Goldbach–Frey Jacobians as Weil Restrictions: Trace Vanishing, Asai L-Functions, and Modularity via Q(i)

For the Goldbach–Frey curve C: y² = x(x²−p²)(x²−q²) with p ≠ q distinct odd primes, we prove that the Frobenius trace a_r = 0 at every good prime r ≡ 3 (mod 4). This trace vanishing law is the signature of an induced Galois representation: the involution (x,y) → (−x, iy) defined over Q(i) forces Jac(C) to be isogenous to the Weil restriction ResQ(i)/Q(E) of an elliptic curve E/Q(i), with E₂ ≅ E₁^(−1) as a quadratic twist by −1. The degree-4 L-function is an Asai L-function (Langlands lift from GL₂(Q(i)) to GSp₄(Q)), and the associated Siegel paramodular form is an endoscopic lift. We compute refined local L-factors at all bad odd primes—including split/non-split node analysis via quadratic residues—and establish modularity via automorphic induction from GL₂(Q(i)).