Parametrization of Supersingular Cohomology Modules

Definitions and mathematical statement

The supplied record uses $H_s^{(1)}$ for an algebraic object acting on modules and $B_s^b$ for an object whose cohomology contains modules described as supersingular. It does not define these objects, the indices $s,b$, the parameter $t$, or supersingularity in this setting.

Source question. Can the supersingular $H_s^{(1)}$-modules appearing in the cohomology of $B_s^b$ be parametrized by $t$?

Unresolved formulation. A precise parametrization statement requires the definition and coefficient ring of $H_s^{(1)}$, the construction of $B_s^b$, the cohomology theory and coefficients, the relevant degrees, and the meaning of a module appearing in cohomology. It also requires the domain of $t$, the permitted values and dependence of $s,b$, the equivalence relation on modules, and the intended meaning of parametrization. In particular, the supplied text does not specify whether parametrization means a bijection with isomorphism classes, a surjective indexing, or another correspondence. No such interpretation is imposed here.

Supplied qualification. The dated curation labels the problem partially solved and states that the report proves realization with multiplicity one for simple supersingular modules in dimensions $n\leq 3$, while no general parametrization by $t$ is established. The record does not define $n$, state the hypotheses of that realization result, or explain its precise relationship to the question above. This is a supplied literature assessment, not an independently verified theorem statement or a new assessment of the problem's status.

Short English statement

The question asks whether the supersingular $H_s^{(1)}$-modules occurring in the cohomology of $B_s^b$ can be indexed by $t$. The supplied record leaves the objects, parameter domain, and required correspondence undefined, so a fully precise formulation requires additional source context. Its dated curation reports partial progress for simple supersingular modules in dimensions $n\leq 3$.

Sources

• Local UnsolvedMath registry: permanent ID $30003880$, source code catalogue entry, title Parametrization of Supersingular Cohomology Modules; supplied record from the Oberwolfach Reports Open Problems collection. Repository registry locations: data/unsolvedmath/problems.json and catalog/problems.json.
• Supplied report citation: Algebraische Zahlentheorie (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/30, pp. 1785--1844. Supplied source URL and literature-source link, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2018/30. The collection describes these reports as documenting mathematical workshops and their open-problem discussions.
• Literature checked at 2026-08-21: supplied dated curation, not independently verified. Its status is "partially solved"; it reports realization with multiplicity one for simple supersingular modules in dimensions $n\leq 3$, but no established general parametrization by $t$. The background expressly requests independent verification of time-sensitive status and cited claims.
• Attribution and verification qualifications: no individual proposer is supplied. The proposed year and workshop-year tag are $2018$, whereas the supplied report citation says $2019$. The registry labels the statement corrected_verified and says it was checked against the official report and preserves its hypotheses, notation, and target. These are supplied registry assertions; no independent source inspection is claimed here.