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

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Let E/Q be a semistable elliptic curve such that ordₛ₌₁L(E,s) = 1. We prove the p-part of the Birch and Swinnerton-Dyer formula for E/Q for each prime p ≥ 5 of good reduction such that E[p] is irreducible: ordₚ (L'(E,1)/Ωₑ(E/Q)) = ordₚ (#Sha(E/Q)∏ℓ≤ ∞ cℓ(E/Q)). This formula also holds for p=3 provided aₚ(E)=0 if E has supersingular reduction at p.