What question did this study set out to answer?

To establish a complete proof of the Birch and Swinnerton-Dyer conjecture and the finiteness of the Tate-Shafarevich group for all elliptic curves defined over the rational numbers.

August 14, 2026Open Access

A Cogenetic Proof of the Birch and Swinnerton–Dyer Conjecture

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To establish a complete proof of the Birch and Swinnerton-Dyer conjecture and the finiteness of the Tate-Shafarevich group for all elliptic curves defined over the rational numbers.
Constructed an elliptic-curve carrier that unifies local Euler factors, analytic germs at s=1, rational points, and Selmer groups.
Formulated a finite perfect Selmer complex through Poitou-Tate duality with analytic registration via modular zeta elements and arithmetic registration via Kummer-Selmer sequences.
Employed cogenetic determinant non-collapse across finite descent refinements to preserve determinant elements, transition maps, and filtration orders.
Proved the exact equality between the analytic order of vanishing of the L-function at s=1 and the Mordell-Weil rank.
Recovered the complete Birch and Swinnerton-Dyer leading-coefficient formula, including the Néron period, regulator, Tamagawa factors, and torsion denominator.
Demonstrated the finiteness of the Tate-Shafarevich group across all rational elliptic curves.

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Implication

Theoretical proof demonstrates the Birch and Swinnerton-Dyer conjecture for rational elliptic curves, resolving the equivalence of analytic rank and arithmetic invariants.

Cite This Study

Petrik et al. (2026) studied this question.

synapsesocial.com/papers/6a7ec78db70b84ec8b913f87 https://doi.org/10.5281/zenodo.21907044

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Implication

Theoretical proof demonstrates the Birch and Swinnerton-Dyer conjecture for rational elliptic curves, resolving the equivalence of analytic rank and arithmetic invariants.

A Cogenetic Proof of the Birch and Swinnerton–Dyer Conjecture

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Discussion

Implication

Cite This Study

A Cogenetic Proof of the Birch and Swinnerton–Dyer Conjecture

Key Points

Discussion

Implication

Cite This Study

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