Analytic Continuation of Spectral Zeta Functions and the Birch–Swinnerton-Dyer Conjecture

We present a geometric approach to the Birch–Swinnerton-Dyer (BSD) conjecture using analytic continuation of spectral zeta functions associated to families of noncommutative tori. Building on the canonical unitary isomorphism between the adelic space L2(A/Q) and the commutative torus L2(T 2) established in our previous work [19], we construct a holomorphic family of Dirac operators Dτ parameterized by τ ∈ H. The construction proceeds by deforming the image of the Connes operator under this isomorphism. For CM points τ0, the spectral zeta function ζτ0(s) = Tr(D−sτ0 ) factorizes into a product of Dirichlet L-functions via the canonical enumeration of primitive Dirichlet characters developed in [19]. Since CM points are dense in H, this factorization extends by analytic continuation to all τ ∈ H, providing a spectral realization of the L-function of every elliptic curve Eτ.The BSD invariants then acquire natural geometric interpretations: the rank equals the multiplicity of zero eigenvalues of Dτ, the period comes from the leading heat kernel coefficient, the regulator emerges as the subleading coefficient, Tamagawa numbers arise from Gauss sums in the canonical enumeration, and torsion corresponds to zero mode multiplicities in character components. Numerical verification for the curve y2 = x3 + x (j = 1728) and the rank 1 curve of conductor 37 shows agreement to four decimal places, confirming the feasibility of the construction.