Class Number Relations in Abelian Extensions of Global Fields

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.