Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series


Author: Alvaro Alvarez-Parrilla
Journal: Trans. Amer. Math. Soc. 352 (2000), 5563-5582
MSC (1991): Primary 11F30; Secondary 11N37
DOI: https://doi.org/10.1090/S0002-9947-00-02502-2
Published electronically: August 21, 2000
MathSciNet review: 1675233
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series $\varphi_{ir}$ with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of $\varphi_{ir}$; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter $\lambda (\lambda=\frac{1}{4}+r^2$is the eigenvalue of $\varphi_{ir}$) for a cubic phase.

As applications we find sets of asymptotic relations for divisor functions.


References [Enhancements On Off] (What's this?)

  • [Bor97] Armand Borel, Automorphic forms on ${S}{L}_2({{\mathbb R}})$, Cambridge University Press, 1997. MR 98j:11028
  • [Bru94] R. W. Bruggeman, Families of Automorphic Forms, Birkhäuser-Verlag, 1994. MR 95k:11060
  • [CFU57] C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Cambridge Philos. Soc. 53 (1957), 599-611. MR 19:853a
  • [GS77] V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys. Number 14, American Mathematical Society, 1977. MR 58:24404
  • [Hej83] D. A. Hejhal, The Selberg Trace Formula for ${P}{S}{L}_2({{\mathbb R}})$ Volume $2$, Lecture Notes in Mathematics $1001$, Springer-Verlag, 1983. MR 86e:11040
  • [HW79] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, $5^{th}$ edition, Oxford University Press, 1979. MR 81i:10002
  • [Iwa97] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, 1997. MR 98e:11051
  • [Olv97] F. W. J. Olver, Asymptotics and Special Functions, A. K. Peters, 1997. MR 97i:41001
  • [Wol99] S. A. Wolpert, Asymptotic relations among Fourier coefficients of automorphic eigenfunctions, Preprint sent to publication, 1998.
  • [Zem65] A. H. Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill Book Company, New York, 1965. MR 31:1556

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11F30, 11N37

Retrieve articles in all journals with MSC (1991): 11F30, 11N37


Additional Information

Alvaro Alvarez-Parrilla
Affiliation: Department of Mathematics, University of Maryland at College Park, College Park, Maryland 20740
Address at time of publication: P.O. Box 435294, San Ysidro, California 92173
Email: aap@math.umd.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02502-2
Keywords: Automorphic forms, Eisenstein series, microlocal analysis, divisor functions
Received by editor(s): September 29, 1998
Received by editor(s) in revised form: November 24, 1998, and January 29, 1999
Published electronically: August 21, 2000
Additional Notes: Thanks to Scott Wolpert for suggesting the problem, many very insightful talks and helpful ideas, and for providing copies of his preprint
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society