Points on Fermat curves over finite fields

On the Solutions of the Equation x m + y m - z m = 1 in a Finite Field

“…Put m = (q − 1)/k. In [11] and [12] an explicit formula is given for the number of solutions of the equation (1.1) x m + y m − z m = 1 in the Galois field F = GF(q) under the following hypothesis: the unique multiplicative subgroup Φ of order k in F satisfies |Φa ∩ (Φb + c)| ≤ 2 for all a, b, c ∈ F * = F \ {0}. In this case, the pair (F, Φ) (known as a Ferrero pair, see [4]) as well as the pair (q, k), is said to be circular.…”

“…That is, (q, k) is circular if and only if (p r ′ , k) is circular for some r ′ ∈ N such that k | (p r ′ − 1). Thus, we say that the pair (p, k) is circular if (p r ′ , k) is circular for some r ′ ∈ N. Now, the formula for the number of solutions of (1.1) given in [11] and [12] was actually under another constraint when k is odd. Namely, when k is odd, it was conveniently required that (p, 2k) is also circular.…”

“…Remark 4. In [13], Korchmáros and Szönyi also use a geometric approach to find the number of points on the Fermat curve given by the equation x m + y m + z m = 0, which we have dealt with in [11] and [12] as well. They assume that the group Φ contains the multiplicative group of some subfield K of F .…”