Self-Duality of Quasi-Supersingular Group Schemes

Definitions and mathematical statement

Let $p$ be a prime and let $n\geq 3$ be an integer. The supplied statement denotes a ring of Witt vectors by $W$, but does not specify its coefficient field or the hypotheses on that field. Write $k$ for the intended residue field and $\overline G=G\times_{\operatorname{Spec}W}\operatorname{Spec}k$ for the special fiber. The choice of $W$ and $k$ remains an unresolved part of the formulation.

Let $G$ be a finite flat commutative group scheme over $W$, of constant rank $p^{2n}$, annihilated by $p$, so that $[p]_G=0$. Assume that $\overline G$ is quasi-supersingular in the sense intended by the source. The supplied record does not define quasi-supersingularity, so this hypothesis cannot be made explicit from the available evidence.

For a finite locally free commutative group scheme $H$ over a base scheme $S$, its Cartier dual $H^D$ represents the sheaf $\underline{\operatorname{Hom}}_S(H,\mathbb G_{m,S})$. Interpret self-duality as the existence of an isomorphism $H\cong H^D$ over $S$, with no additional compatibility stipulated.

Provisional question from the supplied clean statement. For every such $G$, must there exist isomorphisms \[ G\cong G^D\quad\text{over }W, \qquad \overline G\cong(\overline G)^D\quad\text{over }k? \] The intended coefficient field, the meaning of quasi-supersingularity, and the precise meaning of self-duality require confirmation. Moreover, the supplied original excerpt discusses equivalence of unspecified conditions (1) and (2) in Proposition 1 under an additional self-duality assumption; it does not itself state the universal question above. Thus the correspondence between this question and the original report remains unresolved.

Short English statement

The supplied clean statement asks whether every finite flat commutative group scheme of rank $p^{2n}$, killed by $p$, over the intended Witt-vector ring, with quasi-supersingular special fiber, is self-dual together with its special fiber when $n\geq3$. The base field, quasi-supersingularity hypothesis, and correspondence with the supplied original excerpt need clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30005649$, exact source problem code catalogue entry, title Self-Duality of Quasi-Supersingular Group Schemes. The supplied record labels the statement corrected_verified and reports comparison with the official report; this is supplied provenance, not independent verification here. No individual proposer is supplied.
• Supplied report citation: MFO--RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/42, pp. 2377--2488. https://doi.org/10.4171/owr/2023/42. The record separately gives proposed year 2023 and an owr-year:2023 tag; the relationship to the citation's 2024 date is unspecified.
• Supplied literature source: "Takao, Coleman conjecture report." https://doi.org/10.4171/owr/2023/42.
• Supplied literature source: "Hoshi, finite-flat group-scheme results." https://www.kurims.kyoto-u.ac.jp/~yuichiro/papers_e.html. No specific paper or theorem is identified in the supplied record.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. The record labels the problem partially solved and states that the genus-two case of an associated supersingular torsion-point problem is proved affirmative, while higher-genus Raynaud-type cases remain open strategies and questions in the report. This assessment concerns an associated problem; its precise implications for the self-duality question are not supplied. The record describes this assessment as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.