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.