Functional Equations for Higher Bloch Groups

Definitions and mathematical statement

Let $n$ denote the integer weight of a higher polylogarithm, with particular emphasis on $n>3$, and write $B_n(F)$ for the higher Bloch group intended by the source. The source does not specify the admissible base fields $F$, the definition of $B_n(F)$, or the normalization of the polylogarithms. A functional equation here is an identity among polylogarithm values; the precise admissible arguments, coefficients, and interpretation of these identities remain unspecified.

Problem (source-level formulation). Construct functional equations for higher polylogarithms, especially in weights $n>3$, that generate the universal relations needed to define $B_n(F)$, and prove the corresponding form of Zagier's conjecture.

The phrase "generate the universal relations" requires a specified relation system and specified operations for deriving relations. Neither is supplied. Likewise, the corresponding form of Zagier's conjecture is not stated. Consequently, this is a research question with unresolved formulation data, rather than a fully specified quantified conjecture. In particular, the source does not determine whether one construction is required uniformly for every integer $n>3$, or whether separate constructions in individual weights are intended.

Short English statement

Find functional equations for higher polylogarithms, especially for integer weights $n>3$, that generate the relations defining the intended higher Bloch groups $B_n(F)$, and establish the intended version of Zagier's conjecture. The field domain, group conventions, meaning of generation, and precise conjectural conclusion require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001843$, source problem code catalogue entry, title "Functional Equations for Higher Bloch Groups," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2011$ and names no proposer.
• Supplied source citation: Explicit Methods in Number Theory (2012), Oberwolfach Reports, pp. 1971--2044, DOI 10.4171/owr/2011/35. Supplied source and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2011/35. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source link, labelled "functional equations of higher logarithms": https://arxiv.org/abs/math/0207222. No authors or additional bibliographic details for this reference are supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and states that Goncharov's higher-weight conjectural programme supplies the $n=4$ target, while no general candidate generating relation for every $n>3$ has been established. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note claims that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification claim, not verification performed for this draft.