Galois-invariant Néron--Severi ranks of Fermat surfaces over number fields: a Galois module, closed forms, a threshold, and exact tables

Abstract

For a Fermat surface X_d: x_0^d+x_1^d+x_2^d+x_3^d=0 we compute the Galois-invariant Neron-Severi (Picard) rank rho_K(X_d) = dim (NS(X) \otimes Q)^{Gal(Qbar/K)}: for every subfield K of Q(zeta_d) by an explicit character average -- in closed form for gcd(d,6)=1, with the rational ranks tabulated for 4 <= d <= 30 -- and for an arbitrary number field K through the reduction rho_K = rho_{K \cap L} to the field of definition L. The geometric Picard number is classical (Shioda, Aoki), and L is due to Gvirtz-Chen and Skorobogatov; here we supply the rank layer. As a Gal(Q(zeta_d)/Q)-module the zeta_d-rational Neron-Severi group is Qh \oplus M with M \otimes Q(zeta_d) monomial on the Shioda eigenlines; writing chi_NS = 1 + chi_M, we get rho_K = (1/|H|) sum_{t in H} chi_NS(t). For degree coprime to 6 we prove rho_Q(X_d) = 1 + 3(Psi_2(d) - 3 tau(d) + 2) with Psi_2 multiplicative; we establish a field-of-definition threshold with explicit Hasse-Davenport witnesses, an assembled orbit rule for even d, and the exact table for 4 <= d <= 30, where the exceptional entries at d = 14, 24, 28, 30 are proved internally by Hasse-Davenport identities and a stabilizer descent, and corroborated by exact Z[zeta_m] evaluation and by the per-character field-of-definition computation of Gvirtz-Chen--Skorobogatov. The average order is sum_{d <= x, gcd(d,6)=1} rho_Q(X_d) ~ (3/5) x^2. All results are unconditional: on a surface the algebraic classes are the rational (1,1)-classes by Lefschetz. Ancillary files provide 15 standalone exact-arithmetic scripts, a 1270-row orbit-by-orbit certificate of the exceptional layer, and the full character tables, with pinned dependencies and chained checksums.