What does this research mean for the field?

Infinitely many positive quadratic twists of optimal elliptic curves with analytic rank zero and controlled additive reduction have analytic and algebraic rank zero, finite Tate-Shafarevich group, and satisfy the exact strong Birch-Swinnerton-Dyer formula at every prime. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Establish the exact strong Birch–Swinnerton-Dyer (BSD) formula at every prime for infinitely many positive quadratic twists of elliptic curves with controlled additive reduction.

September 2, 20260 citationsOpen Access

Strong Birch-Swinnerton-Dyer Formula in Quadratic-Twist Families with Controlled Additive Reduction

Strong BSD in Quadratic-Twist Families with Controlled Additive Reduction

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Authors

LBLiam Birkett

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Overview

Mathematical analysis proves the exact strong Birch–Swinnerton-Dyer formula across infinite quadratic twists of elliptic curves, extending rank-zero verification beyond semistable settings.

Key Points

Establish the exact strong Birch–Swinnerton-Dyer (BSD) formula at every prime for infinitely many positive quadratic twists of elliptic curves with controlled additive reduction.
Constructed a unified Chebotarev family combining exact two-primary formulas across ordinary, supersingular, and reducible-residual primes away from the twist discriminant.
Treated ramified twist primes via Kato's zeta elements, ramified finite-character specialization in Skinner–Urban Eisenstein data, and primitive finite Kato Kolyvagin systems with integral Néron normalization.
Demonstrated that infinitely many positive quadratic twists of the given curve have analytic rank zero, algebraic rank zero, and a finite Tate–Shafarevich group.
Verified the exact strong Birch–Swinnerton-Dyer equality at every prime in a controlled nonsemistable setting featuring potentially good supersingular reduction at 11.

Cite This Study

Liam Birkett (2026) studied this question.

synapsesocial.com/papers/6a97e29ec562ede874ec6d06 https://doi.org/10.5281/zenodo.22209855

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September 2, 20260 citationsOpen Access

Strong Birch-Swinnerton-Dyer Formula in Quadratic-Twist Families with Controlled Additive Reduction

Strong BSD in Quadratic-Twist Families with Controlled Additive Reduction

View Full Paper

Ask AI

Bookmark

Authors

LBLiam Birkett

Discussion

Member takes

Overview

Mathematical analysis proves the exact strong Birch–Swinnerton-Dyer formula across infinite quadratic twists of elliptic curves, extending rank-zero verification beyond semistable settings.

Key Points

Establish the exact strong Birch–Swinnerton-Dyer (BSD) formula at every prime for infinitely many positive quadratic twists of elliptic curves with controlled additive reduction.
Constructed a unified Chebotarev family combining exact two-primary formulas across ordinary, supersingular, and reducible-residual primes away from the twist discriminant.
Treated ramified twist primes via Kato's zeta elements, ramified finite-character specialization in Skinner–Urban Eisenstein data, and primitive finite Kato Kolyvagin systems with integral Néron normalization.
Demonstrated that infinitely many positive quadratic twists of the given curve have analytic rank zero, algebraic rank zero, and a finite Tate–Shafarevich group.
Verified the exact strong Birch–Swinnerton-Dyer equality at every prime in a controlled nonsemistable setting featuring potentially good supersingular reduction at 11.

Cite This Study

Liam Birkett (2026) studied this question.

synapsesocial.com/papers/6a97e29ec562ede874ec6d06 https://doi.org/10.5281/zenodo.22209855

View Full Paper

Ask AI

Bookmark