The Terminal D4 Packet in Rank-Six Arithmetic Magic: A Gaussian Weil Prym Fourfold and a Dihedral Weight-One Form

Abstract

The rank-six power-reciprocal Wigner family studied previously leaves a terminal rank-four finite-monodromy local system D₄ whose image was only known to lie in Aut(D₄) = W(F₄), and whose intersection-cohomology rank was not identified. We close this packet exactly. Two explicit height-one sections fix an orthogonal plane in D₄*, while the remaining minimal sections are governed by a quartic cover with geometric Galois group W(B₂), the dihedral group of order eight. The true branch divisor has twelve points, and Grothendieck–Ogg–Shafarevich gives dim IH¹(P¹, D₄) = 8. We realize this cohomology as the first cohomology of an explicit Prym fourfold P_AM = Prym(C₄/C₂), where C₂ is the genus-three curve w² = (3t⁴ − 14t² + 27)(3t⁴ + 18t² − 5), and C₄ → C₂ is a double cover branched at four points. The induced polarization has type (1,2,2,2). An order-four automorphism satisfies J² = −1 on the Prym and makes P_AM a Q(i)-Weil fourfold of signature (2,2) and Weil discriminant one. Exact Frobenius calculations at p = 13 and p = 17 prove absolute simplicity and End⁰(P_AM over Q̄) = Q(i). Consequently, the Hodge group is the associated special unitary group, with real form SU(2,2). The connected compact algebraic Sato–Tate group is SU(4), while its component quotient is dihedral of order eight. The connectedness field is Q(i, ⁴√(−135)). The exceptional Weil classes in H⁴(P_AM) carry the two-dimensional Artin representation induced from the Gaussian quartic character (−135/·)₄. This Artin representation has conductor 900, root number +1, and dihedral image. Its normalized weight-one modular form is LMFDB 900.1.c.a, equivalently a quartic-character twist of 180.1.f.a. Prime equidistribution in the resulting compact Sato–Tate group is not claimed; the finite Artin and algebraic monodromy statements are exact.