Abstract :
Finite upper half planes have been studied by Terras, Poulos, Celniker, Trimble, and Velasquez. Motivated by Starkʹsp-adic upper half plane as ap-adic analog of the Poincaré upper half plane, a finite field of odd characteristic was used as the finite analog of the real line. The analog of the upper half plane was constructed by adjoining the square root of a non-square to the finite field. Since this is not possible with a finite field of even characteristic, this paper will introduce a modification which will enable all finite fields to be considered. After constructing the finite upper half planes, graphs on these planes will be considered. We are interested in the spectrum, girth, diameter, and other combinatorial properties of these graphs.
