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

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 theelliptic curve L-function. We establish a strict spectral correspondence between the eigenvalues ofthese matrices and the squares of the distances from the critical point s = 1 to the zeros of L(E, s).Using mathematical induction and the monotone convergence theorem for self-adjoint operators, weextend 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-termformula relating the first non-vanishing coefficient of the Taylor expansion of L(E, s) at s = 1 tothe 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. Key words and phrases. Birch–Swinnerton–Dyer conjecture; elliptic curve; L-function; self-adjoint operator; spectral correspondence; Mordell–Weil rank. 2020 Mathematics Subject Classification. 11G05, 11M41, 47A10, 14H52.