Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane $\mathcal{K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x}$, $z=x+iy$, $\lambda=\frac14+r^2$ the eigenvalue, $s=2\pi m\lambda^{-1/2}$ and $K_{ir}$ the Macdonald-Bessel function. The phase velocity of $\mathcal{K}$ on $\{\vert s\vert Im z\le1\}$ is a double-valued vector field, the tangent field to the pencil of geodesics $\mathcal{G}$ tangent to the horocycle $\{\vert s\vert Im z =1 \}$. For $A\in SL(2;\mathbb{R} )$ a multi-term stationary phase expansion is presented in $\lambda$ for $\mathcal{K}(Az)e^{2\pi in\,Re z}$ uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for $\psi$automorphic with coefficients $\{a_n\}$ and eigenvalue $\lambda$ it is shown for the special range $n\sim \lambda^{1/2}$ that $a_n$ is $O(\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$ for $\lambda$ large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound $O(\vert n\vert^{1/2}\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$. An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.