### Greenberg’s conjecture for ${\mathbb{Z}}_{p}^{d}$-extensions

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We explicitly perform some steps of a 3-descent algorithm for the curves ${y}^{2}={x}^{3}+a$, $a$ a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.

Let K/k be a ℤₚ-extension of a number field k, and denote by kₙ its layers. We prove some stabilization properties for the orders and the p-ranks of the higher Iwasawa modules arising from the lower central series of the Galois group of the maximal unramified pro-p-extension of K (resp. of the kₙ).

Let $F$ be a function field of characteristic $p\>0$, $\mathcal{F}/F$ a ${\mathbb{Z}}_{l}^{d}$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $\mathcal{F}$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $\mathcal{F}/F$.

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