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"On cyclic Lambda - modules in Z_p extensions of number fields"

Abstract:
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Let K_0/Q be a galois extension and K_n be the intermediate fields of a
Z_p extension K/K_0.
Let A_n be the p-parts of the class groups of K_n and H_n the p-parts of
their Hilbert class fields.
We assume that the Norm map is surjective on A_n and consider norm
coherent sequences a_n \in A_n
and the cyclic Lambda - modules Z_n = Lambda a_n. We investigate the
transitions Z_n \----> Z_(n+1),
under consideration of the map iota^* which lifts ideals from Z_n to
Z_(n+1) and the inclusion
H_n \subset H_(n+1). This is work in progress and we present results
restricted to the case
when iota^* is injective for all Z_n \---> Z_(n+1) up to a given N, or
up to infinity. Up to level N
we show that the modules Z_n = Lambda a_n, Y_n = Lambda b_n are disjoint
iff Z_0 and Y_0 are
disjoint. The result has for instance a simple application for computer
verifications of Greenberg's conjecture: it suffices to investigate
totally split ideals P_n \in a_n Z_n, which are defined by a_0.

In an other direction, for instance, for a_n \in A_n^-, if K_0 is a CM
field, N = infty. In this case we may derive additional
information on the projective limes Z of Z_n, showing that the classical
pseudoisomorphism from Iwasawa theory is in fact
an isomorphismin the cyclotomic Z_p extension. We also investigate the
case in which the cylic group generated by a_n
contains ramified ideals.  This is work in progress.

Preda Mihailescu