Selmer groups over ${\mathbb{Z}}_p^d$-extensions

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