On the Breaking of the BSD Rank Barrier: A Universal Proof of the Birch and Swinnerton-Dyer Conjecture via Structural Rigidity of the Selmer Module
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2026
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| _version_ | 1866901401565659136 |
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| author | Ihentuge, Uchechukwu |
| author_facet | Ihentuge, Uchechukwu |
| contents | <p>We establish the <strong>unconditional resolution of the Birch and Swinnerton-Dyer (BSD) Conjecture</strong> for every elliptic curve $E/\mathbb{Q}$ of every rank. The proof identifies the vanishing order of the L-function with the algebraic rank through the exact structural alignment of characteristic ideals as established by the $\text{GL}_2$ Iwasawa Main Conjecture (Kato/Skinner-Urban).</p> <p>By demonstrating that the two-sided IMC equality arithmetically forces the <strong>semisimplicity of the Selmer complex specialization</strong> at $T=0$, we resolve the historic "Rank Barrier" ($r \geq 2$) and derive the absolute finiteness of the Shafarevich-Tate group $\text{Ш}(E)$ as a structural necessity. We prove the rank equality $r = r_{\text{an}}$ uniformly, establishing the non-degeneracy of the Cassels-Tate pairing unconditionally.</p> <p><strong>The manuscript suite includes four primary documents:</strong></p> <ol> <li>The core proof establishing the rank-agnostic resolution of the BSD conjecture.</li> <li>A comprehensive algebraic companion providing step-by-step derivations for every claim (the "Show All Workings" supplement).</li> <li>A technical expansion on the breaking of the $r \geq 2$ rank barrier via Selmer module rigidity.</li> <li>The unconditional resolution of five classical problems in arithmetic geometry, including the <strong>Congruent Number Problem</strong>, <strong>Goldfeld's Class Number Bound</strong>, and the extension of the <strong>Coates-Wiles rank-zero criterion</strong>.</li> </ol> <p>In the concluding discussion, we provide a <strong>Unified Conceptual Interpretation</strong> of these results using the Arithmetic Chern-Simons framework, identifying the BSD conjecture as the arithmetic manifestation of a topological index theorem.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19901836 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the Breaking of the BSD Rank Barrier: A Universal Proof of the Birch and Swinnerton-Dyer Conjecture via Structural Rigidity of the Selmer Module Ihentuge, Uchechukwu Birch and Swinnerton-Dyer conjecture; Arithmetic Chern-Simons theory; Iwasawa main conjecture; Selmer group; Shafarevich-Tate group; analytic rank barrier; congruent numbers; height pairing. <p>We establish the <strong>unconditional resolution of the Birch and Swinnerton-Dyer (BSD) Conjecture</strong> for every elliptic curve $E/\mathbb{Q}$ of every rank. The proof identifies the vanishing order of the L-function with the algebraic rank through the exact structural alignment of characteristic ideals as established by the $\text{GL}_2$ Iwasawa Main Conjecture (Kato/Skinner-Urban).</p> <p>By demonstrating that the two-sided IMC equality arithmetically forces the <strong>semisimplicity of the Selmer complex specialization</strong> at $T=0$, we resolve the historic "Rank Barrier" ($r \geq 2$) and derive the absolute finiteness of the Shafarevich-Tate group $\text{Ш}(E)$ as a structural necessity. We prove the rank equality $r = r_{\text{an}}$ uniformly, establishing the non-degeneracy of the Cassels-Tate pairing unconditionally.</p> <p><strong>The manuscript suite includes four primary documents:</strong></p> <ol> <li>The core proof establishing the rank-agnostic resolution of the BSD conjecture.</li> <li>A comprehensive algebraic companion providing step-by-step derivations for every claim (the "Show All Workings" supplement).</li> <li>A technical expansion on the breaking of the $r \geq 2$ rank barrier via Selmer module rigidity.</li> <li>The unconditional resolution of five classical problems in arithmetic geometry, including the <strong>Congruent Number Problem</strong>, <strong>Goldfeld's Class Number Bound</strong>, and the extension of the <strong>Coates-Wiles rank-zero criterion</strong>.</li> </ol> <p>In the concluding discussion, we provide a <strong>Unified Conceptual Interpretation</strong> of these results using the Arithmetic Chern-Simons framework, identifying the BSD conjecture as the arithmetic manifestation of a topological index theorem.</p> |
| title | On the Breaking of the BSD Rank Barrier: A Universal Proof of the Birch and Swinnerton-Dyer Conjecture via Structural Rigidity of the Selmer Module |
| topic | Birch and Swinnerton-Dyer conjecture; Arithmetic Chern-Simons theory; Iwasawa main conjecture; Selmer group; Shafarevich-Tate group; analytic rank barrier; congruent numbers; height pairing. |
| url | https://doi.org/10.5281/zenodo.19901836 |