The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One

Abstract

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 $p\ge 5$ of good reduction such that $E[p]$ is irreducible: $${\mathrm{ord}}_p\left(\frac{L^{\prime }(E,1)}{{\Omega }_E\cdot \mathrm{Reg}(E/\mathbb{Q})}\right)={\mathrm{ord}}_p\left(\#\mathrm{Sha}(E/\mathbb{Q})\prod _{\ell \le \infty }{c}_{\ell }(E/\mathbb{Q})\right).$$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.

51 pages

References in corpus (1)

• Indivisibility of Heegner points in the multiplicative case

Cited by in corpus (7)

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• Arithmetic of p-irregular modular forms: families and p-adic L-functions
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• Triple product p-adic L-functions for Shimura curves over totally real number fields
• Central values of $L$-functions of cubic twists