Global L-Functions from Adelic Cyclic Homology

Definitions and mathematical statement

Let $s$ denote a complex variable, and let $M$ denote the object whose completed $L$-function is sought. The supplied record writes this function as $\Lambda(s,M)$, but does not specify the nature or admissible class of $M$, the underlying global field, the adelic coefficient object, or the normalization of the completion. Thus $\Lambda(s,M)$ is target notation here, not an independently defined function.

The requested construction problem is to develop a version or extension of Connes--Consani cyclic homology with adelic coefficients that provides, for the intended objects $M$, all of the following:

1. a global definition of the completed $L$-function $\Lambda(s,M)$;
2. analytic continuation of that function;
3. local epsilon factors;
4. a global duality yielding its functional equation.

The source does not specify whether the continuation is to be holomorphic or meromorphic, its required domain, the normalization and auxiliary choices for the local epsilon factors, or the dual object and transformation of $s$ in the functional equation. No particular functional-equation formula or universal quantifier over a class of $M$ can therefore be fixed from this record. These omissions leave a research-program formulation requiring further hypotheses before it becomes a fully specified mathematical assertion.

The supplied dated assessment states that cyclic-homology constructions recover important archimedean local factors, while the complete global construction requested here is not established. This is supplied curation, not an independently verified mathematical or literature conclusion.

Short English statement

The problem asks for an adelic extension of Connes--Consani cyclic homology that constructs a completed $L$-function together with analytic continuation, local epsilon factors, and a duality explaining its functional equation. The objects $M$, coefficient system, and analytic and normalization requirements remain unspecified. The supplied record labels the problem partially solved; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002408$, title "Global L-Functions from Adelic Cyclic Homology," Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. The record gives proposed year 2013 and supplies no named proposer.
• Supplied report citation: Noncommutative Geometry (2014), Oberwolfach Reports, pp. 2553--2629, DOI 10.4171/owr/2013/45: https://doi.org/10.4171/owr/2013/45. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2014 designation and the record's 2013 year are retained without reconciliation.
• Supplied literature source, "Oberwolfach Report": https://ems.press/content/serial-article-files/46476.
• Supplied literature source, "Connes--Consani": https://arxiv.org/abs/1211.4239.
• Supplied literature source, "Published version": https://doi.org/10.1017/is014006012jkt270.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that important archimedean local factors are recovered by cyclic-homology constructions, but that the requested complete global definition is not established. This is supplied dated curation derived from the OWR record, not independently verified here; its status and cited claims require independent checking.