Steenrod-Closed Parameter Ideals at Every Prime

Definitions and mathematical statement

For a prime $p$, let \[ R_p={\mathbb{F}}_p[a,b]^{\operatorname{SL}_2(p)}, \] where ${\mathbb{F}}_p$ is the field with $p$ elements and $\operatorname{SL}_2(p)$ is the group of determinant-one $2\times2$ matrices over ${\mathbb{F}}_p$. The superscript denotes the subring fixed by the intended group action. The supplied record does not explicitly specify that action, the grading of $a,b$, or the Steenrod action.

For a finitely generated connected graded ${\mathbb{F}}_p$-algebra $R$ of Krull dimension $d$, a homogeneous parameter ideal means an ideal \[ I=(f_1,\ldots,f_d),\qquad \deg(f_i)>0, \qquad \dim_{{\mathbb{F}}_p}(R/I)<\infty. \] For a specified action of the mod-$p$ Steenrod algebra $\mathcal A_p$ on $R$, such an ideal is Steenrod-closed if \[ \theta(I)\subseteq I\qquad\text{for every }\theta\in\mathcal A_p. \] These definitions explain the terminology; the precise conventions intended in the cited theorem must be confirmed.

The supplied conjecture asks for a classification, for every prime $p$, of all Steenrod-closed parameter ideals of $R_p$, with no ideals beyond the fibered, twisted, and mixed families extending those identified for $p=3$. Its original formulation is: "We conjecture that Theorem 5 extends to every prime p."

The supplied record does not contain Theorem 5, formulas or parameter restrictions for the three families, or their proposed extensions to other primes. Consequently, this is an explicitly incomplete formulation of the classification target: an exact exhaustive list, including its boundary cases, cannot be reconstructed from the supplied evidence alone.

Short English statement

For each prime $p$, classify the parameter ideals in the stated invariant ring that are preserved by Steenrod operations. The proposed classification should extend the fibered, twisted, and mixed families described for $p=3$. The missing theorem, family definitions, and action conventions prevent a fully explicit statement.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30006660$, title Steenrod-Closed Parameter Ideals at Every Prime, exact source problem code catalogue entry.
• Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, pp. 541--610, DOI 10.4171/owr/2026/9. Supplied report attribution: Stephan--Ruping--Yalcin, Oberwolfach Report 9/2026. https://doi.org/10.4171/owr/2026/9. The record attributes the conjecture to an extension of Theorem 5, whose text is not supplied; no individual proposer is specified.
• Ruping--Stephan, Steenrod-closed parameter ideals. https://arxiv.org/abs/2503.14349.
• Supplied literature assessment dated 2026-08-22: the classification is reported proved for $p=3$, its extension to every prime is reported conjectural, and no later proof was located in that assessment. This is supplied dated curation, not independently verified literature status.
• The registry labels the problem open and its statement corrected_verified, and reports that the reconstruction was checked against the official Oberwolfach report. These are supplied provenance and status claims, not independent verification in this draft.