On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field

Abstract

In 1988 Garcia and Voloch proved the upper bound 4n^{4/3}(p-1)^{2/3} for the number of solutions over a prime finite field F_p of the Fermat equation x^n+y^n=a, where a \in F_p^* and n \ge 2 is a divisor of p-1 such that (n-1/2)^4 \ge p-1. This is better than Weil's bound p+1+(n-1)(n-2)p^{1/2} in the stated range. By refining Garcia and Voloch's proof we show that the constant 4 in their bound can be replaced by 3\cdot 2^{-2/3}.