What question did this study set out to answer?

To provide a comprehensive proof of the Birch–Swinnerton–Dyer conjecture concerning elliptic curves over the rational numbers.

March 2, 2026Open Access

The Birch–Swinnerton–Dyer conjecture - Complete Proof

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To provide a comprehensive proof of the Birch–Swinnerton–Dyer conjecture concerning elliptic curves over the rational numbers.
Utilizes a modular approach with proof nodes NODE0 to NODE19.
Determines the valuation of the Tate–Shafarevich group via two-channel mechanisms.
Employs Selmer/Iwasawa structures for upper control and explicit reciprocity for lower control.
Establishes that ordₛ₌₁ L(E, s) equals the rank of E(ℚ).
Demonstrates the finiteness of the Tate–Shafarevich group.
Presents the full leading-term formula at s = 1 incorporating standard arithmetic factors.

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Establishes the Birch–Swinnerton–Dyer conjecture for elliptic curves over ℚ, suggesting concrete implications for related mathematical structures.

Maximus Shlygin (2026) studied this question.

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Establishes the Birch–Swinnerton–Dyer conjecture for elliptic curves over ℚ, suggesting concrete implications for related mathematical structures.