Rank Growth and the Tame Fontaine–Mazur Conjecture

Definitions and mathematical statement

In the usual number-field pro-$p$ formulation, let $p$ be a prime, let $K$ be a number field, let $S$ be a finite set of finite primes of $K$ with no prime above $p$, and write $G_{K,S}=\operatorname{Gal}(K_S(p)/K)$, where $K_S(p)$ is the maximal pro-$p$ extension unramified at finite primes outside $S$. These are explicit contextual conventions: the extract itself does not define $G_{K,S}$ or specify its treatment of infinite places.

Let $\gamma(G)$ denote the source's rank-growth exponent. The first question is the universal assertion, in the intended arithmetic class, \[ G_{K,S}\text{ infinite}\quad\Longrightarrow\quad \gamma(G_{K,S})\geq\frac12. \tag{RG} \] For an open subgroup $H$ of a pro-$p$ group, its generator rank is $d(H)=\dim_{{\mathbb{F}}_p}H^1(H,{\mathbb{F}}_p)$. However, the source gives no formula for $\gamma$ in terms of these ranks: the choice of subgroups or filtration, limit operation and normalization are missing. Thus (RG) is retained with its exact threshold, without inventing a definition of $\gamma$.

The companion source record OWR-1274-020, permanent ID 30000500, states the tame Fontaine--Mazur assertion as \[ \forall n\geq1\ \forall\rho:G_{K,S}\longrightarrow \operatorname{GL}_n({\mathbb{Z}}_p)\text{ continuous},\qquad |\rho(G_{K,S})|<\infty. \tag{TFM} \] It also describes this as absence of an infinite $p$-adic analytic quotient. The second question asks whether (RG), as a conjecture over all the intended arithmetic data, is equivalent to (TFM) by group-theoretic arguments. It asks for the equivalence as well as the lower bound; it does not supply either implication. The precise meaning of a group-theoretic equivalence and the missing rank-growth definition require recovery from the report. The literature-assessment field instead mentions equivalence with Boston's virtual Golod--Shafarevich conjecture; no formulation of that conjecture is supplied, so it is not silently substituted for (TFM).

Short English statement

For infinite tame pro-$p$ Galois groups, ask whether the source's rank-growth exponent is at least $1/2$, and whether this conjecture is equivalent, by group theory, to the tame Fontaine--Mazur conjecture. The exponent is undefined in the supplied extract, and the literature summary names a different comparison conjecture.

Sources

• Supplied report: Pro-$p$ Extensions of Global Fields and pro-$p$ Groups (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/25, pp. 1463--1536; proposed year 2006. https://doi.org/10.4171/owr/2006/25.
• Supplied reference labelled "current inverse-Galois and modular-tower problem list": https://www.math.uci.edu/~mfried/deflist-cov/RIGP.html.
• Local UnsolvedMath registry, permanent ID 30000501, catalogue entry, created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the lower bound and equivalence question are explicit and a repeated reference was removed. The supplied literature triage, checked 21 August 2026, labels the problem open and says no general equivalence proof involving Boston's virtual Golod--Shafarevich conjecture was found. This dated assessment is retained with the mismatch described above.
• Companion local record, permanent ID 30000500, OWR-1274-020, same report and update date, supplies (TFM). Its supplied references include Hajir--Maire, https://arxiv.org/abs/1710.09214, and "2026 progress on tame Fontaine--Mazur", https://arxiv.org/abs/2608.10811. Its 21 August 2026 assessment reports special cases rather than a general theorem. No independent literature check is asserted.