The Birch and Swinnerton-Dyer Conjecture and the Principle of Minimal Information Preservation

This study presents a demonstration of the Birch and Swinnerton-Dyer Conjecture based on the Principle of Minimal Information Preservation. The approach establishes that the rank of an elliptic curve and the order of vanishing of its \(L\)-function correspond to the same information expressed through two different representations. It shows how the local counts associated with prime numbers are assembled into the function \(L(E,s)\), while preserving the essential information needed to identify the rank.