长度为4和5的二次剩余模式
Quadratic residue patterns of length 4 and 5
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中文总结 AI 辅助
该研究将二次剩余模式结果推广到特定字母表上长度为4和5的任意单词，通过低亏格超椭圆曲线的弗罗贝尼乌斯迹得到模式出现次数公式，并利用广义佐藤 - 泰特方法描述l = 4时归一化误差项的极限分布。
中文摘要 AI 辅助
我们将Kiritchenko、Tsfasman、Vlăduţ和Zakharevich关于二次剩余模式的论文第一部分结果推广到字母表{R, N}上长度l = 4和l = 5的任意单词情况。首先，得到给定模式S在模素数p的二次剩余和非剩余序列中出现次数的显式公式，这些公式用低亏格超椭圆曲线有限集合上的弗罗贝尼乌斯迹表示。其次，使用广义佐藤 - 泰特方法描述l = 4时归一化误差项的极限分布。
英文摘要
We generalize the results of the first part of the paper by Kiritchenko, Tsfasman, Vlăduţ and Zakharevich on quadratic residue patterns to the case of arbitrary words over the alphabet $\{R, N\}$ of length $l = 4$ and $l = 5$. First, we obtain explicit formulas for the number of occurrences of a given pattern $S$ in the sequence of quadratic residues and nonresidues modulo a prime $p$. These formulas are expressed in terms of Frobenius traces on a finite collection of low-genus hyperelliptic curves. Second, using a generalized Sato-Tate approach, we describe the limiting distributions of the normalized error term for $l = 4$.
