On the p-Part of the Birch–Swinnerton–Dyer Formula via Selmer Descent

This paper supplies the Gate-3 interface in a modular Selmer descent program for elliptic curves over ℚ. Assuming Selmer rank control and finiteness of the p-primary Shafarevich–Tate group, it identifies the p-part of the Birch–Swinnerton–Dyer (BSD) formula using existing p-adic height and Iwasawa-theoretic results. Specifically, the paper assumes the Selmer rank bound established in “Selmer-Structure Descent and Rank-2 Upper Bounds via Active Primes” (Zenodo DOI: https://doi.org/10.5281/zenodo.18344561) and finiteness of the p-primary Shafarevich–Tate group deduced in “Poitou–Tate Descent and Finiteness of Sha[p∞] from Selmer Rank Bounds” (Zenodo, 2026). Under these hypotheses, classical results of Perrin–Riou, Kato, Skinner–Urban, and others imply the p-adic valuation of the leading term in the BSD formula in ranks 0, 1, and 2. No new analytic input is introduced. The contribution of this paper is to show that, once Selmer-theoretic obstructions are removed via descent, the remaining p-part of BSD follows from existing arithmetic machinery. This work completes the Selmer-side of the BSD program at the p-primary level under standard hypotheses, without claiming a proof of BSD itself.