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Let A be an abelian variety defined over a global field F of positive characteristic p and let F / F be a Z p N -extension, unramified outside a finite set of places of F . Assuming that all ramified places are totally ramified, we define a pro-characteristic ideal associated to the Pontrjagin dual of the p -primary Selmer group of A . To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a Z p d -extension F d / F and for any Z p d − 1 -extension contained in F d , and then use a limit process. Finally, we give an application to an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra Z p [ [ Gal ( F / F ) ] ] in the case A is a constant abelian variety. [ABSTRACT FROM AUTHOR]

Abstract: Let p be an odd prime. Let k be an algebraic number field and let be the compositum of all the -extensions of k, so that for some finite d. We shall consider fields k with . Building on known results for quadratic fields, we shall show that the Galois group of the maximal abelian unramified pro-p-extension of is pseudo-null for several such k''s, thus confirming a conjecture of Greenberg. Moreover we shall see that pseudo-nullity can be achieved quite early, namely in a -extension, and explain the consequences of this on the capitulation of ideals in such extensions. [Copyright &y& Elsevier]