$2$-Adic Analyticity of the Groups $G(A)$ and $G(B)$

Definitions and mathematical statement

Retain $G(A)$ and $G(B)$ for the two closures in the source. The conjecture has two analytic assertions and one more specific identification: \[ G(A)\text{ is }2\text{-adic analytic},\qquad G(B)\text{ is }2\text{-adic analytic}, \] \[ G(B)\text{ is a pro-}2\text{ Sylow subgroup of }SL_2({\mathbb{Z}}_2). \] The symbols $A,B$, the constructions producing the groups, the ambient groups and the topologies defining their closures are not supplied. In particular $A,B$ are not assumed to be specific matrices or presentations invented for this draft.

Here ${\mathbb{Z}}_2=\varprojlim_n{\mathbb{Z}}/2^n{\mathbb{Z}}$ and $SL_2({\mathbb{Z}}_2)=\{g\in\operatorname{Mat}_2({\mathbb{Z}}_2):\det g=1\}$ has its natural profinite topology. A pro-$2$ group is a profinite inverse limit of finite $2$-groups, and a pro-$2$ Sylow subgroup is a maximal closed pro-$2$ subgroup of the ambient profinite group. In the usual terminology, a $2$-adic analytic group has a finite-dimensional $2$-adic analytic manifold structure compatible with its topology, for which multiplication and inversion are analytic. The notation ${\mathbb{Z}}_2$-analytic in the source is retained in this conventional sense, rather than as a newly specified integral group scheme.

If the source's $G(B)$ is not already given as a subgroup of $SL_2({\mathbb{Z}}_2)$, the identification should be read as asking for a topological group isomorphism \[ G(B)\cong P,\qquad P\leq SL_2({\mathbb{Z}}_2)\text{ a pro-}2\text{ Sylow subgroup}. \] An abstract group isomorphism or agreement of selected finite quotients without compatible topology would be a different claim. The intended literal embedding versus topological identification is not specified. The more specific $G(B)$ statement does not by itself settle the separate assertion about $G(A)$.

Although the cleaned wording is declarative, the original explicitly labels these statements a conjecture. The literature triage checked 2026-08-21 says the pro-$2$ Sylow identification remains conjectural and that the cited paper supplies constructions and evidence for related unramified $3$-extension quotients, not a proof of this $2$-adic identification. This distinction is retained. The open label and editorial verification note remain dated metadata, not an independently checked theorem or status conclusion.

Short English statement

The conjecture predicts $2$-adic analytic structures for both closures, and specifically identifies $G(B)$ with a pro-$2$ Sylow subgroup of $SL_2({\mathbb{Z}}_2)$. The groups and closure topologies are not defined in the record. Related evidence about $3$-extension quotients does not supply the missing $2$-adic identification.

Sources

• Local UnsolvedMath registry, permanent ID 30000491, source code catalogue entry, "$2$-Adic Analyticity of the Groups $G(A)$ and $G(B)$," including the original conjecture label and cleaned assertions.
• Pro-$p$ Extensions of Global Fields and pro-$p$ Groups (2007), Oberwolfach Reports, pp. 1463--1536, DOI 10.4171/owr/2006/25: https://doi.org/10.4171/owr/2006/25. The citation uses 2007, while the DOI and proposed-year metadata use 2006.
• Supplied reference "Bartholdi--Bush primary paper": https://doi.org/10.1016/j.jnt.2006.08.008.
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. No external source has been visited or independently checked here.