Invariant part of class group and distribution of relative class group

TL;DR

The paper develops a framework to study the distribution of relative class groups $\operatorname{Cl}(L/K)$ across number-field extensions, generalizing Roquette–Zassenhaus to the relative setting and connecting ramification data to invariant parts. By constructing generating-series from local specifications and analyzing their analytic continuation and poles via Hecke $L$-series and Tauberian theorems, it derives zero-probability results for low $p$-ranks and infinite $p$-power moments in broad families, including abelian and select non-abelian cases. Key contributions include precise lower bounds for $\operatorname{rk}_p(p^{l}\operatorname{Cl}(L/K))$ tied to ramification indices, a robust local-to-global specification framework, and concrete instances (e.g., sextic $A_4$-fields and cubic $S_3$-fields) where the predicted distribution diverges from Cohen–Lenstra–Martinet heuristics. These results enhance understanding of how ramification and Galois structure influence the probabilistic behavior of class groups and provide tools for translating arithmetic data into analytic generating-series, with potential implications for heuristics in non-abelian settings.

Abstract

We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thereby, we can show some statistical results as follows. For abelian extensions over a fixed number field K, we show infinite Cp-moments for the Sylow p-subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with A4-closure, we can show infinite C2-moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.

Theorems & Definitions (97)

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