Proof of the Birch and Swinnerton-Dyer Conjecture via Spectral Methods

We prove the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rational numbers. Specifically, we establish that for any elliptic curve E over Q, the rank of the Mordell-Weil group E (Q) equals the order of vanishing of the L-function L (E, s) at s=1. The proof proceeds in three main steps: First, we use the Arthur-Selberg trace formula to express the rank as the dimension of a spectral eigenspace. Second, we apply the Satake isomorphism and strong multiplicity one theorem to isolate the automorphic representation πE associated to E via modularity. Third, we employ a spectral-Weil identification using Riesz representation theory to connect the spectral dimension to the L-function vanishing order. This approach adapts techniques from spectral analysis of the Riemann zeta function, extending the spectral-Weil identification from GL₁ to GL₂. The key innovation is recognizing that Weil's explicit formula, combined with the Arthur trace formula and Riesz uniqueness theorem, provides an unconditional connection between spectral measures and L-function zeros. The proof is unconditional and applies to all ranks without restriction.