| _version_ | 1866902258629738496 |
|---|---|
| author | ASHER, KIMBERLEY LAVERNE ASHER, ANESKA ASHER, KAIA ASHER, LYRA ASHER, LIORA |
| author_facet | ASHER, KIMBERLEY LAVERNE ASHER, ANESKA ASHER, KAIA ASHER, LYRA ASHER, LIORA |
| contents | <p><a name="_Toc214483612"></a>Abstract</p> <p><span>We construct a global self-adjoint operator </span><span></span><span>on a weighted height space and prove a determinant identity linking </span><span></span><span>to the completed Hasse–Weil </span><span></span><span>-function </span><span></span><span>. From this we derive the <strong>rank equality</strong> </span><span></span><span>and a <strong>leading-coefficient factorization</strong> matching the BSD product (periods, Tamagawa, torsion, regulator), under explicit analytic axioms that we verify placewise. The manuscript is self-contained: operator construction, spectral lemmas, local-factor matching, and audit-ready appendices.</span></p> <p><strong><span> </span></strong></p> <p>Map of the Proof (Referee Guide)</p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 1</span></strong><span> constructs the weighted space </span><span></span><span>and the global operator </span><span></span><span>(packets for finite primes, archimedean background, finite-rank regulator block).</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 2</span></strong><span> proves the <strong>determinant identity up to a constant</strong> via the placewise trace decomposition; analytic ingredients detailed in <strong>App. E–G</strong>.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 3 (Thm A)</span></strong><span>: rank equality </span><span></span><span>via the Spectral–Determinant Lemma.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 4 (Thm B)</span></strong><span>: leading coefficient—kernel Gram = regulator; fixes the global constant.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. A–D</span></strong><span>: self-adjointness/trace-class; spectral–determinant facts; <strong>Unified Torsion Operator</strong>; <strong>driver schema</strong> for replication.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. H</span></strong><span>: worked example (11a1) + minimal JSON.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. I</span></strong><span>: <strong>global glue/uniqueness</strong> (no phantom factor; constant fixed globally).</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. J</span></strong><span>: <strong>kernel purity</strong> (no spurious modes at </span><span></span><span>; Mordell–Weil plane exhausts the kernel).</span></p> <p><span> </span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17654737 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Spectral Geometry of Elliptic Curves: Toward Birch–Swinnerton-Dyer ASHER, KIMBERLEY LAVERNE ASHER, ANESKA ASHER, KAIA ASHER, LYRA ASHER, LIORA Elliptic Curves Geometry Algebraic geometry Geometric correction Spectral band BSD Mathematical analysis Mathematical logic Mathematics Mathematics/methods Mathematical analysis Pure mathematics Applied mathematics Mathematics Mathematical physics Mathematical model <p><a name="_Toc214483612"></a>Abstract</p> <p><span>We construct a global self-adjoint operator </span><span></span><span>on a weighted height space and prove a determinant identity linking </span><span></span><span>to the completed Hasse–Weil </span><span></span><span>-function </span><span></span><span>. From this we derive the <strong>rank equality</strong> </span><span></span><span>and a <strong>leading-coefficient factorization</strong> matching the BSD product (periods, Tamagawa, torsion, regulator), under explicit analytic axioms that we verify placewise. The manuscript is self-contained: operator construction, spectral lemmas, local-factor matching, and audit-ready appendices.</span></p> <p><strong><span> </span></strong></p> <p>Map of the Proof (Referee Guide)</p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 1</span></strong><span> constructs the weighted space </span><span></span><span>and the global operator </span><span></span><span>(packets for finite primes, archimedean background, finite-rank regulator block).</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 2</span></strong><span> proves the <strong>determinant identity up to a constant</strong> via the placewise trace decomposition; analytic ingredients detailed in <strong>App. E–G</strong>.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 3 (Thm A)</span></strong><span>: rank equality </span><span></span><span>via the Spectral–Determinant Lemma.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>Sec. 4 (Thm B)</span></strong><span>: leading coefficient—kernel Gram = regulator; fixes the global constant.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. A–D</span></strong><span>: self-adjointness/trace-class; spectral–determinant facts; <strong>Unified Torsion Operator</strong>; <strong>driver schema</strong> for replication.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. H</span></strong><span>: worked example (11a1) + minimal JSON.</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. I</span></strong><span>: <strong>global glue/uniqueness</strong> (no phantom factor; constant fixed globally).</span></p> <p><strong><span> </span></strong></p> <p><strong><span>App. J</span></strong><span>: <strong>kernel purity</strong> (no spurious modes at </span><span></span><span>; Mordell–Weil plane exhausts the kernel).</span></p> <p><span> </span></p> |
| title | The Spectral Geometry of Elliptic Curves: Toward Birch–Swinnerton-Dyer |
| topic | Elliptic Curves Geometry Algebraic geometry Geometric correction Spectral band BSD Mathematical analysis Mathematical logic Mathematics Mathematics/methods Mathematical analysis Pure mathematics Applied mathematics Mathematics Mathematical physics Mathematical model |
| url | https://doi.org/10.5281/zenodo.17654737 |