Holomorphic differentials of generalized Fermat curves
Holomorphic differentials of generalized Fermat curves
复制标题
广义费马曲线的全纯微分
DOI：
10.1016/j.jnt.2020.05.014
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发表时间：
2017
影响因子：
0.7
通讯作者：
R. Hidalgo
中科院分区：
文献类型：
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作者：
R. Hidalgo
定义在代数闭域 K 上的非奇异完全不可约代数曲线 F k, n 称为 (k, n) 类型的广义费马曲线，其中 n, k≥ 2 是整数，并且 k 与 K 的特征 p 互质，如果它允许自同构群 H≅ Z k n 使得 F k, n/H 同构于 P K 1 并且它恰好具有 (n+ 1) 个圆锥点，每一个都是 k 阶。根据 Riemann-Hurwitz-Hasse 公式，F k, n 至少有一个当且仅当 (k− 1)(n− 1)> 1 时。在这种情况下，我们构造一个正则形式的空间 H 1, 0 (F k, n) 的基，称为标准基，包含基数 n+ 1 的子集，该子集提供 F k, n 嵌入到 P K n 中，其图像是 (n− 的纤维乘积) 1) k 次经典费马曲线。对于 p= 2，我们获得精确一式空间维数的下界（对于 n= 2, 3 是尖锐的），即 Cartier 算子的核。我们也对 (p, k, n)=(3, 2, 4) 执行此操作。
A non-singular complete irreducible algebraic curve F k, n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k, n), where n, k≥ 2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅ Z k n of automorphisms such that F k, n/H is isomorphic to P K 1 and it has exactly (n+ 1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, F k, n has genus at least one if and only if (k− 1)(n− 1)> 1. In such a situation, we construct a basis, called a standard basis, of its space H 1, 0 (F k, n) of regular forms, containing a subset of cardinality n+ 1 that provides an embedding of F k, n into P K n whose image is the fiber product of (n− 1) classical Fermat curves of degree k. For p= 2, we obtain a lower bound (which is sharp for n= 2, 3) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator. We also do this for (p, k, n)=(3, 2, 4).