Selmer groups over $\Z_p^d$-extensions

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=Γ$. Let $Λ(Γ):=\Z_p[[Γ]]$ denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the $Λ(Γ)$-module $X_L$, the dual $p$-primary Selmer group, varies when $L/K$ is replaced by a intermediate $\Z_p^e$-extension.

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