On Greenberg's generalized conjecture

Abstract

For a number field F and an odd prime number p, let F˜ be the compositum of all Zp-extensions of F and Λ˜ the associated Iwasawa algebra. Let GS(F˜) be the Galois group over F˜ of the maximal extension which is unramified outside p-adic and infinite places. In this paper we study the Λ˜-module XS(−i)(F˜):=H1(GS(F˜),Zp(−i)) and its relationship with X(F˜(μp))(i−1)Δ, the Δ:=Gal(F˜(μp)/F˜)-invariant of the Galois group over F˜(μp) of the maximal abelian unramified pro-p-extension of F˜(μp). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ˜-module X(F˜(μp))(i−1)Δ is implied by the existence of a Zpd-extension L with XS(−i)(L):=H1(GS(L),Zp(−i)) being without torsion over the Iwasawa algebra associated to L, and which contains a Zp-extension F∞ satisfying H2(GS(F∞),Qp/Zp(i))=0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i≡1mod[F(μp):F]. This existence is fulfilled for (p,i)-regular fields.