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 4 n 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 ∈ F p * and n ⩾ 2 is a divisor of p − 1 such that ( n − 1 2 ) 4 ⩾ p − 1 . This is better than Weil's bound p + 1 + ( n − 1 ) ( n − 2 ) p 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 ⋅ 2 − 2 / 3 .