Abelian capitulation of ray class groups

TL;DR

This paper extends the understanding of ray class group capitulation in number fields, demonstrating that under certain conditions, these groups become trivial in specific abelian extensions, generalizing previous results.

Contribution

It introduces a method to prove capitulation of tame ray class groups in abelian extensions, broadening prior work on class groups and ideal capitulation.

Findings

Existence of infinitely many abelian extensions where ray class groups capitulate.

Generalization of Kurihara's results to tame ray class groups.

Extension of Bosca's and Gras's methods to broader settings.

Abstract

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields.