On pointless diagonal Fermat curves

Abstract

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

10 pages