What does this research mean for the field?

Computational evaluation of local prime Euler products confirms exact arithmetic-analytic rank equality for rank 0 and 1 elliptic curves, while demonstrating that finite prime truncation cannot universally prove the Birch and Swinnerton-Dyer conjecture for higher-rank curves. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

To evaluate the Birch and Swinnerton-Dyer conjecture using numerical audits across elliptic curves and examine the computational limitations of finite Euler products for higher ranks.

September 14, 2026Open Access

Dual-Path Arithmetic and Analytic Audit of the Birch and Swinnerton-Dyer Conjecture: Elliptic Curve L-Function Zeros, Local Euler Products (p < 1000), and Higher-Rank Arithmetic Barriers

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To evaluate the Birch and Swinnerton-Dyer conjecture using numerical audits across elliptic curves and examine the computational limitations of finite Euler products for higher ranks.
Implemented a dual-path framework auditing elliptic curves with conductors N_E ∈ [11, 5077] using local prime Euler products truncated at p < 1000.
Evaluated exact L-function values and first derivatives at central point s = 1 for rank 0 and rank 1 curves to verify theoretical rank equality.
Analyzed partial Euler product convergence delays and Tate-Shafarevich group finiteness barriers for higher-rank curves (r ≥ 2).
Demonstrated 100% exact rank equality for evaluated rank r = 0 curves (e.g., 11a1, L(E, 1) = 0.25384) and rank r = 1 curves (e.g., 37a1, L(E, 1) = 0.00000, L'(E, 1) = 0.30599).
Identified severe logarithmic convergence delays in partial Euler products truncated at p < 1000 when applied to higher-rank curves such as 5077a1 (r = 3).
Confirmed that establishing universal finiteness for the Tate-Shafarevich group remains fundamentally out of reach using finite computational datasets.

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Implication

Computational audit demonstrates exact rank equality in low-rank elliptic curves, highlighting severe finite convergence barriers for higher-rank Birch and Swinnerton-Dyer verification.

Cite This Study

Gastón Alejandro Rovetta Benvenuto (2026) studied this question.

synapsesocial.com/papers/6aa7b2f70926e14a848b199d https://doi.org/10.5281/zenodo.22713139

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Implication

Computational audit demonstrates exact rank equality in low-rank elliptic curves, highlighting severe finite convergence barriers for higher-rank Birch and Swinnerton-Dyer verification.

Dual-Path Arithmetic and Analytic Audit of the Birch and Swinnerton-Dyer Conjecture: Elliptic Curve L-Function Zeros, Local Euler Products (p < 1000), and Higher-Rank Arithmetic Barriers

Key Points

Discussion

Implication

Cite This Study

Dual-Path Arithmetic and Analytic Audit of the Birch and Swinnerton-Dyer Conjecture: Elliptic Curve L-Function Zeros, Local Euler Products (p < 1000), and Higher-Rank Arithmetic Barriers

Key Points

Discussion

Implication

Cite This Study

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