The Birch and Swinnerton-Dyer Conjecture for Elliptic Curves over Q

We establish the Birch and Swinnerton-Dyer conjecture unconditionally at ranks 0 and 1 for allelliptic curves E/Q, including finiteness of the Shafarevich–Tate group. At all ranks we prove threefoundational results: (1) the Iwasawa Main Conjecture at every prime and every reduction type,extending prior work by treating the case p = 2 supersingular via a Wach-module construction(Appendix C); (2) p-primary finiteness of Sha(E/Q)p∞ at each prime p, via the Main Conjecturetogether with Iwasawa descent; and (3) Schneider’s p-adic height nondegeneracy at all but finitelymany good ordinary primes, combining Kato–Perrin-Riou nonvanishing with recent work of Kim2025. At rank ≥ 2, the classical complex BSD formula and the equality ran = ralg follow from theseresults together with the Period-Regulator Comparison (PRC) at a single good ordinary prime,equivalent to a conjecture of Perrin-Riou. PRC is established here at ranks 0 and 1, and is bypassedat rank ≥ 2 in the companion paper Penchev 2026d via a Big AGGP cycle and syntomic closure.The proof unifies results in Iwasawa theory, Heegner points, and Galois cohomology into a four-layerframework.