Palindromic Genus-2 Curves as Weil Restrictions: Trace Vanishing, Asai L-Functions, and Modularity via Q(i)

For the palindromic genus-2 curve C₀, ₁: y² = x (x²−a²) (x²−b²) over Q, where a, b ∈ Q* with a² ≠ b², we prove that the Frobenius trace aᵣ = 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 the ℓ-adic representation of Jac (C₀, ₁) to be induced from Gₐ (₈) to GQ, so that Jac (C₀, ₁) is isogenous over Q to the Weil restriction of an elliptic curve E/Q (i). The degree-4 L-function is therefore an Asai L-function, and modularity reduces to that of E/Q (i). We compute explicit local L-factors at all bad odd primes and present computational evidence (130, 000 Dirichlet coefficients) for the wild conductor exponent at p=2. Version 2 changes: Title generalized from "Goldbach–Frey" to "Palindromic Genus-2 Curves" to reflect that the Goldbach constraint p+q=2N plays no role in any proof. Endomorphism ring proof (Theorem 3. 1) completed in full. Modularity proof (Theorem 8. 1) restructured as a conditional theorem with explicit hypotheses verified for (a, b) = (3, 7). Functional equation section recharacterized as computational evidence. Automorphism group, discriminant formula, and local L-factor signs corrected.