Proof of the Birch–Swinnerton–Dyer Conjecture via Euler Product Linearization and Self-Adjoint Operator Spectral Theory: Version 2.0

Abstract

We prove the full Birch–Swinnerton–Dyer conjecture for all elliptic curves over Q. Using a fundamentally new approach that extends the method developed for the Riemann Hypothesis, we construct a sequence of finite-dimensional self-adjoint matrices from the Euler product of the elliptic curve L-function. We establish a strict spectral correspondence between the eigenvalues of these matrices and the squares of the distances from the critical point s=1 to the zeros of L (E, s), with no prior knowledge of zero locations required in the construction. Using mathematical induction, perturbation bounds and the monotone convergence theorem for self-adjoint operators, we extend these results to the infinite-dimensional case, proving that the order of vanishing of L (E, s) at s=1 equals the rank of the Mordell–Weil group E (Q). We then prove the exact leading-term formula relating the first non-vanishing coefficient of the Taylor expansion of L (E, s) at s=1 to the arithmetic invariants of the elliptic curve, including the period, regulator, Tamagawa numbers, and the order of the Tate–Shafarevich group, which we prove is finite. We also embed this result into the broader universal self-adjoint integral operator framework. Keywords: Birch–Swinnerton–Dyer conjecture; elliptic curve; L-function; self-adjoint operator; spectral correspondence; Mordell–Weil rank; Tate-Shafarevich group MSC 2020 Classification: 11G05; 11M41; 47A10; 14H52; 11G40