Remarks on Kato's Euler systems for elliptic curves with additive reduction

Abstract

Extending the former work for the good reduction case, we provide a numerical criterion to verify a large portion of the "Iwasawa main conjecture without $p$-adic $L$-functions" for elliptic curves with additive reduction at an odd prime $p$ over the cyclotomic $\mathbb{Z}_p$-extension. We also deduce the corresponding $p$-part of the Birch and Swinnerton-Dyer formula for elliptic curves of rank zero from the same numerical criterion. We give explicit examples at the end and specify our choice of Kato's Euler system in the appendix.