Paramodular Conjecture for the Goldbach–Frey Jacobian: From GSp(4) to Bianchi Modularity via Weil Restriction

The Brumer-Kramer paramodular conjecture predicts that every abelian surface over Q with EndQ (A) = Z and conductor N corresponds to a weight-2 Siegel paramodular newform of level N. We apply this conjecture to the Goldbach-Frey Jacobian Jac (C₍, ₏) from our companion paper, where C₍, ₏: y² = x (x²−p²) (x²− (2N−p) ²). We prove three unconditional results: (1) the explicit conductor of Jac (C₍, ₏) from the discriminant Δ = 2¹² p⁶ (2N−p) ⁶ (N−p) ⁴N⁴, with fᵣ = 2 at both static and dynamic conduit primes (two nodes on an irreducible curve give b₁ = 2 and local Euler factor (1−r⁻ˢ) ⁻²) ; (2) the Weil restriction structure Jac ~ Res₊/ₐ (Eₚ) reduces the paramodular conjecture to the modularity of Eₚ over K = Q (√−1) via the Asai transfer, with symplectic descent guaranteed by the exterior square L-function criterion of Jacquet-Shalika; (3) the residual representation ρ̄₄䂹, ₂ is universally reducible (the 2-2 coincidence: all 2-torsion is Q-rational), while ρ̄₄䂹, ₃ is generically absolutely irreducible, identifying ℓ = 3 as the first prime where the ten-author modularity lifting theorem may apply. For fixed N, the paramodular conjecture holds for Jac (C₍, ₏) for all but finitely many Goldbach primes p (Corollary 5. 2). The paper does not prove the Goldbach conjecture; it provides a bridge from GSp (4) to GL (2) /K where existing tools apply.