Symplectic Models for Relative Class Group Distributions

Definitions and mathematical statement

For a number field $K$, let $\operatorname{Cl}(K)$ denote its ideal class group and let $\mathcal O_K^{\times}$ denote its unit group. For a prime $\ell$ and a finite abelian group $A$, written additively, its $\ell$-primary part is \[ A[\ell^{\infty}]=\{a\in A:\ell^n a=0\text{ for some integer }n\geq 0\}. \] The supplied record does not specify which primes $\ell$ are admissible or what definition of relative class group is intended.

The source proposes the following conjectural comparison: after incorporating the contribution of unit groups, the joint distribution over number fields of the relative class-group $\ell$-part and two additional class-group invariants agrees with the distribution predicted by a random symplectic-similitude model. Its intended mathematical form can therefore be recorded only schematically as \[ \mu_{\mathrm{arithmetic}}=\mu_{\mathrm{symplectic\text{-}similitude}}, \] where both sides are intended to describe the same triple of invariants, with the unit-group contribution incorporated as required by the source.

This is not yet a fully specified equality of probability measures: the record supplies neither the family of number fields and underlying extensions, nor their ordering or probability law, nor the relative class-group convention, nor the two accompanying invariants. It also does not define the unit-group contribution, the parameters and probability law of the random symplectic-similitude model, or the procedure extracting the triple from that model. These omissions prevent a precise quantified conjecture from being recovered from the supplied record alone.

Short English statement

The proposed conjecture compares the joint distribution of three class-group invariants over number fields with a random symplectic-similitude prediction, taking unit groups into account. The supplied record leaves the field family, two of the invariants, and the model unspecified, so it determines only a schematic statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $30003905$, title Symplectic Models for Relative Class Group Distributions, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Explicit Methods in Number Theory (2019), Oberwolfach Reports, pp. 2033--2076, DOI 10.4171/owr/2018/34. Supplied source link: https://doi.org/10.4171/owr/2018/34. The registry gives proposed year 2018 and names no proposer.
• Supplied literature source, labeled "OWR report": https://ems.press/content/serial-article-files/46754.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the joint distribution over number fields is explicitly conjectural, with function-field verification and numerical evidence in the report but no proof for number fields. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial qualification does not resolve the missing mathematical specifications or establish mathematical correctness.