Congruent elliptic curves and ${\mathbb{Z}}_p^d$-extensions

Abstract Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ , with good ordinary reduction at $p$ . We compare the $\Lambda$ -ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$ -extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$ . This generalizes and complements previous work over ${\mathbb{Z}}_p$ -extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$ -extensions of $K$ which are contained in $\mathbb{L}_\infty$ .