Number of points of Fermat surfaces $X^n + Y^n - U^n - V^n = 0$

Question

Let $n$ be a positive integer such that $n^2 + n + 1$ is a prime, and consider the Fermat surface $F$ given by the equation $X^n + Y^n - U^n - V^n = 0$ (where we work with homogeneous coordinates $(x : y : u : v)$ in the projective space $\mathbb{P}^3(\mathbb{F}_p)$).

What can be said about the number of $\mathbb{F}_p$-rational points of $F$ (for instance, using Deligne's solution of the third Weil conjecture) ? Are interesting lower or upper bounds known, or even equalities (depending on $p$) ?

Answer 1

A special case of your question for the case $p=n^2+n+1$ (was this intended?) would be to ask if the subgroup $G$ of order $n+1$ of $\mathbb F_p^\star$ is a difference set, that is for each $x\in\mathbb F_p^\star$ there are unique $a,b\in G$ with $x=a-b$.

It is an open question whether this can be the case for $n>8$. This question came up first in the work of Kantor and Feit about flag-transitive projective planes, and was later thoroughly studied by Thas and Zagier in Finite projective planes, Fermat curves, and Gaussian periods.

Note that there is the wrong paper Sharply flag-transitive projective planes and power residue difference sets by Ott which allegedly settles this question. The mistake is explained e.g. in A note on power residue difference sets.