Descent of Freeness for Orthogonal Modular Form Algebras

Definitions and mathematical statement

For a subgroup $H$ of the orthogonal group appearing in the source, let $M_*(H)$ denote its graded algebra of modular forms, with multiplication given by multiplication of forms and grading by weight. Under the polynomial-algebra interpretation of "free," a graded algebra $A$ over a coefficient field $K$ is free if there are homogeneous elements $f_i$, indexed by a set $I$, such that the graded homomorphism \[ K[X_i\mid i\in I]\longrightarrow A,\qquad X_i\longmapsto f_i, \] is an isomorphism, where $\deg X_i=\deg f_i$. The source does not specify the coefficient field, the allowed weights, whether finitely many generators are required, or the precise modular-form conventions.

The object $M$ and the notation $O^+(M)$ are also not defined in the supplied record: in particular, the lattice hypotheses, signature, modular domain, and meaning of the superscript $+$ remain unspecified. For fixed compatible choices of these data, the supplied question is as follows.

Let $\Gamma\leq O^+(M)$ and $\Gamma'\leq\Gamma$ satisfy \[ [O^+(M):\Gamma]<\infty, \qquad [\Gamma:\Gamma']<\infty, \] and suppose that $\Gamma$ is generated by reflections. Must the implication \[ M_*(\Gamma')\text{ is a free graded algebra} \quad\Longrightarrow\quad M_*(\Gamma)\text{ is a free graded algebra} \] hold for every such pair? With compatible modular-form conventions, the subgroup inclusion gives $M_*(\Gamma)\subseteq M_*(\Gamma')$. No normality assumption on $\Gamma'$, and no assumption that $\Gamma'$ is generated by reflections, appears in the supplied statement. The missing conventions above must be resolved before this becomes a fully specified conjecture.

Short English statement

Does polynomial freeness of the modular-form algebra of a finite-index subgroup $\Gamma'$ imply polynomial freeness for its containing group $\Gamma$, when $\Gamma$ is a finite-index reflection-generated subgroup of $O^+(M)$? The supplied record leaves the underlying orthogonal and modular-form conventions unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004615$, source problem code catalogue entry, title "Descent of Freeness for Orthogonal Modular Form Algebras," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2021 and does not identify a proposer.
• Supplied report citation: Moduli spaces and Modular forms (2022), Oberwolfach Reports, pp. 301--359, DOI 10.4171/owr/2021/6. https://doi.org/10.4171/owr/2021/6. The citation's title year is 2022, while the DOI and proposed-year metadata indicate 2021; this discrepancy is unresolved here.
• Supplied literature reference: "Wang, classification of free algebras." https://arxiv.org/abs/2006.02291. The supplied curation associates the question with Wang's conjecture and says that this classification paper formulates the conjecture without proving it; these claims have not been independently verified.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and lists the report and Wang reference above as checked sources. This is supplied dated curation, not an independently verified literature assessment. The registry also labels the statement corrected_verified; that label records supplied editorial review and does not resolve the missing mathematical conventions identified here.