Algebraicity of Rigid Cocycle Values at Real Multiplication Points

Definitions and mathematical statement

The source uses the terms rigid meromorphic cocycle $J$, real-multiplication point $\tau$, and its value $J[\tau]$, but does not specify their underlying arithmetic and analytic framework. Write $H_\tau$ for the narrow ring class field attached to $\tau$, as designated by the source. The associated real quadratic field, order, and field embeddings are not specified. For subfields $K,L$ of a common ambient field, their compositum $KL$ is the smallest subfield containing both.

The supplied conjectural assertion is: for every rigid meromorphic cocycle $J$, there exists a finite extension $H_J/{\mathbb{Q}}$ such that \[ J[\tau]\in H_JH_\tau \qquad\text{for every real-multiplication point }\tau. \] The extension $H_J$ may depend on $J$, but must work simultaneously for every $\tau$. The assertion requires a common ambient field in which the values and the two number fields can be compared.

This is a qualified transcription rather than a complete intrinsic formulation: the source does not define the cocycle space, the domain of real-multiplication points, the evaluation and its normalization, or the order determining $H_\tau$. In particular, it does not explain whether evaluations are defined and finite at every point quantified over, or how poles and other undefined evaluations are handled. These omissions require mathematical review; no restriction of the stated universal quantifier is imposed here.

Short English statement

The conjecture asserts that each rigid meromorphic cocycle has a single finite number field extension which, when composed with the narrow ring class field of any real-multiplication point, contains its value there. The supplied record leaves the precise cocycle and evaluation framework unspecified.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004346$, source code catalogue entry, title: Algebraicity of Rigid Cocycle Values at Real Multiplication Points. The supplied record gives proposed year 2019 and names no individual proposer.
• Supplied report citation: Modular Forms (2020), Oberwolfach Reports, pp. 3529--3606, DOI 10.4171/owr/2019/57. https://doi.org/10.4171/owr/2019/57. The background attributes the extraction to workshop open-problem discussions documented in Oberwolfach Reports.
• Darmon, rigid meromorphic cocycles: https://www.math.mcgill.ca/darmon/pub/Articles/Expository/15.Rigid-Meromorphic-Cocycles/paper.pdf.
• Darmon--Pozzi--Vonk, Dedekind--Rademacher cocycle: https://doi.org/10.4171/JEMS/1344.
• Literature assessment dated 2026-08-22: supplied dated curation, not independently verified. It labels the full assertion for arbitrary rigid meromorphic cocycles and all real-multiplication points as open, and reports algebraicity results for special Dedekind--Rademacher values rather than the full extracted assertion. The background describes this assessment as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.