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

TL;DR

This paper extends previous work on Kato's Euler systems to elliptic curves with additive reduction, providing a numerical criterion to verify parts of the Iwasawa main conjecture and the BSD formula at odd primes.

Contribution

It introduces a new numerical criterion for elliptic curves with additive reduction to verify the Iwasawa main conjecture and BSD formula, expanding the applicability of prior results.

Findings

Numerical criterion for Iwasawa main conjecture verification

Verification of BSD p-part for rank zero elliptic curves

Explicit examples illustrating the criterion

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.