Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Author: Scott A. Wolpert
Journal: Trans. Amer. Math. Soc. 356 (2004), 427-456
MSC (2000): Primary 11F30, 33C10; Secondary 11M06, 42A16
Published electronically: September 22, 2003
MathSciNet review: 2022706
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Abstract: 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.

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Additional Information

Scott A. Wolpert
Affiliation: 3400 AV Williams Building, University of Maryland, College Park, Maryland 20742

Keywords: Automorphic eigenfunctions, Macdonald-Bessel functions, Fourier coefficients
Received by editor(s): December 15, 1999
Received by editor(s) in revised form: October 13, 2000
Published electronically: September 22, 2003
Additional Notes: This research was supported in part by NSF Grants DMS-9504176 and DMS-9800701
Article copyright: © Copyright 2003 American Mathematical Society