# On the $p$-part of the Birch-Swinnerton-Dyer formula for multiplicative primes

## TL;DR

This paper proves the $p$-part of the Birch-Swinnerton-Dyer formula for semistable elliptic curves with multiplicative reduction at primes greater than 3, using Iwasawa theory and extending previous results.

## Contribution

It extends the $p$-part of the BSD formula to primes of multiplicative reduction for semistable elliptic curves, building on and modifying existing methods.

## Key findings

- Established the $p$-part of BSD for multiplicative primes
- Extended previous results to a broader class of primes
- Utilized Iwasawa theory techniques in the proof

## Abstract

Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to establish the $p$-part of the Birch and Swinnerton-Dyer formula for $E$. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.