关于广义Ree曲线
On the generalized Ree curve
- Shahid Chamran University of Ahvaz（阿瓦士沙希德·恰姆朗大学）
- IMECC/UNICAMP（坎皮纳斯州立大学计算数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究广义Ree曲线R_{p,s}，确定其自同构群与分歧结构，证明其Jacobian分解及超奇异性质，并给出自同构群阶的渐近公式。
AI中文摘要：
设p为奇素数，令q_0:=p^s，其中s≥1，并置q:=p q_0^2。我们研究光滑射影曲线R_{p,s}，其函数域为F_q(x,y,z)，满足y^q - y = x^{q_0}(x^q - x)，z^q - z = x^{2q_0}(x^q - x)。这两个方程的层已经出现在射线类和big-action文献中；当p=3时，它给出经典的Ree曲线。与广义Suzuki曲线类似，我们将R_{p,s}称为与p和s相关的广义Ree曲线。对于p>3，我们的主要几何结果确定了完整的几何自同构群：Aut_{F_q}(R_{p,s})同构于U和F_q^*的半直积，其中|U|=q^3。自然p-群U给出一个big action；我们确定了初等阿贝尔覆盖R_{p,s} -> P^1_x的完全分歧滤过，并证明(R_{p,s}, P_∞)是Castle的。在特征3中，同一子群只是完整Ree群中P_∞的稳定化子，因此自同构结构呈现出尖锐的特征三二分法。在算术方面，记Y_{p,s}: y^q - y = x^{q_0}(x^q - x)，Z_{p,s}: z^q - z = x^{2q_0}(x^q - x)，我们证明F_q-同源Jac(R_{p,s}) ~ Jac(Y_{p,s}) × Jac(Z_{p,s})^q。第一个因子是超奇异的，且所有三条曲线的p-秩为零。对于p>3，Z_{p,s}的第一个Newton斜率满足1/(p q_0 + 2) ≤ λ_min(Z_{p,s}) ≤ 1/3，因此Z_{p,s}和R_{p,s}不是超奇异的。对于固定的p>3且s→∞，完整自同构群的阶渐近于2^(8/5) p^(-4/5) q^(8/5)。
英文摘要：
Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).
