On the $\mathfrak{M}_H(G)$-property for Selmer groups at supersingular reduction

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ which has good supersingular reduction at the odd prime $p$. We study the variation of Iwasawa invariants and the $\mathfrak{M}_H(G)$-property for signed Selmer groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic number field $K$ that lie inside the $\mathbb{Z}_p^2$-extension $\mathbb{K}_\infty$ of $K$ and are not necessarily cyclotomic. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property which involve the growth of $\mu$-invariants of the signed Selmer groups over intermediate shifted $\mathbb{Z}_p$-extensions in $\mathbb{K}_\infty$, and the boundedness of $\lambda$-invariants as one runs over $\mathbb{Z}_p$-extensions of $K$ inside $\mathbb{K}_\infty$. We give examples where the $\mathfrak{M}_H(G)$-property holds, and also examples where we can prove that it does not hold. It is striking that although the case of supersingular reduction is much more difficult than the case of ordinary reduction, we get finer results here; moreover, we are able to derive analogous criteria for the validity of the $\mathfrak{M}_H(G)$-property of the classical Selmer group, as well as the fine Selmer group. Many of the properties that we investigate have not been studied before in this non-torsion setting. Further, we study various implications between the $\mathfrak{M}_H(G)$-properties for Selmer groups, signed Selmer groups and fine Selmer groups. We apply our results to a conjecture of Mazur, and prove implications between the $\mathfrak{M}_H(G)$-property and Conjectures A and B of Coates and Sujatha.