Asymptotic relations among Fourier coefficients of automorphic eigenfunctions
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- by Scott A. Wolpert PDF
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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 $\mathcal {K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x}$, $z=x+iy$, $\lambda =\frac 14+r^2$ the eigenvalue, $s=2\pi m\lambda ^{-1/2}$ and $K_{ir}$ the Macdonald-Bessel function. The phase velocity of $\mathcal {K}$ on $\{|s|Im z\le 1\}$ is a double-valued vector field, the tangent field to the pencil of geodesics $\mathcal {G}$ tangent to the horocycle $\{|s|Im z =1 \}$. 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 $O(|n|^{1/2}\lambda ^{1/4} e^{\pi \lambda ^{1/2}/2})$. An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.References
- Charles B. Balogh, Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math. 15 (1967), 1315–1323. MR 222354, DOI 10.1137/0115114
- Daniel Bump, The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 49–109. MR 993311
- Daniel Bump, W. Duke, Jeffrey Hoffstein, and Henryk Iwaniec, An estimate for the Hecke eigenvalues of Maass forms, Internat. Math. Res. Notices 4 (1992), 75–81. MR 1159448, DOI 10.1155/S1073792892000084
- J.-M. Deshouillers and H. Iwaniec, The nonvanishing of Rankin-Selberg zeta-functions at special points, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 51–95. MR 853553, DOI 10.1090/conm/053/853553
- Stephen Gelbart and Freydoon Shahidi, Analytic properties of automorphic $L$-functions, Perspectives in Mathematics, vol. 6, Academic Press, Inc., Boston, MA, 1988. MR 951897
- Alain Grigis and Johannes Sjöstrand, Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107, DOI 10.1017/CBO9780511721441
- D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for $\textrm {PSL}(2,\mathbf Z)$, Math. Comp. 61 (1993), no. 203, 245–267, S11–S16. MR 1199991, DOI 10.1090/S0025-5718-1993-1199991-8
- Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,\,\textbf {R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197, DOI 10.1007/BFb0061302
- Dennis A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992), no. 469, vi+165. MR 1106989, DOI 10.1090/memo/0469
- Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
- Henryk Iwaniec, Spectral theory of automorphic functions and recent developments in analytic number theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 444–456. MR 934244
- Henryk Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matemática Iberoamericana. [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1995. MR 1325466
- N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
- Yiannis N. Petridis, On squares of eigenfunctions for the hyperbolic plane and a new bound on certain $L$-series, Internat. Math. Res. Notices 3 (1995), 111–127. MR 1321699, DOI 10.1155/S1073792895000092
- Yiannis N. Petridis, Fourier coefficients of cusp forms, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 297–307. MR 1684610, DOI 10.1090/crmp/019/26
- Peter Sarnak, Arithmetic quantum chaos, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 183–236. MR 1321639
- Peter Sarnak, Integrals of products of eigenfunctions, Internat. Math. Res. Notices 6 (1994), 251 ff., approx. 10 pp.}, issn=1073-7928, review= MR 1277052, doi=10.1155/S1073792894000280, DOI 10.1155/S1073792894000280
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15. MR 0182610
- Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406, DOI 10.1007/978-1-4612-5128-6
- E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
- Alexei B. Venkov, Spectral theory of automorphic functions and its applications, Mathematics and its Applications (Soviet Series), vol. 51, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian by N. B. Lebedinskaya. MR 1135112, DOI 10.1007/978-94-009-1892-4
- Scott A. Wolpert, Semiclassical limits for the hyperbolic plane, Duke Math. J. 108 (2001), no. 3, 449–509. MR 1838659, DOI 10.1215/S0012-7094-01-10833-8
Additional Information
- Scott A. Wolpert
- Affiliation: 3400 AV Williams Building, University of Maryland, College Park, Maryland 20742
- MR Author ID: 184255
- Email: saw@math.umd.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 427-456
- MSC (2000): Primary 11F30, 33C10; Secondary 11M06, 42A16
- DOI: https://doi.org/10.1090/S0002-9947-03-03154-4
- MathSciNet review: 2022706