Pro-l Congruence Subgroup Problem for Mapping Class Groups

Definitions and mathematical statement

For a prime $\ell$, a pro-$\ell$ group is an inverse limit of finite groups whose orders are powers of $\ell$. The supplied record asks about a representation denoted by $\rho^{\ell\text{-csp}}_{g,r}$, described as the pro-$\ell$ congruence-subgroup representation associated with the congruence subgroup problem for mapping class groups. However, the record does not define this representation, its domain or codomain, or the admissible values and hypotheses for $g,r,\ell$. Thus the following records the requested property without supplying a missing construction.

Write $D=\operatorname{dom}(\rho^{\ell\text{-csp}}_{g,r})$ solely as placeholder notation for the unspecified domain. The question is whether \[ \forall x,y\in D,\qquad \rho^{\ell\text{-csp}}_{g,r}(x)=\rho^{\ell\text{-csp}}_{g,r}(y) \ \Longrightarrow\ x=y. \] A fully specified mathematical problem requires the definition of $\rho^{\ell\text{-csp}}_{g,r}$ and the intended parameter range; neither can be recovered from the supplied statement alone.

The supplied dated curation labels the problem partially solved and reports injectivity in several cases, including genus zero and genus at most two for $\ell=2$, and non-injectivity in genus one for $\ell>7$. It also reports that no general answer is known. These are supplied literature assessments, not independently verified assertions here. Their full hypotheses, including any restrictions involving $r$, are absent from the record.

Short English statement

The question asks whether the representation $\rho^{\ell\text{-csp}}_{g,r}$ distinguishes every pair of elements in its domain. The representation and its parameter restrictions are missing, so its precise formulation remains unresolved. The supplied curation reports both positive and negative cases.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005636$, source problem code catalogue entry. The supplied record gives proposed year 2023 and names no individual proposer.
• 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. This is also the supplied literature source labelled "OWR report with the positive and negative theorems." The record associates the problem with 2023 while citing the workshop as 2024; that dating has not been reconciled.
• Supplied literature source labelled "Hoshi--Iijima, genus-one pro-$\ell$ problem": https://www.sciencedirect.com/science/article/pii/S0021869318306410. No further bibliographic details or precise theorem statements are supplied.
• Literature assessment date: 2026-08-22. The partially solved label, reported positive and negative cases, and statement that no general answer is known are supplied dated curation derived from the OWR record, not independently verified here. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim; the present draft does not independently verify that claim or consult either linked source.