Hostname: page-component-7f64f4797f-bnl7t Total loading time: 0 Render date: 2025-11-10T06:17:11.412Z Has data issue: false hasContentIssue false
  • English
  • Français

A Functional Equation for Degree two Local Factors

Published online by Cambridge University Press:  20 November 2018

Paul Gérardin  and
Wen-Ch'ing Winnie Li
Paul Gérardin
Affiliation:
U.E.R. de Mathématique et, Informatique, Université Paris VII 2, Place Jussieu, 75251 Paris Cedex 5, France and Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.
Wen-Ch'ing Winnie Li
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Fourier transforms of the admissible irreducible representations of the group GL2 over a nonarchimedian local field F are characterized by a functional equation (MF). We also prove that the functions satisfying (MF) and having at most one pole are exactly the Fourier transforms of the irreducible representations of the quaternion group H over F. The Jacquet-Langlands correspondence between irreducible representations of H and discrete series of GL2 then follows immediately from our criteria.

Keywords

12B2712B30

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 355 - 371
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Gérardin, P. and Kutzko, P., Facteurs locaux pour GL(2), Ann. Se. Ec. Norm. Sup. 13 (1980), pp. 349–384.Google Scholar
2. Gérardin, P. and Li, W.-C.W., Fourier transforms of representations of quaternions J. reine u. angewandte Math, (to appear).Google Scholar
3. Godement, R. and Jacquet, H., Zeta functions of simple algebras, Lecture Notes in Mathematics 260, Springer-Verlag, Berlin-Heidelberg-New York, 1972.Google Scholar
4. Jacquet, H. and Langlands, R.P., Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
5. Li, W.-C.W., On the representations of GL(2). I ∊-factors and n-closeness, J. reine angew. Math. 313 (1980), pp. 27–42.Google Scholar
6. Li, W.-C. W., Barnes’ identities and representations of GL(2). II Nonarchimedean local case, J. reine angew. Math. 345 (1983), pp. 69–92.Google Scholar