Ax–Schanuel and Zilber–Pink for Abelian-Differential Strata

Definitions and mathematical statement

An abelian differential on a smooth compact connected complex curve $X$ is a nonzero holomorphic one-form $\omega$ on $X$. In the usual unprojectivized convention, a stratum of abelian differentials parametrizes isomorphism classes of pairs $(X,\omega)$ with prescribed multiplicities of the zeros of $\omega$. The supplied record denotes its intended stratum by $S_\alpha$, but does not specify the genus, the admissible zero data $\alpha$, whether zeros are labelled, whether a connected component is intended, or whether differentials are considered up to scaling. This description therefore does not fix a convention for the source's $S_\alpha$.

The source poses two conjectural targets:

1. Establish the Ax--Schanuel conjecture for $S_\alpha$. The supplied record does not state that conjecture's maps, ambient spaces, exceptional loci, hypotheses, or dimension inequality, so a precise Ax--Schanuel assertion cannot be recovered from this record alone.
2. Establish the stated Zilber--Pink finiteness assertion. Write \[ \mathcal A_{\geq 3}(S_\alpha) :=\{s\in S_\alpha : s\text{ is an arithmetic point of rank at least }3\text{ in the source's sense}\}. \] The requested assertion is \[ \#\mathcal A_{\geq 3}(S_\alpha)<\infty. \] Here "arithmetic point" and its "rank" are unresolved source terms: the supplied record gives neither definition. This displayed formulation records the target but is not a complete intrinsic mathematical statement until those definitions and the convention for $S_\alpha$ are supplied. The record also leaves the range of strata to which the assertion is intended to apply implicit.

For context, the original excerpt attributes to the works labelled [Ler21] and [KL22] the first steps of a bi-algebraic heuristic for $S_\alpha$, motivated by its usefulness for diophantine questions on Shimura varieties. It reports a complete geometric characterization of arithmetic points; density of rank-one arithmetic points in the analytification $S_\alpha^{\mathrm{an}}$; and finiteness of arithmetic points on each algebraically primitive Teichmüller curve when the genus satisfies $g\geq 3$. It further reports linearity of bi-algebraic subvarieties in many cases, without specifying those cases. Finally, contrary to a hope attributed to [KL22], it attributes to [DM] examples of bi-algebraic subvarieties of $S_\alpha$ that are not linear. These are supplied contextual claims, not independently verified results or additional hypotheses that resolve the missing definitions.

Short English statement

The source asks for an Ax--Schanuel theorem for a stratum of abelian differentials and conjectures that the stratum has only finitely many arithmetic points of rank at least $3$. The relevant Ax--Schanuel formulation, the meanings of arithmetic point and rank, and the precise stratum convention are missing. Both targets therefore require clarification before a complete mathematical statement can be reviewed.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30005488$, exact source problem code catalogue entry, title "Ax--Schanuel and Zilber--Pink for Abelian-Differential Strata." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2023; it identifies no proposer.
• Supplied report citation: Komplexe Analysis -- Differential and Algebraic methods in Kähler spaces (2023), Oberwolfach Reports, pp. 965--1030, DOI 10.4171/owr/2023/18. Supplied link: https://doi.org/10.4171/owr/2023/18. This is the authoritative report citation supplied by the record; its contents have not been independently consulted here.
• Supplied literature source, labelled "OWR 2023 report": https://ems.press/content/serial-article-files/47013.
• Supplied literature source, labelled "Deroin and Matheus": https://arxiv.org/abs/2303.00642. The original excerpt instead spells the attribution "Deroin-Mattheus" and labels the work [DM]; that correspondence and spelling require source review.
• The excerpt also cites [Ler21] and [KL22], without full bibliographic information or links. Their authors, titles, and precise statements cannot be recovered from the supplied record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and assesses the Ax--Schanuel and Zilber--Pink directions as conjectural, with existing results covering important special bi-algebraic settings rather than the stated general problem. This is supplied dated curation, not an independently verified assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that it was checked against the official report and preserves its hypotheses, notation, and target. That verification claim is supplied metadata, not verification performed for this draft.