Signed-Selmer groups over the \(\mathbb{Z}_p^2\)-extension of an imaginary quadratic field (Q2876530)

Let \(E\) be an elliptic curve over \(\mathbb Q\) with good supersingular reduction at \(p>3\). Let \(K\) be an imaginary quadratic field such that \(p\) splits in \(K\). Generalizing work of \textit{S.-i. Kobayashi} [Invent. Math. 152, No. 1, 1--36 (2003; Zbl 1047.11105)] the author defines four modified Selmer groups \(\mathrm{Sel}_p^{\pm / \pm}(E/K(\mathfrak f p^{\infty}))\), where \(\mathfrak f\) is an integral ideal of \(K\) prime to \(p\). He proves a control theorem for these Selmer groups in analogy to a theorem of \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)].NEWLINENEWLINENow suppose that \(\mathfrak f\) is trivial. Then \(\mathrm{Gal}(K(p^{\infty}) / K(p)) \simeq \mathbb Z_p^2\) and the work of \textit{D. Loeffler} and \textit{S. Zerbes} [Int. J. Number Theory 10, No. 8, 2045--2095 (2014; Zbl 1314.11066)] predicts the existence of four \(p\)-adic \(L\)-functions NEWLINE\[NEWLINEL_p^{\pm / \pm} \in \mathbb Z_p [[\mathrm{Gal}(K(p^{\infty}) / K(p))]] \otimes \overline{\mathbb Q}_p.NEWLINE\]NEWLINE These \(p\)-adic \(L\)-functions have recently been constructed by \textit{D. Loeffler} (see [``\(p\)-adic integration on ray class groups and non-ordinary \(p\)-adic \(L\)-functions'', Preprint, \url{arXiv:1304.4042}]). The author then states the following variant of the main conjecture of Iwasawa theory: NEWLINE\[NEWLINE\mathrm{char}(X^{\pm / \pm}) = (L_p^{\pm / \pm}),NEWLINE\]NEWLINE where \(X^{\pm / \pm}\) denotes the Pontryagin dual of \(\mathrm{Sel}_p^{\pm / \pm}(E/K(p^{\infty}))\). In fact, the author proposes a conjecture even in the case when \(\mathfrak f\) is non-trivial.