×
Goldfeld, Dorian; Stade, Eric; Woodbury, Michael

The first coefficient of Langlands Eisenstein series for \(\mathrm{SL}(n,\mathbb{Z})\). (English) Zbl 07903811

Acta Arith. 214, 179-189 (2024).
Summary: Fourier coefficients of Eisenstein series figure prominently in the study of automorphic \(\mathrm{L}\)-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations.
In this paper, we define Langlands Eisenstein series for \(\mathrm{SL}(n,\mathbb{Z})\) in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)

Keywords:

Langlands Eisenstein series; Fourier-Whittaker coefficients; automorphic forms; automorphic \(\mathrm{L}\)-functions

Software:

GL(n)pack
Cite Review PDF
Full Text: DOI arXiv

References:

[1] S. Gelbart and F. Shahidi, Analytic Properties of Automorphic L-Functions, Perspect. Math. 6, Academic Press, Boston, MA, 1988. · Zbl 0654.10028
[2] D. Goldfeld, Automorphic Forms and L-Functions for the Group GL(n, R) (with an appendix by K. A. Broughan), Cambridge Stud. Adv. Math. 99, Cam-bridge Univ. Press, Cambridge, 2006. · Zbl 1108.11039
[3] D. Goldfeld, S. D. Miller, and M. Woodbury, A template method for Fourier coefficients of Langlands Eisenstein series, Riv. Mat. Univ. Parma (N.S.) 12 (2021), 63-117. · Zbl 1473.11215
[4] D. Goldfeld, E. Stade, and M. Woodbury, An orthogonality relation for GL(4, R) (with an appendix by B. Huang), Forum Math. Sigma 9 (2021), art. e47, 83 pp. · Zbl 1478.11078
[5] H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 (with appendix 1 by D. Ramakrishnan and appendix 2 by Kim and P. Sarnak), J. Amer. Math. Soc. 16 (2003), 139-183. · Zbl 1018.11024
[6] H. H. Kim, Langlands-Shahidi method and poles of automorphic L-functions. III Exceptional groups, J. Number Theory 128 (2008), 354-376. · Zbl 1185.11035
[7] H. H. Kim and F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann. of Math. (2) 150 (1999), 645-662. · Zbl 0957.11026
[8] H. H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177-197. · Zbl 1074.11027
[9] H. H. Kim and F. Shahidi, Functorial products for GL2×GL3 and the symmetric cube for GL2 (with an appendix by C. J. Bushnell and G. Henniart), Ann. of Math. (2) 155 (2002), 837-893. · Zbl 1040.11036
[10] R. P. Langlands, Euler Products, A James K. Whittemore Lecture in Mathe-matics given at Yale University, 1967, Yale Math. Monogr. 1, Yale Univ. Press, New Haven, CT, 1971.
[11] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer, Berlin, 1976. · Zbl 0332.10018
[12] S. D. Miller, The highest lowest zero and other applications of positivity, Duke Math. J. 112 (2002), 83-116. · Zbl 1014.11036
[13] F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-355. · Zbl 0467.12013
[14] F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973-1007. · Zbl 0674.10027
[15] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; comple-mentary series for p-adic groups, Ann. of Math. (2) 132 (1990), 273-330. · Zbl 0780.22005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.