On the parity bridge between algebraic rank and root numbers for elliptic curves of rank one

Let E/ℚ be an elliptic curve. The Birch–Swinnerton-Dyer conjecture predicts that the order of vanishing of the L-function L(E,s) at s=1 coincides with the algebraic rank of E(ℚ). While the implication from analytic rank to algebraic rank is known in analytic rank ≤ 1 through the work of Gross–Zagier and Kolyvagin, the converse direction remains largely open. This work focuses on the rank one case of the rank part of the Birch–Swinnerton-Dyer conjecture, namely the implication rank E(ℚ) = 1 ⇒ L(E,1) = 0. We provide a precise and self-contained exposition of the arithmetic–analytic bridge underlying this statement via parity arguments. Under standard arithmetic hypotheses, including the finiteness of the p-primary part of the Tate–Shafarevich group and good ordinary reduction at an odd prime p, we explain how the parity of the algebraic rank is transmitted through the p-adic Selmer group and identified with the global root number using results of Nekovář and Dokchitser–Dokchitser. The functional equation then forces the vanishing of the central L-value. The contribution is not the establishment of new unconditional cases of the Birch–Swinnerton-Dyer conjecture, but a rigorous synthesis clarifying the logical structure, hypotheses, and limitations of parity-based arguments in the rank one setting.