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The aim is to prove the Birch and Swinnerton-Dyer conjecture for all elliptic curves over rational numbers.

February 2, 2026Open Access

The Complete Resolution of the Birch and Swinnerton-Dyer Conjecture via Iwasawa-Theoretic Descent

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The aim is to prove the Birch and Swinnerton-Dyer conjecture for all elliptic curves over rational numbers.
Combined the Main Conjecture of Iwasawa Theory with the vanishing of the μ-invariant.
Utilized Iwasawa descent to analyze p-adic L-functions and Selmer groups.
Established the finitude of the Tate-Shafarevich group through descent arguments.
Confirmed that rank(E(ℚ)) equals the order of vanishing of the L-function at s=1 for all elliptic curves.
Demonstrated that the Tate-Shafarevich group is finite for all considered curves.
Achieved perfect agreement between algebraic and analytic ranks across extensive testing of over 500,000 curves.

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Proves the Birch and Swinnerton-Dyer conjecture for elliptic curves over rational numbers, indicating a major breakthrough in number theory.

Douglas H. M. FULBER (2026) studied this question.

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Proves the Birch and Swinnerton-Dyer conjecture for elliptic curves over rational numbers, indicating a major breakthrough in number theory.