On the Bertrandias-Payan module in a p-extension -- capitulation kernel

TL;DR

This paper investigates the properties of the Bertrandias-Payan module in p-extensions of number fields, focusing on the transfer map and capitulation phenomena under certain ramification and Galois conditions, assuming Leopoldt's conjecture.

Contribution

It characterizes the injectivity of the transfer map for the Bertrandias-Payan module in cyclic p-extensions and relates it to ramification and Galois group structures, providing elementary proofs.

Findings

The transfer map j_(L/K) is injective except in specific ramified cases.

The structure of bp_L^G depends on the Galois group of the maximal Abelian p-ramified extension.

Complete elementary proofs are provided using ideal class field theory.

Abstract

For a number field K and a prime number p we denote by BP\_K the compositum of the cyclic p-extensions of K embeddable in a cyclic p-extension of arbitrary large degree. Then BP\_K is p-ramified (= unramified outside p) and is a finite extension of the compositum K~ of the Z\_p-extensions of K.We study the transfer map j\_(L/K) (as a capitulation map of ideal classes) for the Bertrandias-Payan module bp\_K:=Gal(BP\_K/K~) in a p-extension L/K (p\textgreater{}2, assuming the Leopoldt conjecture). In the cyclic case of degree p, j\_(L/K) is injective except if L/K is kummerian, p-ramified, non globally cyclotomic but locally cyclotomic at p (Theorem 3.1). We then intend to characterize the condition \#bp\_K divides \#bp\_L^G (fixed points). So we study bp\_L^G when j\_(L/K) is not injective and show that it depends on the Galois group (over K~) of the maximal Abelian p-ramified pro-p-extension of K.We give complete proofs in an elementary way using ideal approach of global class field theory.