Unified Selmer and Cohen-Lenstra-Martinet Heuristics

Definitions and mathematical statement

A number field is a finite extension $K/{\mathbb{Q}}$. Its ordinary ideal class group is \[ \operatorname{Cl}(K)=I_K/P_K, \] where $I_K$ is the group of nonzero fractional ideals of the ring of integers $\mathcal O_K$, and $P_K$ is its subgroup of nonzero principal fractional ideals. This recalls the ordinary class-group convention; the supplied record does not specify whether the intended heuristics concern ordinary class groups, another class-group variant, or particular primary components.

The notation $n$-Selmer refers to Selmer theory associated with multiplication by a positive integer $n$, in a setting where that construction is defined. A precise definition requires hypotheses on the algebraic group and its base, together with a cohomological construction and local conditions. The source does not supply these choices for its proposed class of possibly nonproper algebraic groups. Consequently, no single definition of an $n$-Selmer group for the entire intended scope is imposed here. Likewise, distributional heuristics require specified families, an ordering or probability measure, and limiting predictions; these are not supplied.

Research question. Can one formulate unified heuristics for $n$-Selmer groups of possibly nonproper algebraic groups that also imply the Cohen--Lenstra--Martinet heuristics for class groups of number fields?

The requested output is a coherent formulation of heuristics and an explanation of how the relevant Cohen--Lenstra--Martinet predictions follow from it. The source does not specify the allowed indices $n$, the base fields, the class of algebraic groups, the local conditions, the varying families, or the precise versions of the Cohen--Lenstra--Martinet predictions to be recovered. Thus this is a research-program question whose full quantifiers and hypotheses remain to be formulated, rather than a completely specified conjectural assertion.

Short English statement

Seek a common framework predicting the distributions of $n$-Selmer groups for algebraic groups, including nonproper groups, that also yields the Cohen--Lenstra--Martinet class-group heuristics. The source leaves the groups, fields, local conditions, families, and precise predictions unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $30004750$, source problem code catalogue entry, title Unified Selmer and Cohen-Lenstra-Martinet Heuristics; collection: Oberwolfach Reports Open Problems. No individual proposer is identified. The supplied proposed year is 2021.
• Supplied report citation: Explicit Methods in Number Theory (2022), Oberwolfach Reports, pp. 1805--1858, DOI 10.4171/owr/2021/34. https://doi.org/10.4171/owr/2021/34. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature source, labelled "Landesman": https://arxiv.org/abs/2106.10357. No further bibliographic details or precise supporting passage are supplied.
• Supplied literature source, labelled "Oberwolfach report": https://ems.press/content/serial-article-files/46913.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that unified heuristics for Selmer groups of general algebraic groups, with the requested implication for Cohen--Lenstra--Martinet heuristics, remain a research question. This is supplied dated curation, not an independently verified literature assessment; the source explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial claims, not independent verification here of mathematical precision or current status.