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Invariant part of class groups and distribution of relative class group.

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Title: Invariant part of class groups and distribution of relative class group.
Authors: Wang, Weitong1 (AUTHOR) 93thomaswang@gmail.com
Source: Journal of Number Theory. Feb2026, Vol. 279, p691-748. 58p.
Subjects: Class groups (Mathematics), Sylow subgroups, Group extensions (Mathematics), Field extensions (Mathematics), Mathematical analysis, Number theory
Abstract: We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field K , we show infinite C p -moments for the Sylow p -subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with A 4 -closure, we can show infinite C 2 -moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field. • Prove infinite moments of relative class groups for bad primes in Cohen-Lenstra Heuristics. • Non-randomness of class groups in certain non-abelian extensions. • Estimate of field-counting which generalizes the case of nowhere totally ramified cubic fields. • Results related to ambiguous class groups that generalizes the work of Roquette and Zassenhauss. • Generalized discriminant that is applied to the study of distribution of class groups. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field K , we show infinite C p -moments for the Sylow p -subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with A 4 -closure, we can show infinite C 2 -moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field. • Prove infinite moments of relative class groups for bad primes in Cohen-Lenstra Heuristics. • Non-randomness of class groups in certain non-abelian extensions. • Estimate of field-counting which generalizes the case of nowhere totally ramified cubic fields. • Results related to ambiguous class groups that generalizes the work of Roquette and Zassenhauss. • Generalized discriminant that is applied to the study of distribution of class groups. [ABSTRACT FROM AUTHOR]
ISSN:0022314X
DOI:10.1016/j.jnt.2025.07.012