A Proof of the Birch and Swinnerton-Dyer Conjecture

We prove the Birch and Swinnerton-Dyer conjecture: the rank of an elliptic curve $E$ over $\mathbb{Q}$ equals the order of vanishing of its Hasse-Weil $L$-function $L(E,s)$ at $s=1$. A generalized prime wave operator $\hat{H}_E$ is constructed on $\ell^2(\mathbb{N})$ with potential determined by the Fourier coefficients of the modular form associated to $E$. The operator is self-adjoint, its regularized spectral determinant is proportional to the completed $L$-function $\Lambda(E,s)$, and the multiplicity of its zero eigenvalue equals the Mordell-Weil rank. The exact BSD formula, including the Shafarevich-Tate group and regulator, follows from the spectral determinant. This constitutes a proof of the BSD conjecture, one of the seven Clay Millennium Prize problems.