Eigenfunctions on the Finite Poincaré Plane

Abstract

Let $F$ be a finite field of odd number of elements. Let $F(\sqrtδ)$ be its quadratic extension. $F(\sqrtδ)-F$ is the so-called finite Poincare plane. This paper relates the bases of eigenfunctions constructed by Evans and by Kuang. The finite Poincare plane can be viewed as a Ramanujan graph. This paper also provides evidence for Terras' conjecture regarding the asymptotic distribution of the eigenvalus of the adjacency matrices.