On the arithmetic of generalized Fekete polynomials

TL;DR

This paper extends the study of Fekete polynomials by defining generalized versions linked to quadratic characters, investigates their cyclotomic factors, explores their Galois groups using modular methods, and proposes a conjecture on their structure.

Contribution

It introduces generalized Fekete polynomials for composite conductors, analyzes their cyclotomic factors, and studies their Galois groups with new symmetries and conjectures.

Findings

Identification of cyclotomic factors in generalized Fekete polynomials

Discovery of extra symmetries affecting Galois groups

Proposal of a conjecture on Galois group structures

Abstract

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.