On a class of finite upper half-planes Original Research Article

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 Fq2 − Fq with respect to the Frobenius action, and an extra point ∞, which appears as a collapse of the field Fq. We also introduce, for every odd prime power q, the ‘length spectrum’ Σq, 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 ϵ Σq, for which the ‘finite parts’ Vo(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 + (q2 − q)/2 vertices. Finally, we show how these results generalize to any higher dimension n. The corresponding metric space Vn(q) is obtained out of the set of the orbits of the Frobenius action on Fqn over Fq, by making appropriate identifications.