Primitive Exterior-Square Frobenius Moments for a Paired-Root Genus-Two Pencil

Description This revised preprint studies exact Frobenius moments of the rank-five primitive exterior-square étale local system associated with a paired-root genus-two pencil. The main result presents an explicit degree-fifteen Frobenius polynomial and exact primitive exterior-square moment identities over finite fields and their extensions. The proof combines simultaneously based Picard–Lefschetz monodromy, prime-to-characteristic specialization of Jacobian 2-power torsion, and semistable reduction of the five singular fibres. At infinity, a quartic tame base change produces two elliptic components whose Galois-equivariant cohomology determines the local Frobenius factor. A boundary-defect identity then establishes the vanishing of the associated parabolic cohomology and yields the global formula from five explicit local factors. Changes in version v1.2r2: Replaces the three geometric comparison hypotheses of the earlier conditional preprint with explicit arguments using based monodromy, SGA 1 specialization, and Galois-equivariant semistable reduction. Presents a proof of the family-specific Frobenius moment identity without additional geometric comparison hypotheses. Corrects the extension-field moment formula from the earlier version. Expands the discussion of related work and includes reproducible symbolic and finite-field verification materials. Incorporates editorial revisions for journal submission. This version supersedes the mathematical presentation of v0.1r1 while preserving the earlier version as part of the research record. Status: Research preprint, not yet peer-reviewed. The manuscript presents an unconditional proof of the stated family-specific results, subject to independent mathematical review. It does not claim automorphy, a general Sato–Tate theorem, or results for arbitrary genus-two families. Keywords Genus-two curves; Hyperelliptic curves; Frobenius moments; Primitive exterior square; Étale cohomology; Parabolic cohomology; Geometric monodromy; Picard–Lefschetz theory; Semistable reduction; Galois representations; Frobenius polynomials; Character sums; Finite fields.