We extend the work of Lai et al. (Algebra Number Theory 15(4):863–907, 2021) and study the change of \(\mu \) -invariants, with respect to a finite Galois p-extension \(K'/K\) , of an ordinary abelian variety A over a \(\mathbb {Z}_p^d\) -extension of global fields L/K that ramifies at a finite number of places at which A has ordinary reduction. In characteristic \(p>0\) , we define a local invariant \(\delta _v\) (see Sect. 1.3 for its precise definition), associated with the size of the first local Galois cohomology of the Mordell–Weil group of A with respect to a p-extension ramified at a supersingular place v, and derive an explicit bound for \(\delta _v\) . Next, in all characteristics, we describe the asymptotic growth of \(\delta _v\) along a multiple \({\mathbb {Z}}_p\) -extension L/K and provide a lower bound for the change of \(\mu \) -invariants of A from the tower L/K to the tower \(LK'/K'\) . Finally, we present numerical evidence supporting these results.
