Signed Selmer Groups over p-adic Lie Extensions

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\ge 3$ and $a_p=0$. We generalise the definition of Kobayashi’s plus/minus Selmer groups over $\mathbb{Q}({\mu }_{p^{\mathrm{\infty }}})$ to $p$-adic Lie extensions $K_{\mathrm{\infty }}$ of $\mathbb{Q}$ containing $\mathbb{Q}({\mu }_{p^{\mathrm{\infty }}})$, using the theory of $(\varphi ,\mathrm{\Gamma })$-modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using the “jumping conditions” of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case.