On pointless diagonal Fermat curves

TL;DR

This paper improves the asymptotic upper bounds on the number of diagonal Fermat curves over finite fields that lack rational points, advancing understanding of their distribution and properties.

Contribution

It provides a sharper asymptotic upper bound on the count of point-free diagonal Fermat curves over finite fields, refining previous estimates.

Findings

Established an improved asymptotic upper bound for point-free diagonal Fermat curves.

Enhanced understanding of the distribution of such curves over finite fields.

Results applicable to primes dividing q-1 in finite field settings.

Abstract

We give an improved asymptotic upper bound on the number of diagonal Fermat curves $Ax^{\ell}+By^{\ell}=z^{\ell}$ over $\mathbb{F}_{q}$ with no $\mathbb{F}_{q}$-rational points, where $\ell$ is a prime number dividing $q-1$.