Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Abstract

A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane K ( z ) = y 1 / 2 K i r ( 2 π m y ) e 2 π i m x \mathcal {K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x} , z = x + i y z=x+iy , λ = 1 4 + r 2 \lambda =\frac 14+r^2 the eigenvalue, s = 2 π m λ − 1 / 2 s=2\pi m\lambda ^{-1/2} and K i r K_{ir} the Macdonald-Bessel function. The phase velocity of K \mathcal {K} on { | s | I m z ≤ 1 } \{|s|Im z\le 1\} is a double-valued vector field, the tangent field to the pencil of geodesics G \mathcal {G} tangent to the horocycle { | s | I m z = 1 } \{|s|Im z =1 \} . For A ∈ S L ( 2 ; R ) A\in SL(2;\mathbb {R}) a multi-term stationary phase expansion is presented in λ \lambda for K ( A z ) e 2 π i n R e z \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 ψ \psi automorphic with coefficients { a n } \{a_n\} and eigenvalue λ \lambda it is shown for the special range n ∼ λ 1 / 2 n\sim \lambda ^{1/2} that a n a_n is O ( λ 1 / 4 e π λ 1 / 2 / 2 ) O(\lambda ^{1/4}\,e^{\pi \lambda ^{1/2}/2}) for λ \lambda large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound O ( | n | 1 / 2 λ 1 / 4 e π λ 1 / 2 / 2 ) O(|n|^{1/2}\lambda ^{1/4}\,e^{\pi \lambda ^{1/2}/2}) . An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.