Khubaib Haider

Topological Torus Dynamics and Rank 3 Geometric Quantization in Elliptic Curves: A Physical Vortex Model for the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer (BSD) Conjecture bridges the arithmetic rank of an elliptic curve E(Q) to the analytical order of vanishing of its complex L-function at the central critical point s=1. While algebraic and modular methods face substantial barriers for curves of rank...

Topological Torus Dynamics and Rank 3 Geometric Quantization in Elliptic Curves: A Physical Vortex Model for the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer (BSD) Conjecture bridges the arithmetic rank of an elliptic curve E(Q) to the analytical order of vanishing of its complex L-function at the central critical point s=1. While algebraic and modular methods face substantial barriers for curves of rank...

A Topological Hydrodynamic Framework for High-Rank Elliptic Curves: Geometric Realization of the Birch and Swinnerton-Dyer Invariants up to Rank 5

We introduce a deterministic topological and fluid-mechanical correspondence for rational points on elliptic curves over Q, extending the framework systematically from Rank 1 through Rank 5. By establishing a Bohr-Sommerfeld-type circulation quantization condition (\Gamma_k = n_k...