THE PRIME GEODOME AND THE SHELBYDON FRAMEWORK
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2025
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| _version_ | 1866901210321125376 |
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| author | Shelby, Cory |
| author_facet | Shelby, Cory |
| contents | <p>This article develops a new geometric framework for interpreting the arithmetic of elliptic curves by introducing two visualization tools: the Prime Geodome and the Shelbydon Toroidal Model. Using these structures, the paper provides a comparative analysis of three foundational elliptic curves—a rank-0 non-CM curve, a CM curve, and a rank-1 non-CM curve—revealing how their Frobenius traces generate distinct global geometric signatures.</p> <p>The Prime Geodome maps primes onto a geodesic sphere and colors each region according to the sign and magnitude of the normalized Frobenius trace. This produces three qualitatively different global geometries: (1) a balanced, statistically symmetric field for rank-0 curves; (2) a crystalline equatorial belt for CM curves; and (3) a hemispheric shear pattern for rank-1 curves. These patterns provide a direct, visual representation of Sato–Tate behavior, analytic rank, and functional equation parity.</p> <p>The Shelbydon Toroidal Framework provides a complementary two-dimensional geometric model. By embedding digital-root lattices into a periodic torus, the model interprets deeper arithmetic invariants—such as the regulator and the Tate–Shafarevich group—as geometric deformation modes. Rank-0 curves correspond to toroidal symmetry, CM curves to rigid frozen cores, and rank-1 curves to anti-diagonal shear proportional to the regulator. Internal torsional tension within the torus is used to provide an intuitive geometric analogue for the local–global obstruction measured by .</p> <p>Taken together, these geometric tools offer a unified interpretive language for the arithmetic of elliptic curves. The article proposes that arithmetic invariants—rank, regulator, Frobenius bias, and local–global obstruction—can be coherently viewed as distributed geometric tensions on discrete or spherical surfaces. While the work does not claim new theorems about the Birch–Swinnerton–Dyer conjecture, it provides a conceptual and visual framework through which the conjecture’s components can be interpreted as geometric phenomena.</p> <p>This article therefore sits at the intersection of number theory, geometry, visualization science, and mathematical physics. It presents a novel way to “see” deep arithmetic structure, offering both researchers and educators a powerful new toolkit for understanding global arithmetic phenomena through geometric intuition</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17859991 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | THE PRIME GEODOME AND THE SHELBYDON FRAMEWORK Shelby, Cory elliptic curves, Frobenius trace, Sato–Tate distribution, Birch–Swinnerton–Dyer conjecture, regulator, Tate–Shafarevich group, complex multiplication, arithmetic geometry, L-functions, prime geodome, toroidal deformation, digital root lattice, Shelbydon framework, rank of elliptic curves, local–global obstruction, modular forms, finite field reductions, Frobenius angles, geodesic sphere visualization, torus symmetry, number theory visualization, arithmetic dynamics, global fields, Tamagawa numbers, Mordell–Weil group, functional equation parity, Frobenius bias, geometric representation of arithmetic, arithmetic invariants, toroidal geometry <p>This article develops a new geometric framework for interpreting the arithmetic of elliptic curves by introducing two visualization tools: the Prime Geodome and the Shelbydon Toroidal Model. Using these structures, the paper provides a comparative analysis of three foundational elliptic curves—a rank-0 non-CM curve, a CM curve, and a rank-1 non-CM curve—revealing how their Frobenius traces generate distinct global geometric signatures.</p> <p>The Prime Geodome maps primes onto a geodesic sphere and colors each region according to the sign and magnitude of the normalized Frobenius trace. This produces three qualitatively different global geometries: (1) a balanced, statistically symmetric field for rank-0 curves; (2) a crystalline equatorial belt for CM curves; and (3) a hemispheric shear pattern for rank-1 curves. These patterns provide a direct, visual representation of Sato–Tate behavior, analytic rank, and functional equation parity.</p> <p>The Shelbydon Toroidal Framework provides a complementary two-dimensional geometric model. By embedding digital-root lattices into a periodic torus, the model interprets deeper arithmetic invariants—such as the regulator and the Tate–Shafarevich group—as geometric deformation modes. Rank-0 curves correspond to toroidal symmetry, CM curves to rigid frozen cores, and rank-1 curves to anti-diagonal shear proportional to the regulator. Internal torsional tension within the torus is used to provide an intuitive geometric analogue for the local–global obstruction measured by .</p> <p>Taken together, these geometric tools offer a unified interpretive language for the arithmetic of elliptic curves. The article proposes that arithmetic invariants—rank, regulator, Frobenius bias, and local–global obstruction—can be coherently viewed as distributed geometric tensions on discrete or spherical surfaces. While the work does not claim new theorems about the Birch–Swinnerton–Dyer conjecture, it provides a conceptual and visual framework through which the conjecture’s components can be interpreted as geometric phenomena.</p> <p>This article therefore sits at the intersection of number theory, geometry, visualization science, and mathematical physics. It presents a novel way to “see” deep arithmetic structure, offering both researchers and educators a powerful new toolkit for understanding global arithmetic phenomena through geometric intuition</p> |
| title | THE PRIME GEODOME AND THE SHELBYDON FRAMEWORK |
| topic | elliptic curves, Frobenius trace, Sato–Tate distribution, Birch–Swinnerton–Dyer conjecture, regulator, Tate–Shafarevich group, complex multiplication, arithmetic geometry, L-functions, prime geodome, toroidal deformation, digital root lattice, Shelbydon framework, rank of elliptic curves, local–global obstruction, modular forms, finite field reductions, Frobenius angles, geodesic sphere visualization, torus symmetry, number theory visualization, arithmetic dynamics, global fields, Tamagawa numbers, Mordell–Weil group, functional equation parity, Frobenius bias, geometric representation of arithmetic, arithmetic invariants, toroidal geometry |
| url | https://doi.org/10.5281/zenodo.17859991 |