Modular Construction of Prescribed $\mathbb Z_p$-Extensions

Definitions and mathematical statement

Let $p$ be the prime implicit in the supplied record, and let ${\mathbb{Z}}_p$ denote the additive topological group of $p$-adic integers. The record specifies a cyclotomic ${\mathbb{Z}}_p$-extension $F_\infty/F$; write \[ \Gamma=\operatorname{Gal}(F_\infty/F)\cong{\mathbb{Z}}_p. \] The permitted base fields $F$, restrictions on $p$, and admissible auxiliary prime sets $Q$ are not supplied. Thus the quantification over these data cannot be recovered from this record.

The requested construction, for the intended admissible data, is a field extension $L_Q/F_\infty$ such that $L_Q/F$ is Galois and \[ H=\operatorname{Gal}(L_Q/F_\infty)\cong{\mathbb{Z}}_p \] as topological groups. Writing $G=\operatorname{Gal}(L_Q/F)$, these requirements give an exact sequence of profinite groups \[ 1\longrightarrow H\longrightarrow G\longrightarrow\Gamma\longrightarrow1. \] Since $H$ is abelian, conjugation by a lift $\widetilde\gamma\in G$ of $\gamma\in\Gamma$ defines an action of $\Gamma$ on $H$ independent of the lift. If $\chi:\Gamma\to{\mathbb{Z}}_p^\times$ denotes the intended cyclotomic character, the prescribed cyclotomic action means that there is a topological group isomorphism $\iota:H\to({\mathbb{Z}}_p,+)$ satisfying \[ \iota(\widetilde\gamma h\widetilde\gamma^{-1}) =\chi(\gamma)\,\iota(h) \qquad(\gamma\in\Gamma,\ h\in H). \] The record does not specify the character or hypotheses making this interpretation available. In particular, descent of the full $p$-adic cyclotomic character to $\Gamma$ must not be assumed without suitable hypotheses.

For a prime $w$ of $F_\infty$, let $I_w\subseteq H$ denote its inertia subgroup in $L_Q/F_\infty$; this is independent of the choice of a prime above $w$, since $H$ is abelian. Unramifiedness at $w$ means $I_w=\{1\}$, and ramification means $I_w\ne\{1\}$. The requested extension must be unramified outside the primes prescribed by $Q$, and ramified at $Q$. However, the record does not identify the field whose primes constitute $Q$, their admissibility conditions, or whether ramification is required at every prescribed prime or merely at some prescribed prime. These alternatives are not interchangeable and remain unresolved.

The question asks whether such an extension can be constructed by modular methods. No precise meaning of the required modular construction is supplied. Consequently, the conditions above describe the recoverable part of the requested construction, not a fully specified universally quantified existence conjecture.

The supplied background also asserts an equivalence with a Leopoldt-type finiteness condition and says that a reciprocity input is proved. Neither the finiteness condition nor the reciprocity statement and their hypotheses are supplied, so neither can be incorporated as a precise assertion here.

Short English statement

For the intended cyclotomic extension $F_\infty/F$ and auxiliary primes $Q$, construct by modular methods a further ${\mathbb{Z}}_p$-extension that is Galois over $F$, has the prescribed cyclotomic conjugation action, and has exactly the required ramification behavior. The admissible data, cyclotomic character, ramification quantifiers, and meaning of modular construction remain insufficiently specified.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30002152$, source problem code catalogue entry, title "Modular Construction of Prescribed ${\mathbb{Z}}_p$-Extensions." No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied report citation: Arithmetic Geometry (2013), Oberwolfach Reports, pp. 2335--2388, DOI 10.4171/owr/2012/38. https://doi.org/10.4171/owr/2012/38. The report title/year and DOI are retained as supplied; no problem-specific page or speaker attribution is provided.
• Supplied literature source, labelled "Sharifi, reciprocity conjecture": https://arxiv.org/abs/1205.3226. The supplied record does not identify the exact result used or its hypotheses.
• Supplied literature source, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2012/38.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open, describes the record as a fragment, asserts an equivalence with a Leopoldt-type finiteness condition, and states that the reciprocity input is proved, while noting that omitted hypotheses prevent a resolved classification of the extracted existence claim.
• The registry labels the statement corrected_verified and states that the official report supplies the base cyclotomic extension, Galois action, and ramification conditions. These are supplied verification claims; this draft does not independently verify the report or the mathematical assertions.