Let $S$ be a non-empty finite set of prime numbers and, for each $p$ in $S$ , let $\bm{Z}_p$ denote the ring of $p$ -adic integers. Let $F$ be an abelian extension over the rational field such that the Galois group of $F$ over some subfield of $F$ with finite degree is topologically isomorphic to the additive group of the direct product of $\bm{Z}_p$ for all $p$ in $S$ . We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers $l$ for which the $l$ -class group of $F$ is nontrivial. This result implies our conjecture in [3] that the set of prime numbers $l$ for which the $l$ -class group of $F$ is trivial has natural density $1$ in the set of all prime numbers.
Publié le : 2005-07-14
Classification:
abelian number field,
ideal class group,
Iwasawa theory,
class number formula,
11R29,
11R23,
11R27
@article{1158241937,
author = {HORIE, Kuniaki},
title = {Triviality in ideal class groups of Iwasawa-theoretical abelian number fields},
journal = {J. Math. Soc. Japan},
volume = {57},
number = {4},
year = {2005},
pages = { 827-857},
language = {en},
url = {http://dml.mathdoc.fr/item/1158241937}
}
HORIE, Kuniaki. Triviality in ideal class groups of Iwasawa-theoretical abelian number fields. J. Math. Soc. Japan, Tome 57 (2005) no. 4, pp. 827-857. http://gdmltest.u-ga.fr/item/1158241937/