On the Bertrandias-Payan module in a p-extension -- capitulation kernel
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.
Keywords
Cite
@article{arxiv.1510.05382,
title = {On the Bertrandias-Payan module in a p-extension -- capitulation kernel},
author = {Georges Gras},
journal= {arXiv preprint arXiv:1510.05382},
year = {2021}
}
Comments
in French. This text is the result of a collaborative work with Jean - Francois Jaulent (Bordeaux) and Thong Nguyen Quang Do (Besan{\c c}on). The originality of the approach is that three techniques are proposed (one per author) to study the same subject on Bertrandias-Payan module on the cyclic embeddings (...)