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arXiv · 2607.10326

Class Number Relations in Abelian Extensions of Global Fields

Saad El Boukhari·Ben Forrás
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Abstract

Consider a finite abelian extension $K/k$ of global fields with Galois group $G$. We study the rank-one component of the generalized Stickelberger module associated with $K/k$ and a finite set $S$ of places of $k$. Under explicit splitting conditions on $S$, we compute generalized indices of this module with respect to the torsion-free part of the $S$-unit group of $K$. We also obtain $p$-primary refinements which include the non-semisimple case $p\mid |G|$. As applications, we derive divisibility relations between the $S$-class numbers of $K$ and $k$, both for number fields and for function fields.

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Saad El Boukhari, Ben Forrás. 2026-07-11. Class Number Relations in Abelian Extensions of Global Fields. https://arxiv.org/abs/2607.10326

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