Orbit Structure of Generic A-Number-One Loci

Definitions and mathematical statement

For a prime $p$ and a $p$-divisible group $H$ over an algebraically closed field $k$ of characteristic $p$, the usual $a$-number is \[ a(H)=\dim_k\operatorname{Hom}_{k\text{-group schemes}}(\alpha_p,H[p]), \qquad \alpha_p=\ker(F:\mathbb G_{a,k}\longrightarrow\mathbb G_{a,k}), \] where $F$ is Frobenius. This explains the standard numerical invariant without selecting the unspecified parameter space in the source.

The supplied question concerns strata associated with an affine Deligne--Lusztig invariant denoted $f_g$, and an action denoted $J_b(F)$. It asks whether the locus with generic $a$-number $1$ is a single orbit of strata after removing the hypothesis that $b$ is basic.

The intended equality can be expressed conditionally as follows. Once an ambient parameter space $\mathscr X$, its collection $\mathscr S$ of $f_g$-strata, the action of $J_b(F)$ on those strata, and the intended locus $\mathscr L_1\subseteq\mathscr X$ have been specified, ask whether \[ \exists S_0\in\mathscr S\qquad \mathscr L_1=\bigcup_{j\in J_b(F)}j\cdot S_0 \] continues to hold for nonbasic $b$ in the intended class of data. Here the right-hand side is the union of the strata belonging to one orbit; it does not assert transitivity on individual points.

Unresolved qualifications. The record does not define $f_g$, the ambient parameter space, the strata, $F$, $b$, the underlying group, or the action. It also does not specify whether generic $a$-number $1$ is evaluated on each stratum, each irreducible component, or another family. Thus $\mathscr L_1$ and the class of data quantified over remain unspecified, and the displayed equality is a template for the supplied question, not a fully determined conjecture. The treatment of an empty locus is also unspecified. The supplied curation reports that the equality is proved when $b$ is basic and labels the extension open; these are supplied literature claims, not independently verified conclusions.

Short English statement

The question asks whether the strata representing the intended generic $a$-number-one locus form exactly one $J_b(F)$-orbit even for nonbasic $b$. The source record omits the invariant, parameter space, action, and meaning of genericity needed to make this precise. Its reported basic-case result and open status require independent review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30002950$, source problem code catalogue entry, title "Orbit Structure of Generic A-Number-One Loci." The supplied record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year $2015$, and names no proposer.
• Supplied report citation: Reductions of Shimura Varieties (2016), Oberwolfach Reports, pp. 2265--2328, DOI 10.4171/owr/2015/39. Supplied source URL and literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2015/39.
• Supplied literature reference labeled "full report text": https://ems.press/content/serial-article-files/46591.
• Supplied literature reference labeled "Goertz He and Nie 2022": https://doi.org/10.4153/S0008414X22000608.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports a proof in the basic case, says that no general proof without basicness was located, and says later work continues to identify open cases. 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 its statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied registry annotations; they do not resolve the missing mathematical definitions or establish mathematical correctness.