Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

We compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions $K_\infty $ of number fields K. We prove that in several natural settings the $\pi $-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu $-invariants. In the special case of a $\mathbb {Z}_p$-extension $K_\infty /K$, we also compare the Iwasawa $\lambda $-invariants of the fine Selmer groups, even in situations where the $\mu $-invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$-extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$-extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$-extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.