Relations Among Multiple Weierstrass Function Spaces

Definitions and mathematical statement

The supplied record calls $\mathcal P_k$ spaces of multiple Weierstrass functions and $\mathcal E_k$ Eisenstein-series spaces. It does not define these spaces, their coefficient field, their ambient function space, or their grading conventions. It also does not define the operators $C_n$, the unindexed spaces $\mathcal E$ and $\mathcal P$, the product $\mathcal E\cdot\mathcal P$, or the symbols $\Delta$ and $\wp_2$. These symbols are therefore retained as source notation without assigning additional meanings.

The reconstructed statement supplied by the registry asks for \[ C_n(\mathcal P_{k+n})=\mathcal E_k \qquad\text{for all }k\geq0\text{ and }n\geq3, \] and \[ \frac{d}{dz}(\mathcal E\cdot\mathcal P)\subseteq\mathcal P. \] It further asks for \[ \mathcal E_k\cap\mathcal P_k=\Delta\mathcal E_{k-12}, \qquad \mathcal E_k\cap(\mathcal P_{k-2}\wp_2) =\Delta\mathcal E_{k-14}\wp_2. \] The admissible index sets for $k,n$ are not explicitly specified. No range of $k$ is separately supplied for the intersection identities, and no convention is supplied for spaces with negative indices. The domain of $z$, the meaning of differentiation on these spaces, and the common ambient space needed for the intersections are also unspecified. Consequently, these formulas preserve the supplied target but do not yet constitute a self-contained mathematical conjecture.

The original extracted passage describes a conjecture as an analogue of an unspecified Theorem 4 and refers to a further conjecture concerning $C_0(\mathcal P_k)$ and $C_2(\mathcal P_k)$. The actual conjectural formulas are absent from that passage. It states that dimension conjectures for $\mathcal E_k$ had been given by Okounkov [5] and Bachmann--Kühn [1], and that determining dimension conjectures for the indicated $C_0$ and $C_2$ spaces would yield a dimension conjecture for $\mathcal P_k$. It also states that such conjectures were not available at the time of the report. These are supplied source assertions, not independently verified conclusions.

For completeness, the extracted passage supplies the following numerical table. Its entries are retained as reported data; their computational or conjectural status is not specified. \[ \begin{array}{c|rrrrrrrr} k&2&3&4&5&6&7&8&9\\\hline \dim\mathcal P_k&1&1&2&3&5&7&12&17\\ \dim\mathcal E_k&1&1&2&3&4&7&9&15\\ \dim C_0(\mathcal P_k)&0&0&1&1&2&2&4&5\\ \dim C_2(\mathcal P_{k+2})&1&1&2&2&4&5&8&11 \end{array} \] \[ \begin{array}{c|rrrrrrr} k&10&11&12&13&14&15&16\\\hline \dim\mathcal P_k&27&39&59&88&130&194&286\\ \dim\mathcal E_k&21&32&47&70&104&153&228\\ \dim C_0(\mathcal P_k)&8&11&16&24&33&51&72\\ \dim C_2(\mathcal P_{k+2})&15&24&33&50&71&104&153 \end{array} \]

Short English statement

The supplied reconstructed claim relates coefficient-operator images, differentiation, and intersections of spaces of multiple Weierstrass functions and Eisenstein series. Essential definitions and index conventions are missing, and the original extract omits the conjectural formulas. The registry labels the problem open; this draft preserves that supplied label without independently verifying it.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006594$, exact source code catalogue entry, title "Relations Among Multiple Weierstrass Function Spaces," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2026 and supplies no proposer attribution.
• Multiple Zeta Values and Modular Forms (2026), Oberwolfach Reports, pp. 433--486, DOI 10.4171/owr/2026/7: https://doi.org/10.4171/owr/2026/7. This is the supplied source citation and source URL, also listed under literature sources as "Oberwolfach Report 7/2026."
• Supplied literature assessment, checked exactly 2026-08-22: the extracted record truncates the conjecture, and the underlying report is said to state that the needed dimension conjectures for the indicated spaces were unavailable, leaving the claim unresolved. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims; the report has not been independently consulted for this draft.
• The original passage attributes dimension conjectures for $\mathcal E_k$ to Okounkov [5] and Bachmann--Kühn [1]. Full bibliographic details for these numbered references, and the content of the referenced Theorem 4, are not supplied.