On the Arithmetic of Generalized Fekete Polynomials

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 article, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be composite numbers. We then investigate the appearance of cyclotomic factors in 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.

KEYWORDS:

• Fekete polynomials
• special values of L-functions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 11C08
• 11R09

Acknowledgments

We would like to thank Professor Franz Lemmermeyer for his encouragement to study this topic. The thread [19] initiated by him has been a critical inspiration for us. The second named author would like to thank William Stein for his help with the platform Cocalc where our computations are based. He also thanks to the organizers of the number theory seminars at UIC and Northwestern as well as the organizers of the CTNT 2022 Conference for providing him an opportunity to present parts of this work. We would also like to thank Professors Andrew Granville, Peter Moree, and Ramin Takloo-Bighash for their interest and encouragement related to our work on Fekete polynomials. Last but not least, we are also grateful to the referee for their comments and valuable suggestions which we have used to improve our exposition.

Additional information

Funding

JM is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant R0370A01. He gratefully acknowledges the Western University Faculty of Science Distinguished Professorship 2020-2021. NDT is partially supported by Vietnam National University, Hanoi (VNU) under the project number QG.23.48