Iwasawa Main Conjecture for Supersingular Elliptic Curves

In this paper we prove the $\pm$-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at $p$ such that $a_p=0$, using a new idea of reducing it to another Iwasawa-Greenberg main conjecture, which is more accessible and proved here as a first step. Then we develop some generalized $\pm$ local theory and deduce the main conjecture. The argument uses in an essential way the recent study on explicit reciprocity law for Beilinson-Flach elements by Kings-Loeffler-Zerbes. We also prove as corollaries the $p$-part of the BSD formula at supersingular primes when the analytic rank is $0$ or $1$.