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Higher weight on GL(3). I: The Eisenstein series

  • Jack Buttcane EMAIL logo
Published/Copyright: September 20, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 3

Abstract

The purpose of this paper is to collect and make explicit the results of Langlands [16], Bump [3], Miyazaki [18] and Manabe, Ishii and Oda [17] for the GL⁢(3) Eisenstein series and Whittaker functions which are non-trivial on SO⁢(3,ℝ). The final goal for the series of papers is a complete and completely explicit spectral expansion for L2⁢(SL⁢(3,ℤ)∖SL⁢(3,ℝ)) in the style of Duke, Friedlander and Iwaniec’s paper [8]. We derive a number of new results on the Whittaker functions and Eisenstein series, and give new, concrete proofs of the functional equations and spectral expansion in place of the general constructions of Langlands.

Keywords: Automorphic forms; non-spherical; ramification at infinity; Eisenstein series; Langlands spectral expansion
MSC 2010: 11F72; 11F30

Communicated by Freydoon Shahidi

Acknowledgements

The author would like to thank Valentin Blomer, Stephen D. Miller, Xiaoqing Li, James Cogdell and Joseph Hundley for their comments and helpful discussions along the way.

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Received: 2017-3-25
Revised: 2017-8-5
Published Online: 2017-9-20
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
https://doi.org/10.1515/forum-2017-0060
Keywords for this article
Automorphic forms; non-spherical; ramification at infinity; Eisenstein series; Langlands spectral expansion

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
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