On a class of finite upper half-planes

Using an exponential sum associated to the Legendre character, we introduce a finite 'upper half-plane' V(q), by defining a metric on the set given by the union between the quotient of F-q2-F-q With respect to the Frobenius action, and an extra point infinity, which appears as a collapse of the field F-q. We also introduce, for every odd prime power q, the 'length spectrum' Sigma(q), that is, the set of all possible distances between distinct points of V(q), which plays the role of a 'parameter space' for a class of associated graphs V(q; k), k is an element of Sigma(q), for which the 'finite parts' V-0(q;k) are regular. Up to a normalization, the whole metric space V(q) can be seen as a small perturbation of a complete graph with 1+(q(2)-q)/2 vertices. Finally, we show how these results generalize to any higher dimension n. The corresponding metric space V,(q) is obtained out of the set of the orbits of the Frobenius action on F-q(n) over F-q, by making appropriate identifications.