Uniformly Bounded Ramification in $p$-Frattini-Tower Realizations

Definitions and mathematical statement

Uniform ramification question. For a $p$-perfect finite group $G$ and its $p$-Frattini quotients $G_k$, does there exist $r_0$ such that every $G_k$ satisfies the regular inverse Galois problem using covers with at most $r_0$ ramification classes? Here $p$-perfect conventionally means that $G$ has no nontrivial quotient of order $p$, equivalently $H^1(G,{\mathbb{F}}_p)=0$. No hypothesis $p\mid|G|$, or full group-theoretic perfectness, is silently substituted.

A $p$-Frattini cover is a surjection with pro-$p$ kernel that is Frattini: no proper closed subgroup of the covering group surjects onto the target. A standard tower uses characteristic finite quotients of the universal $p$-Frattini cover of $G$, often obtained by successive Frattini subgroups of its kernel. The source does not supply the chosen universal cover, filtration, or exact indexing, so $G_k$ is retained as its intended tower rather than identified with an arbitrary sequence of finite $p$-extensions.

Over a specified constant field $F$, a regular realization means a finite Galois extension $E_k/F(t)$ with Galois group $G_k$ and $E_k\cap\overline F=F$; equivalently it gives the corresponding geometrically connected Galois cover of the projective line. The source does not name $F$, for example ${\mathbb{Q}}$, or its characteristic. The uniform quantifier is $\exists r_0\,\forall k$: the bound may depend on the fixed $G,p,F$ and stated restrictions but not on the level. The individual realizations may vary with $k$. Requiring them to form a compatible tower of covers would be an additional condition not stated here.

The phrase ramification classes is ambiguous. A bound on branch points, on a branch-cycle conjugacy-class tuple counted with repetitions, or only on the number of distinct inertia classes imposes different restrictions. The precise class rationality, allowed inertia orders, any prime-to-$p$ restriction, and whether classes must lift coherently through levels are missing. In particular, a bound on distinct conjugacy classes alone does not bound the number of branch points. These notions are not silently identified, so no fully quantified bounded-branch theorem can be extracted without the source's convention.

The supplied triage checked 2026-08-21 labels the modular-towers question open and reports no located theorem settling the uniform $r_0$ assertion. It says the report states a restriction, but that restriction is not spelled out beyond the short question here. This dated assessment requires independent verification.

Short English statement

Ask whether every level of a fixed $p$-Frattini tower has a regular Galois realization with one bound on its allowed ramification complexity. The bound must be independent of the level. The constant field, class-counting convention, and inertia restrictions are unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 30000504, code catalogue entry; Uniformly Bounded Ramification in $p$-Frattini-Tower Realizations, Oberwolfach Reports Open Problems.
• Pro-$p$ Extensions of Global Fields and pro-$p$ Groups (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/25, pp. 1463--1536. https://doi.org/10.4171/owr/2006/25. Proposed year and year tag: 2006; no proposer named. The source's escaped title delimiters are rendered as mathematics.
• Supplied literature label original OWR report: https://doi.org/10.4171/owr/2006/25. No separate authoritative tower definition or precise ramification restriction is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field claims editorial and notation cleanup only. Verify dated claims independently.