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

Let $E / \mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p \gt 3$ be a prime forwhich $E[p]$ is irreducible. In this note, following a slight modification of the methods of “The Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one” [Dimitar Jetchev, Christopher Skinner, and Xin Wan, Camb. J. Math. 5 (2017), no. 3, 369–434], 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.

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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152).

Received 21 April 2017

Published 27 March 2018