On some arithmetic properties of Maass forms

Abstract

Iwaniec`s result ([6]) explains the uniform distribution of lattice points on the sphere in $\mathbb{R}^3$ with increasing radius. Our object is to extend this method in order to study the distributions of Heegner points and closed geodesics on $\textrm{PSL}_2(\mathbb{Z})\backslash\mathfrak{H}$. For this purpose we study a non-holomorphic generalization of Iwaniec`s result and an expression of the appropriate Weyl sums in terms of Fourier coefficients of certain non-holomorphic forms of half-integral weight.