Galois co-descent for étale wild kernels and capitulation

Kolster, Manfred, and Movahhedi, Abbas. "Galois co-descent for étale wild kernels and capitulation." Annales de l'institut Fourier 50.1 (2000): 35-65. <http://eudml.org/doc/75419>.

@article{Kolster2000,
abstract = {Let $F$ be a number field with ring of integers $o_F$. For a fixed prime number $p$ and $i \ge 2$ the étale wild kernels $WK^\{\{\rm \acute\{e\}t\}\}_\{2i-2\}(F)$ are defined as kernels of certain localization maps on the $i$-fold twist of the $p$-adic étale cohomology groups of $\{\rm spec\}\,o_F[\{1\over p\}]$. These groups are finite and coincide for $i=2$ with the $p$-part of the classical wild kernel $WK_2(F)$. They play a role similar to the $p$-part of the $p$-class group of $F$. For class groups, Galois co-descent in a cyclic extension $L/F$ is described by the ambiguous class formula given by genus theory. In this formula, the only factor which is not well mastered is the norm index $[U^\{\prime \}_F:U^\{\prime \}_F \cap N_\{L/F\}(L^*)]$ for the $p$-units $U^\{\prime \}_F$. The aim of this paper is the study of the Galois co-descent for wild kernels: Given a cyclic extension $L/F$ of degree $p$ with Galois group $G$, we show that the transfer map $WK^\{\{\rm \acute\{e\}t\}\}_\{2i-2\}(\{L\})_G \rightarrow WK^\{\{\rm \acute\{e\}t\}\}_\{2i-2\}(\{F\})$ is onto except in a very special case, then we determine its kernel as the cokernel of a certain cup-product with values in a Brauer group. This approach also yields a genus formula, analogous to the one for class groups, comparing the sizes of $WK^\{\{\rm \acute\{e\}t\}\}_\{2i-2\}(L)_G$ and $WK^\{\{\rm \acute\{e\}t\}\}_\{2i-2\}(F)$ where $p$-units $U^\{\prime \}_F$ are replaced by odd $K$-theory groups. When $p$ is odd, we illustrate the method by finding all Galois $p$-extensions of $\{\bf Q\}$, for which the $p$-part of the classical wild kernel is trivial. For $p \ge 5$,they turn out to be the layers of the cyclotomic $\{\bf Z\}_p$-extension of $\{\bf Q\}$.},
author = {Kolster, Manfred, Movahhedi, Abbas},
journal = {Annales de l'institut Fourier},
keywords = {wild kernel; codescent; étale capitulation; étale cohomology; étale -theory; Iwasawa theory; Greenberg's conjecture},
language = {eng},
number = {1},
pages = {35-65},
publisher = {Association des Annales de l'Institut Fourier},
title = {Galois co-descent for étale wild kernels and capitulation},
url = {http://eudml.org/doc/75419},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Kolster, Manfred
AU - Movahhedi, Abbas
TI - Galois co-descent for étale wild kernels and capitulation
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 1
SP - 35
EP - 65
AB - Let $F$ be a number field with ring of integers $o_F$. For a fixed prime number $p$ and $i \ge 2$ the étale wild kernels $WK^{{\rm \acute{e}t}}_{2i-2}(F)$ are defined as kernels of certain localization maps on the $i$-fold twist of the $p$-adic étale cohomology groups of ${\rm spec}\,o_F[{1\over p}]$. These groups are finite and coincide for $i=2$ with the $p$-part of the classical wild kernel $WK_2(F)$. They play a role similar to the $p$-part of the $p$-class group of $F$. For class groups, Galois co-descent in a cyclic extension $L/F$ is described by the ambiguous class formula given by genus theory. In this formula, the only factor which is not well mastered is the norm index $[U^{\prime }_F:U^{\prime }_F \cap N_{L/F}(L^*)]$ for the $p$-units $U^{\prime }_F$. The aim of this paper is the study of the Galois co-descent for wild kernels: Given a cyclic extension $L/F$ of degree $p$ with Galois group $G$, we show that the transfer map $WK^{{\rm \acute{e}t}}_{2i-2}({L})_G \rightarrow WK^{{\rm \acute{e}t}}_{2i-2}({F})$ is onto except in a very special case, then we determine its kernel as the cokernel of a certain cup-product with values in a Brauer group. This approach also yields a genus formula, analogous to the one for class groups, comparing the sizes of $WK^{{\rm \acute{e}t}}_{2i-2}(L)_G$ and $WK^{{\rm \acute{e}t}}_{2i-2}(F)$ where $p$-units $U^{\prime }_F$ are replaced by odd $K$-theory groups. When $p$ is odd, we illustrate the method by finding all Galois $p$-extensions of ${\bf Q}$, for which the $p$-part of the classical wild kernel is trivial. For $p \ge 5$,they turn out to be the layers of the cyclotomic ${\bf Z}_p$-extension of ${\bf Q}$.
LA - eng
KW - wild kernel; codescent; étale capitulation; étale cohomology; étale -theory; Iwasawa theory; Greenberg's conjecture
UR - http://eudml.org/doc/75419
ER -