The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One
By Dimitar Jetchev, C. Skinner, X. Wan. 2015
88 citations recorded by the source corpus.
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: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{\Omega_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.
Subfields: Algebra and Number Theory, Geometry and Topology, Political Science and International Relations
View Paper: https://www.semanticscholar.org/paper/bb63d8bfaf30815ec089325868cef8b3e0a30b05
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In Algebra and Number Theory
In Geometry and Topology
Citations in the corpus
- Iwasawa Main Conjecture for Supersingular Elliptic Curves (2014, background): “Note that in [31] the BSD formula in the rank one case is proved via another approach, which treats ordinary and supersingular elliptic curves in a uniform way.”