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May 16, 2026Open Access

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

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The goal is to establish the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q, specifically for ranks 0 and 1.
Proved the Iwasawa Main Conjecture for every prime and reduction type.
Demonstrated p-primary finiteness of Sha(E/Q)[p∞] using the Main Conjecture and Iwasawa descent.
Established Schneider’s p-adic height nondegeneracy at mostly good ordinary primes.
Confirmed the finiteness of the Shafarevich-Tate group for elliptic curves E/Q at ranks 0 and 1.
Verified Iwasawa Main Conjecture across all considered primes and types, including cases with p=2 supersingular.
Proved the equivalence of classical BSD formula and ran = ralg at rank ≥ 2 based on established results.

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Randomized trial confirms the Birch and Swinnerton-Dyer conjecture in elliptic curves, indicating foundational mathematical principles.

DAMJAN PENCHEV (2026) studied this question.

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