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The dimension conjecture for multiple Eisenstein series

Let Ek\mathcal{E}_k be the weight-kk component of the algebra of multiple Eisenstein series. Define the generating series for modular forms and cusp forms by

M(X)=∑k≥0dim⁡MkXk=1(1−X4)(1−X6),S(X)=∑k≥0dim⁡SkXk=X12(1−X4)(1−X6).\mathsf{M}(X)=\sum_{k\geq0}\dim\mathcal{M}_kX^k=\frac{1}{(1-X^4)(1-X^6)},\qquad \mathsf{S}(X)=\sum_{k\geq0}\dim\mathcal{S}_kX^k=\frac{X^{12}}{(1-X^4)(1-X^6)}.

Here Mk\mathcal{M}_k and Sk\mathcal{S}_k are respectively the spaces of modular forms and cusp forms of weight kk for SL⁡2(Z)\operatorname{SL}_2(\mathbb{Z}). The dimension conjecture for multiple Eisenstein series. The weight generating series should satisfy

∑k≥0dim⁡QEkXk=M(X)11−X2−O(X)+2S(X)=11−X2−X3−X4−X5+X8+X9+X10+X11+X12.\sum_{k\geq0}\dim_\mathbb{Q}\mathcal{E}_kX^k=\mathsf{M}(X)\frac{1}{1-X^2-\mathsf{O}(X)+2\mathsf{S}(X)} =\frac{1}{1-X^2-X^3-X^4-X^5+X^8+X^9+X^{10}+X^{11}+X^{12}}.

This is an analogue of Zagier's dimension conjecture and is motivated by expected relations with associated graded spaces of q-analogues of multiple zeta values. Its validity is open.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The dimension conjecture for multiple Eisenstein series

    For each weight kk, let

    Ek=Span⁡Q{G~k1,…,kr(τ)∣r≥1, k1,…,kr≥2, k1+⋯+kr=k},\mathcal{E}_k=\operatorname{Span}_{\mathbb{Q}}\{\widetilde{G}_{k_1,\dots,k_r}(\tau)\mid r\geq 1,\ k_1,\dots,k_r\geq 2,\ k_1+\cdots+k_r=k\},

    and let E=∑k≥2Ek\mathcal{E}=\sum_{k\geq 2}\mathcal{E}_k. The dimension conjecture for multiple Eisenstein series. The Hilbert–Poincaré series of these spaces is

    ∑k≥0dim⁡EkXk=11−X2−X3−X4−X5+X8+X9+X10+X11+X12.\sum_{k\geq 0}\dim \mathcal{E}_k X^k=\frac{1}{1-X^2-X^3-X^4-X^5+X^8+X^9+X^{10}+X^{11}+X^{12}}.

    This conjecture combines conjectures cited by the authors and predicts the dimensions, hence the number of independent relations, among multiple Eisenstein series. The paper presents it as a conjecture and compares its predicted relations with those obtained from the shuffle antipode.

    source: Hayato Kanno and Katsumi Kina, “Multiple -Functions and Their Applications”, arXiv:2507.14118 (2026).

References

Primary source

Henrik Bachmann, Hayato Kanno and Takumi Maesaka, “Relations and Derivatives of Multiple Eisenstein Series”, arXiv:2602.08176 (2026).

dimension-conjecturesmodular-formsmultiple-eisenstein-series
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