Unspecified P-Adic Euler System Conjecture

Definitions and mathematical statement

Unrecoverable mathematical target. The supplied record does not contain an identifiable standalone conjecture. Its extraction begins with the trailing word "conjecture." The supplied verification describes this as the end of a Bloch--Kato reference, followed by statements of proved Euler-system and regulator theorems. Consequently, no precise open assertion can be formulated from this record alone.

Recoverable contextual content. The fragment says that the case $k_2=2$ is allowed "here," unlike in a previous theorem; neither the antecedent of "here" nor that previous theorem is supplied. It subsequently labels a statement "Theorem 3 (L.--Zerbes)" and reports that, if $k_2\geq 3$, then for every $j$ satisfying \[ 1\leq j\leq k_2-2, \] the value $L_p(F)(j)$ is related by a $p$-adic regulator formula to the Euler system for $V_F^{*}(1-j)$. This is contextual content attributed to a theorem in the source, not an extracted conjecture. The domains of $k_2$ and $j$, the hypotheses on $F$ and $p$, and the definition of $V_F$ are absent.

The fragment describes $\eta_F$ as a de Rham class associated to $F$ "as above," and $\log_{\mathrm{BK}}$ as the inverse of the Bloch--Kato exponential map for a local restriction of $V_F^{*}(1-j)$. The required earlier definitions and hypotheses are missing. The displayed regulator formula is corrupted: the placement of its pairing symbols, the marker $[F,j]$, and the Euler-system class cannot be recovered reliably. The local Galois group is extracted as $\operatorname{Gal}({\mathbb{Q}}_p/{\mathbb{Q}}_p)$; its intended extension is unresolved. No corrected formula or Galois group is asserted here.

A mathematical statement requires recovery of the intended assertion, its hypotheses, and its notation from the source context. The supplied title and open-status label do not determine such an assertion.

Short English statement

This record contains an incomplete passage about Euler systems and a $p$-adic regulator theorem attributed to L.--Zerbes. The intended conjecture, necessary hypotheses, and exact formula are unrecoverable from the supplied evidence. The registry's open label is supplied curation, not a verified mathematical status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30005567$, exact source code catalogue entry, title "Unspecified P-Adic Euler System Conjecture." The record has statement_status: unrecoverable, no clean statement, and a supplied status: open label. No proposer is identified; the proposed year is 2023.
• Supplied source citation: Algebraische Zahlentheorie (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/28, pp. 1543--1606. https://doi.org/10.4171/owr/2023/28. This is also the supplied literature-source link labelled "Oberwolfach Report 28/2023." The fragment attributes its Theorem 3 to "L.--Zerbes"; the abbreviation is not expanded in the supplied record. No pinpoint page or complete authoritative statement of the intended conjecture is supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the item open while stating that the extracted word "conjecture" identifies no mathematical assertion and therefore permits no classification as solved or disproved under its uncertainty rule. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the extraction begins with the trailing word of a Bloch--Kato reference and then gives proved Euler-system and regulator theorems; it contains no new standalone conjecture. No separate clean statement was promoted, and the original record was retained publicly for provenance. This verification is supplied metadata, not an independent assessment performed for this draft.