EIGENFUNCTIONS ON THE FINITE POINCARÉ PLANE

Jinghua Kuang

Abstract. Let F be a finite field of odd number of elements. Let F (\sqrt{δ}) be its quadratic extension. F (\sqrt{δ}) − 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.

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