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

TL;DR

This paper advances Iwasawa theory for abelian varieties over $ ext{Z}_p^2$-extensions, applying findings to Diophantine stability, and extends analysis to supersingular elliptic curves, refining existing conjectures.

Contribution

It introduces new results in Iwasawa theory for abelian varieties with potentially good ordinary reduction and extends the analysis to supersingular elliptic curves.

Findings

Refined results on Mazur growth conjecture.

Applications to Diophantine stability and extensions.

Extension of analysis to supersingular elliptic curves.

Abstract

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.