Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture

TL;DR

This paper proves the $ ext{±}$-main conjecture for supersingular elliptic curves at primes with $a_p=0$, and as a consequence, verifies the BSD conjecture for many such curves, advancing understanding of elliptic curves and Iwasawa theory.

Contribution

It establishes the $ ext{±}$-main conjecture for supersingular elliptic curves with $a_p=0$ by reducing it to a more accessible conjecture and proves the BSD conjecture for new infinite families of such curves.

Findings

Proved the $ ext{±}$-main conjecture for supersingular elliptic curves with $a_p=0$.

Verified the $p$-part of BSD conjecture for rank 0 or 1 cases.

Identified explicit infinite families of elliptic curves satisfying BSD without complex multiplication.

Abstract

In this paper we prove the $\pm$-main conjecture of Iwasawa theory formulated by Kobayashi for elliptic curves with supersingular reduction at an odd prime $p$ such that $a_p=0$, using a key new observation that it can be reduced 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 $1$. The main result enables us to present in the Appendix a number of explicit infinite families of elliptic curves without complex multiplications for which we can now prove the full Birch-Swinnerton-Dyer conjecture. No such infinite families of curves without complex multiplication were known previously.