The $\mu$-invariant change for abelian varieties over finite $p$-extensions of global fields

We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K^{\mathrm{\prime }}\mathrm{/}K$, of an ordinary abelian variety $A$ over a ${\mathbb{Z}}_p^d$-extension of global fields $L\mathrm{/}K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size ${\delta }_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of ${\delta }_v$ along a multiple ${\mathbb{Z}}_p$-extension $L\mathrm{/}K$ and provide a lower bound for the change of $\mu$-invariants of $A$ from the tower $L\mathrm{/}K$ to the tower $L{K}^{\mathrm{\prime }}\mathrm{/}{K}^{\mathrm{\prime }}$. Finally, we present numerical evidence supporting these results.