On the $\mathfrak{M}_H(G)$-property

Abstract 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′.

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