On a Family of Twisted Trace Curves over Finite Fields, and Fibonacci Numbers

TL;DR

This paper investigates the number of rational points on a specific family of curves over finite fields, revealing cases with unexpectedly high point counts, and explores connections to Fibonacci numbers and cyclotomic polynomials.

Contribution

It provides new results on rational points of twisted trace curves, highlighting exceptional cases and linking them to Fibonacci numbers and cyclotomic polynomials.

Findings

Some curves have more rational points than expected

Fibonacci numbers appear in the analysis

Connections to cyclotomic polynomials are established

Abstract

We present some results about the number of rational points on a certain family of curves defined over a finite field. In a small number of cases the curves have more rational points than expected. Fibonacci numbers make an appearance, as do cyclotomic polynomials.