Summary
This paper describes our method of pairing automorphic distributions.We present a third technique for obtaining the analytic properties of automorphic L-functions, in addition to the existing methods of integral representations (Rankin–Selberg) and Fourier coefficients of Eisenstein series (Langlands–Shahidi). We recently used this technique to establish new cases of the full analytic continuation of the exterior square L-functions. The paper here gives an exposition of our method in two special, yet representative cases: the Rankin–Selberg tensor product L-functions for PGL(2,Z)∖PGL(2,R), as well as for the exterior square L-functions for GL(4,Z)∖GL(4,R).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Daniel Bump, The Rankin–Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987), 1989, pp. 49–109.
Daniel Bump and Solomon Friedberg, The exterior square automorphic L-functions on GL(n), Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), 1990, pp. 47–65.
W. Casselman, Jacquet modules for real reductive groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 1980, pp. 557–563.
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Canad. J. Math. 41(1989), no. 3, 385–438.
William Casselman, Henryk Hecht, and DraganMiliçić, Bruhat filtrations and Whittaker vectors for real groups, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), 2000, pp. 151–190.
Hervé Jacquet, Automorphic forms on GL(2). Part II, Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 278.
Hervé Jacquet and Joseph Shalika, Exterior square L-functions, Automorphic forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), 1990, pp. 143–226.
Henry H. Kim, Langlands-Shahidi method and poles of automorphic L-functions: application to exterior square L-functions, Canad. J. Math. 51 (1999), no. 4, 835–849.
Henry H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.
R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, 1970, pp. 18–61. Lecture Notes in Math., Vol. 170.
Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn., 1971.
Hans Maass, Über eine neue Art von nichtautomorphen analytische Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Annalen 121 (1949), 141–183.
Stephen D. Miller and Wilfried Schmid, Distributions and analytic continuation of Dirichlet series, J. Funct. Anal. 214 (2004), no. 1, 155–220.
Stephen D, Automorphic Distributions, L-functions, and Voronoi Summation for GL(3), Annals of Mathematics. To appear.
Stephen D, The archimedean theory of the Exterior Square L-functions over Q. preprint, 2005.
A. I. Oksak, Trilinear Lorentz invariant forms, Comm. Math. Phys. 29 (1973), 189–217.
R. A. Rankin, Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetical functions. I. The zeros of the function Σ∞ n =1 τ (n)/ns on the line Rs = 13/2. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372.
Atle Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50 (German).
Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330.
Takuro Shintani, On an explicit formula for class-1 “Whittaker functions” on GL n over P-adic fields, Proc. Japan Acad. 52 (1976), no. 4, 180–182.
Eric Stade, Mellin transforms of GL(n,R) Whittaker functions, Amer. J. Math. 123 (2001), no. 1, 121–161.
Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I (College Park, Md., 1982/1983), 1983, pp. 287–369.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Boston
About this chapter
Cite this chapter
Miller, S.D., Schmid, W. (2008). The Rankin–Selberg Method for Automorphic Distributions. In: Kobayashi, T., Schmid, W., Yang, JH. (eds) Representation Theory and Automorphic Forms. Progress in Mathematics, vol 255. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4646-2_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4646-2_4
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4505-2
Online ISBN: 978-0-8176-4646-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)