数域上费马曲面的伽罗瓦不变奈龙 - 塞维里秩：伽罗瓦模、封闭形式、阈值及精确表格
Galois-invariant Néron--Severi ranks of Fermat surfaces over number fields: a Galois module, closed forms, a threshold, and exact tables
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中文总结 AI 辅助
研究数域上费马曲面的伽罗瓦不变奈龙 - 塞维里秩，通过显式特征平均计算，给出\(\gcd(d,6)=1\)时封闭形式及\(4\leq d\leq30\)的有理秩，证明相关公式，建立阈值和轨道规则，给出精确表格及平均阶，结果无条件且有辅助文件支持。
中文摘要 AI 辅助
对于费马曲面\(X_d:x_0^d + x_1^d + x_2^d + x_3^d = 0\)，我们通过显式特征平均计算伽罗瓦不变的奈龙 - 塞维里（皮卡）秩\(\rho_K(X_d)=\dim(NS(X)\otimes\mathbb{Q})^{Gal(\overline{\mathbb{Q}}/K)}\)，对于\(\mathbb{Q}(\zeta_d)\)的每个子域\(K\)。对于\(\gcd(d,6)=1\)以封闭形式给出，列出了\(4\leq d\leq30\)的有理秩，对于任意数域\(K\)通过归约到定义域\(L\)来计算。给出了伽罗瓦模结构及相关公式，证明了\(\rho_{\mathbb{Q}}(X_d)\)的公式，建立了定义域阈值，给出了偶数\(d\)的组装轨道规则以及\(4\leq d\leq30\)的精确表格，还给出了平均阶。所有结果都是无条件的，辅助文件提供了相关脚本、证书和特征表。
英文摘要
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.
