On Iwasawa theory of abelian varieties over ${\mathbb{Z}}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions

We present certain results on the Iwasawa theory of an abelian variety with potentially good ordinary reduction at all primes above $p$. These are then applied to study Diophantine stability and integally Diophantine extensions. Along the way, we also obtain some results pertaining to Mazur growth conjecture which refine previous results of Gajek-Leonard, Hatley, Kundu and Lei. Finally, we extend our investigation to the case of an elliptic curve with good supersingular reduction at the prime $p$ and make a similar analysis.