论文标题
关于广义fekete多项式的算术
On the arithmetic of generalized Fekete polynomials
论文作者
论文摘要
对于每个质数$ p $，一个人可以将fekete多项式与系数相关联$ -1 $或$ 1 $，除了恒定术语（即0）。这些是经典的多项式，在分析数理论的框架中已进行了广泛的研究。在最近的一篇论文中，我们表明这些多项式还编码有趣的算术信息。在本文中，我们定义了与二次字符相关的广义fekete多项式，其导体可能是复合数。然后，我们研究了这些广义的fekete多项式的循环因子的出现。基于此研究，我们引入了紧凑的Fekete多项式及其痕量多项式。然后，我们使用模块化技术研究了这些Fekete多项式的Galois组。特别是，我们发现了一些令人惊讶的额外对称性，这意味着对相应的Galois组有一些限制。最后，基于理论和数值数据，我们对这些Galois组的结构提出了一个精确的猜想。
For each prime number $p$ one can associate a Fekete polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be a composite number. We then investigate the appearance of cyclotomic factors of these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.
