2026/07/19 by Rifat Jumagulov
#math.NT #math.AG
For a Fermat surface Xd: x0d+x1d+x2d+x3d=0 we compute the Galois-invariant Neron-Severi (Picard) rank rhoK(Xd) = dim (NS(X) ⊗ Q)Gal(Qbar/K): for every subfield K of Q(zetad) 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 rhoK = rhoK ∩ 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(zetad)/Q)-module the zetad-rational Neron-Severi group is Qh ⊕ M with M ⊗ Q(zetad) monomial on the Shioda eigenlines; writing chiNS = 1 + chiM, we get rhoK = (1/|H|) sumt in H chiNS(t). For degree coprime to 6 we prove rhoQ(Xd) = 1 + 3(Psi2(d) - 3 tau(d) + 2) with Psi2 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[zetam] evaluation and by the per-character field-of-definition computation of Gvirtz-Chen--Skorobogatov. The average order is sumd <= x, gcd(d,6)=1 rhoQ(Xd) ~ (3/5) x2. 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.