The degree-28 polynomial of a genus-4 linearised operator, the quadratic-form torsor, and the symplectic monodromy Sp_6(F_2)

Abstract

Let q = 2^k and let 𝔮_λ(x) = T(x) + 2 Tr(λ9 x⁹ + λ5 x⁵ + λ3 x³ + λ1 x) be the Z/4-valued quadratic form attached to the supersingular genus-4 curves y² + y = λ9 x⁹ + λ5 x⁵ + λ3 x³ + λ1 x over F_q, where T is the Witt-vector trace of the Teichmüller representative. Its polar operator T'_λ is a linearised operator with six-dimensional kernel K_λ, and the Gauss sums of 𝔮_λ over F_{q^{2m}} are governed by the 64-point coset C_λ = T'_λ^{-1}(1). We prove that L'_λ = T'_λ^8 satisfies L'_λ(x) = x⁸ B_λ(x)² + B_λ(x) + 1 with the explicit polynomial B_λ(x) = λ9⁴x²⁸ + λ5⁴x¹² + λ9²x¹⁰ + λ3⁴x⁴ + λ5²x² + λ9x + 1, the exact analogue of the sextic of the genus-2 hexad (Parts III–IV of the author's Legendre-interval series); that the Gauss sum over F_{q^{2m}} equals q^m times the sum over the F_{q^m}-rational points w of C_λ of (−1)^{[B_λ(w) ≠ 0]} i^{𝔮^{(m)}(w)}; and that C_λ is canonically the torsor of quadratic forms on K_λ polarising a canonical symplectic form ω_λ, the 28 roots of B_λ being the forms of Arf invariant 1 and the 36 roots of x⁸B_λ + 1 those of Arf invariant 0. An unconditional trace parity theorem (Tr(w) ≡ mk + dim(K_λ ∩ F_{q^m}) mod 2) and a structure theorem for the Gauss sums on each Frobenius type follow; |s_{2m}| = 8 q^m whenever C_λ ⊂ F_{q^m}. The Frobenius acts on K_λ through Sp_6(F_2). By a right-gcd computation in the twisted polynomial ring F_q{τ}, with no field extensions, its GL_6(F_2)-type is determined for all λ with q ≤ 512: all 24 types of Sp_6(F_2) occur, with the frequencies predicted by equidistribution, and the counts are not polynomial in q. An octad structure (the S_8 = O_6^+(2) picture) exists only for the 79% of parameters whose Frobenius type occurs in S_8 and is never canonical; B_λ and the torsor exist for every parameter. The L-polynomials of the rank-8 Witt sheaf are computed for q = 2, 4. The paper (13 pages, 3 figures, 2 tables) is accompanied by the complete proofs, the self-contained Python code and the data for q ≤ 512 in the repository https://github.com/Ruqing1963/genus4-b28-torsor. Related records: Prime polynomials in Legendre intervals over F_{2^k}[t], Parts I–IV, DOI 10.5281/zenodo.23225896, 10.5281/zenodo.23252762, 10.5281/zenodo.23290724, 10.5281/zenodo.23299284.