given an elliptic curve $e/\mathbb{q}$ with good reduction at $p$, iwasawa theory studies its arithmetic over all the fields of $p^n$-th roots of unity. for example, there are nontrivial relations among the participants in the birch–swinnerton-dyer conjectures for $e$ over each layer in this tower of fields. if the reduction of $e$ mod $p$ is ordinary, then have had a satisfactory description of the scenario for quite some time. but if the reduction of $e$ mod $p$ is supersingular, the correct description has required new advances in $p$-adic hodge theory. we will discuss the background and what is now known.