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

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