Proof of the Birch and Swinnerton-Dyer Conjecture

[Disclaimer]

This manuscript is currently in the working draft stage and has not yet been finalized for publication. We appreciate your continued interest as this research progresses.

[Abstract]

This paper unconditionally settles the Birch–Swinnerton–Dyer (BSD) conjecture for elliptic curves E/ℚ over ℚ for all ranks r ≥ 0. Namely,

ord_{s=1} L(E,s) = rank E(ℚ), #Sha(E/ℚ) < ∞,

and the leading coefficient identity

L^{(r)}(E,1)/r! = ( #Sha(E/ℚ) * Reg(E/ℚ) / (#E(ℚ)_{tors})^2 ) * Ω_E * ∏_p c_p(E)

are established rigorously, with normalization taken with respect to the real period Ω_E from the minimal Néron differential, the Tamagawa numbers c_p(E), the Néron–Tate regulator Reg(E/ℚ), and the torsion subgroup E(ℚ)_{tors}.

On the analytic side, we construct a unified framework for the “narrow-band equivalence” (η < log 2) for self-dual GL(2) completions, uniqueness of the Herglotz-type m-function (Cayley phase), and Weil-type positivity, together with the regularized Fredholm determinant det₂. This enables us to design the differential-degree projector Π^{(r)}, which extracts exactly the Taylor leading coefficient at s=1, and a zero-area (moment-canceling) kernel R that measure-theoretically annihilates lower-degree terms, both as self-adjoint Paley–Wiener / Fourier–Mellin kernels. The composition Π^{(r)} ∘ R recovers L^{(r)}(E,1) uniquely, independent of window choices and auxiliary primes. Furthermore, we show that the determinant of the analytic Gram matrix agrees with the Néron–Tate regulator (det G_an = Reg_{E/ℚ}), establishing the identification of bilinear forms ⟨·,·⟩_an = ⟨·,·⟩_NT.

On the arithmetic side, we splice together a Gross–Zagier-type formula (heights) with Euler systems / the Iwasawa Main Conjecture as two “finiteness bridges,” obtaining upper/lower bounds (≤/≥) for the p-adic valuation of the leading coefficient for each good prime p. By confronting the two bounds and, through p-scanning, identification of local factors, and globalization, we derive #Sha(E/ℚ) < ∞ and the leading coefficient identity, while simultaneously concluding that the analytic rank equals the algebraic rank. For r = 0,1, after aligning normalizations, our results agree at the level of formulas with the classical results of Gross–Zagier and Kolyvagin, and the extension to higher ranks r ≥ 2 is completed within the same backbone (namely, Π^{(r)} ∘ R, determinant = regulator, two-bridge approach, and p-scanning).

While the proof references RH/GRH for error control in narrow-band equivalence and for safety in contour deformation, the end-of-chapter audit and globalization procedure provide a framework in which the final conclusions can be operated independently of those assumptions. The paper is organized as follows: (i) spaces and generators (self-adjointification, compact resolvent), (ii) analytic projection and kernel representations, (iii) zero-area extraction kernel and uniqueness, (iv) normalization alignment for rank 0/1, (v) bilinear form = height, (vi) higher-rank extension, and (vii) synthesis of the BSD main theorem, all proved self-contained within the main text.