Arithmetic Geometry of Polynomial Curves: Decomposition of the Jacobian and the Tate-Shafarevich Group in Quadratic Twists of y² = Q₇(x)

The polynomials Qq (x) = xq − (x−1) q are of significant interest in computational number theory due to their prime-generating capacities under the Bateman–Horn heuristic. In this paper, we study the arithmetic geometry of the hyperelliptic curves Cq: y² = Qq (x) over ℚ. For q = 7, the curve C₇ has genus g = 2. We prove that the splitting field of Q₇ (x) exhibits a strict C₆ Galois cyclic symmetry, allowing a non-trivial hyperelliptic involution. Through this involution, we explicitly construct a decomposition of the Jacobian, obtaining a quotient elliptic curve E₁. Furthermore, by studying the family of square-free quadratic twists E₁^ (D), we perform a 2-descent analysis. At D = 167, we demonstrate a failure of the Hasse principle by explicitly computing a non-trivial Tate–Shafarevich group Ш (E₁^ (167) /ℚ) 2 ≅ (ℤ/2ℤ) ². By constructing the explicit quartic torsor and relying on the unconditional theorem of Kolyvagin, we provide a proof of this local-global obstruction that is independent of the Birch and Swinnerton-Dyer conjecture.