gauss-sums

Algebraic invariant from the sixth triangular number, appearing across Gauss sums, Dirichlet class numbers, zeta ratios, and modular forms.

Part III of the Legendre-interval series: ker L' is the pairwise-sum set of the roots of the sextic u·A(u)+1 (hexad, S6 ≅ Sp4(F2)); Witt–Artin–Schreier sheaves give W' = −q^7 for 5-cycles unconditionally and reduce X = q^6 + q^5 to class counts and Brown invariants. Paper, code, exact data.

Genus-4 Gold family y²+y = λ9x⁹+λ5x⁵+λ3x³+λ1x over F_{2^k}: the canonical degree-28 polynomial B with L' = x⁸B²+B+1, the coset-sum theorem for Z/4 Gauss sums, the quadratic-form torsor on K = F₂⁶ (28 Arf-1 + 36 Arf-0 points), trace parity, Sp6(F2) Frobenius types. Paper, Python code, data. DOI 10.5281/zenodo.23306002

Part IV of "Prime polynomials in Legendre intervals over F_{2^k}[t]": hexad formula and trace parity for Z/4 Gauss sums; the sign law is proved in all 11 Frobenius classes (v2: classes 3 2 1 and 6); X = q^6 + q^5 now depends only on the class counts. Paper, Python code, data for q = 8, 32, 128. DOI 10.5281/zenodo.23308682