_w$ consists of homogeneous polynomials of degree $w$ satisfying depth-$2$ linear shuffle relations. This space coincides, via explicit $\mathrm{GL}_2$-actions, with the Fay–shuffle space $FSh_w^\mathrm{pol}$, which encodes relations satisfied by elliptic double zeta values:

• For even $w$, the odd-period-polynomial space $W_w^\odd$ is contained in $LSh^{(2)}_w$.
• For odd $w$, $FSh^\mathrm{pol}_w \cong LSh^{(2)}_w$ via explicit isomorphisms.

This duality means the algebraic relations (linear shuffle) determining symmetric double Eisenstein series mirror the Fay–shuffle structure for elliptic double zeta values, reinforcing deep connections between double Eisenstein series, multiple zeta values, and their elliptic generalizations (Hara et al., 20 Jan 2026).

6. $q$-Expansion, Modularity, and Limit Behaviors

For all indices, symmetric double Eisenstein series admit $q$-expansions with rational coefficients determined by differences of divisor-type sums. In numerous cases (e.g., $n_1=n_2$ even, $n_1+n_2$ even, and $n_i>1$), these symmetrized series correspond to (quasi-)modular forms of weight $n_1+n_2$:

$$\mathcal{E}^{\mathrm{sym}}_{n_1,n_2}(q) = E_{n_1,n_2}(q) + E_{n_2,n_1}(q) - \delta_{n_1,n_2}E_{n_1+n_2}(q)$$

where for $n_1\neq n_2$, $\mathcal{E}^{\mathrm{sym}}_{n_1,n_2}=2E_{n_1,n_2}$, and for $n_1=n_2$, $\mathcal{E}^{\mathrm{sym}}_{n,n}=2E_{n,n}-E_{2n}$.

These series interpolate between a rational double-shuffle solution at $q\to0$ (limit gives a rational value) and classical double zeta values at $q\to1$ (limit recovers regularized multiple zeta values): $\lim_{q \to 0} E_{n_1,n_2}(q) = \beta(n_1,n_2)$

$$\lim_{q \to 1} (1-q)^{n_1+n_2}\, E_{n_1,n_2}(q) = \zeta^*(n_1,n_2)$$

representing a $q$-analogue spanning the combinatorial and modular world (Bachmann et al., 2022).

7. Concrete Expressions and Examples

Classical Eisenstein series $E_k(\tau)=G^\shuffle_k(\tau)$ satisfy the following representations in low weights:

• Weight 4:

$G^{\shuffle,S}_{2,2}=4G_4^\shuffle-G_2'(\tau),\quad G^{\shuffle,S}_{1,3}=\tfrac14 G_2'(\tau)-G_4^\shuffle$

and $E_4(\tau)=G_4^\shuffle(\tau)=\frac14(G^{\shuffle,S}_{2,2}+G_2'(\tau))$.

• Weight 6:

$G^{\shuffle,S}_{3,3}=2G_3^\shuffle\,G_3^\shuffle-G_6^\shuffle=-G_6^\shuffle, \quad E_6(\tau)= -G^{\shuffle,S}_{3,3}(\tau)$

with, e.g., $G^{\shuffle,S}_{2,4}=2E_2E_4-E_6$ and thus $E_6=2E_2E_4-G^{\shuffle,S}_{2,4}$.

• Weight 8:

The four symmetric double series are $G^{\shuffle,S}_{1,7}, G^{\shuffle,S}_{2,6}, G^{\shuffle,S}_{3,5}, G^{\shuffle,S}_{4,4}$, with, e.g., $G^{\shuffle,S}_{2,6}=2E_2E_6-E_8$ and $G^{\shuffle,S}_{4,4}=2E_4^2-E_8$.

For odd weights, the basis elements $\widetilde G^{\shuffle,S}_{j,k-j}$ for $j=1,\dots,\lfloor k/3\rfloor$ form an explicit spanning set (Hara et al., 20 Jan 2026). These formulae illustrate the central role of double Eisenstein symmetries in decomposing modular forms and constructing algebraic bases.

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Symmetric double Eisenstein series thus sit at the confluence of multiple zeta values, the algebraic theory of modular forms, double-shuffle algebraic relations, and the emerging structure of elliptic multiple zeta values. They provide a framework that unifies and extends classical and modern objects in arithmetic geometry, with their algebraic and analytic properties encoded by explicitly symmetric combinations and linked to both $q$-series and period polynomials of modular groups (Bachmann et al., 2022, Kaneko et al., 2011, Hara et al., 20 Jan 2026).