This repository contains the formalization of the Birch–Swinnerton-Dyer (BSD) Conjecture using the Universal Object Reference (UOR) framework. The work develops a unified embedding of elliptic curve data (including rational points, the group law, and the (L)-function) into a Clifford algebra setting and constructs associated operators whose spectral properties mirror the analytic behavior of (L(E,s)). The ultimate goal is to provide a conceptual and computational framework in which the equality
[ \mathrm{rank},E(\mathbb{Q}) = \mathrm{ord}_{s=1} L(E,s) ]
emerges naturally.
Key features of this work include:
- UOR Embedding: A unified construction embedding the Mordell–Weil group (E(\mathbb{Q})) and the zeros of the (L)-function (L(E,s)) into a real Clifford algebra (\Cl(V)) with an associated Lie group (G) of symmetries.
- Operator Construction: The formulation of an operator (H_E) whose spectrum (in particular, the multiplicity of the zero eigenvalue) reflects the analytic rank of (E).
- Trace Formula: A derivation of a trace formula within the UOR framework that links the spectral data of (H_E) with geometric data (such as Frobenius actions and rational point counts).
- Worked Example: A detailed example is provided for the elliptic curve [ y^2 + y = x^3 - x, ] (LMFDB label 37.a1) which has conductor 37, rank 1, trivial torsion, and a simple zero of (L(E,s)) at (s=1).
- Numerical Computations: Numerical verifications (using SageMath and LMFDB data) confirm that the computed (L)-function derivatives, regulators, and periods match the BSD prediction.
- Connections to Physics: Expanded discussion on analogies with quantum mechanics (e.g. the Berry–Keating model), string theory, and condensed matter physics.
- Future Directions and Open Questions: Suggestions for extending the framework to higher-dimensional abelian varieties, refining the operator (H_E), and exploring further computational and theoretical problems.
- README.md: This file.
- uor-bsd1.tex: The main document.
- uor-bsd2.tex: LaTeX source file for Appendix 1 (Existence of UOR Embedding).
- uor-bsd3.tex: LaTeX source file for Appendix 2 (Detailed Examples, Numerical Computations, and Future Directions).
This project opens up several exciting directions:
- Generalizations: Extending the UOR framework to abelian varieties, motives, and other (L)-functions.
- Operator Refinement: Constructing an explicit and canonical operator (H_E) within the Clifford algebra.
- Trace Formula: Developing a rigorous trace formula in the UOR context that directly proves the equality of analytic and algebraic ranks.
- Computational Exploration: Building a UOR module in SageMath to streamline BSD verification and explore potential new algorithms.
- Interdisciplinary Links: Investigating deeper connections to quantum mechanics, string theory, and condensed matter physics to leverage methods from these fields in solving deep arithmetic problems.
This work is released under the MIT License.