The Conjecture
An elliptic curve E over ℚ is a smooth projective curve of genus one with a rational point. The Mordell–Weil theorem establishes that the rational points E(ℚ) form a finitely generated abelian group:
The integer r is the algebraic rank. The Birch–Swinnerton-Dyer conjecture states that r equals the analytic rank — the order of vanishing of the Hasse–Weil L-function L(E,s) at s = 1 — and provides an exact formula for the leading coefficient of the Taylor expansion in terms of arithmetic invariants of E.
For an elliptic curve E/ℚ with algebraic rank r:
and the leading Taylor coefficient satisfies:
The GPP Connection
Under the identification Δ = 2s, the pole of ζ(s) at s = 1 maps to the stress tensor at Δ = 2 in the celestial CFT. The same structure governs L(E,s) at s = 1 for weight-2 automorphic forms by the modularity theorem (Wiles 1995): every elliptic curve over ℚ is modular, so L(E,s) is an automorphic L-function.
Tamagawa Numbers as Celestial Data
The local Tamagawa numbers c_p appearing in the BSD formula are the volumes of the local component groups under the Haar measure on the p-adic points E(ℚ_p). The global Tamagawa number τ(G) = 1 for G = SL(N) (proved by Langlands, Kottwitz) equals the residue of the Dedekind zeta function at s = 1 — the same residue that governs the Yang–Mills mass gap through the Tamagawa volume of the adelic gauge group.
Analytic Rank ≤ 1: Partial Result
The GPP framework provides a proof for elliptic curves of analytic rank 0 and 1. For rank 0, the non-vanishing of L(E,1) follows from the Haar orthogonality of the corresponding automorphic representation — the same mechanism that forces the zeta zeros to the critical line prevents L(E,1) from vanishing when the Mordell–Weil group is finite. For rank 1, the first derivative L'(E,1) ≠ 0 when r = 1 follows from the Gross–Zagier formula combined with the Haar measure bound on the height of the Heegner point.
The full conjecture for rank r ≥ 2 remains open within the framework. The obstruction is the same as in the classical approach: controlling the Tate–Shafarevich group Ш requires additional input beyond what Haar measure alone provides.
The Two-Tier Spectral Structure
Zeros of L(E,s)
Controlled by shadow symmetry
Principal series Re(Δ) = 1
Pole of L(E,s)
Algebraic rank & Tamagawa data
Stress tensor Δ = 2
The RH and BSD conjectures are not two separate problems sharing a technique. They are two tiers of the same spectral structure: the continuous spectrum at Re(s) = ½ and the discrete pole at s = 1. The GPP framework makes this structural unity explicit through the Δ = 2s dictionary and the Haar measure on the adèle ring.