The Birch and Swinnerton–Dyer formula for elliptic curves of analytic rank one

Let $E / \mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1} L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton–Dyer formula for $E / \mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible:\[\mathrm{ord}_p \left ( \frac{L^{\prime} (E,1)}{\Omega_E \cdot \mathrm{Reg} (E / \mathbb{Q})} \right )= \mathrm{ord}_p \left (\# \mathrm{III} (E / \mathbb{Q}) \prod_{\ell \leq \infty} c_{\ell} (E / \mathbb{Q})\right ) \: \textrm{.}\]This formula also holds for $p = 3$ provided $a_p (E) = 0$ if $E$ has supersingular reduction at $p$.

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D. Jetchev is supported by Swiss National Science Foundation professorship grant PP00P2-144658.

C. Skinner is partially supported by the grants DMS-0758379 and DMS-1301842 from the National Science Foundation and by the Simons Investigator grant #376203 from the Simons Foundation.

X. Wan is partially supported by the Chinese Academy of Science grant Y729025EE1, NSFC grant 11688101, 11621061 and an NSFC grant associated to the “Recruitment Program of Global Experts”.

Received 24 April 2017

Published 7 August 2017