On a class of finite upper half-planes
Inspired by the finite upper half plane graphs considered in [\textit{A. Terras}, Exp. Math. 5, 15-32 (1996; Zbl 0871.05044)]\ with their finite-field valued analogue of Poincaré's distance, the author produces a real-valued distance on the finite upper half-plane over the finite field with q elements and some new associated graphs \(V(q;k)\). This distance is shown to be a character sum. Basic results on numbers of points on a complete elliptic curve over a finite field or character sum estimates of Weil are required. Unlike the \((q+1)\)-regular finite upper half-plane graphs of the reviewer, these \(V(q;k)\) graphs may have no edges. But it seems that when renormalized, the nonempty \(V(q;k)\) graphs are `small perturbations of complete graphs'. The original finite upper half-plane graphs are of interest because of their connections with the finite Gelfand pair \((G,K)\), where \(G\) is the general linear group of \(2\times 2\) non-singular matrices with entries in a finite field \(F\), and \(K\) is analogous to the rotation group \(O(2)\). Moreover, they provide examples of Ramanujan graphs in the sense of Lubotzky, Phillips and Sarnak. [See \textit{A. Lubotzky}, Discrete groups, expanding graphs, and invariant measures. Birkhäuser (1994; Zbl 0826.22012)]. But no study of the spectra of the adjacency matrices of these new finite upper half-plane graphs seems to have been made. Generalizations of finite upper half-planes to higher dimensions are also considered by considering higher degree extensions of the base field. (Also submitted to MR).
scientific article; zbMATH DE number 1019020
scientific article