Aspects of non-commutative Iwasawa theory at supersingular primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ and let $K$ be a false Tate extension of $\mathbb{Q}$. On the one hand, I will explain how to define plus and minus Selmer groups of $E$ over $K$ using ideas from $p$-adic Hodge theory, generalising works of Kobayashi. On the other hand, I will talk about some congruences of $L$-values of $E$ which might give rise to a possible definition of plus and minus $p$-adic $L$-functions for $K$, generalising works of Pollack.