Capitulation, unit groups, and the cohomology of $S$-id\`{e}le classes

TL;DR

This paper investigates the Galois cohomology of $S$-idele classes in cyclic extensions of number fields, linking it to capitulation phenomena and the cohomology of $S$-units, advancing understanding of class group behavior.

Contribution

It introduces new relationships between Galois cohomology of $S$-idele classes, capitulation maps, and $S$-unit cohomology in cyclic number field extensions.

Findings

Established connections between Galois cohomology and capitulation maps.

Derived formulas relating $S$-idele class cohomology to $S$-unit cohomology.

Enhanced understanding of class group capitulation in cyclic extensions.

Abstract

Let $L/K$ be a finite, cyclic extension of number fields with Galois group $G$, and let $S$ be a finite set of primes of $K$ that includes all the infinite primes. In this paper, we study the $G$-cohomology of the $S$-id\`{e}le classes of $L$ and relate it to the $S$-capitulation map as well as to the $G$-cohomology of the $S$-unit group $U_{L,S}$.