Israel Journal of Mathematics

, Volume 123, Issue 1, pp 317–340 | Cite as

Quadratic base change of θ10

  • Ju-Lee KimEmail author
  • Ilya I. Piatetski-Shapiro


In case ofGL n overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ10. In the second part, we compute the localL-factor of θ10. Unlike many other supercuspidal representations, we find that theL-factor of θ10 has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence.


Discrete Series Quadratic Extension Cuspidal Representation Admissible Representation Quadratic Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AC] J. Arthur and L. Clozel,Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Vol. 120, Annals of Mathematics Studies, Princeton University Press, 1989.Google Scholar
  2. [As] C. Asmuth,Weil representations of symplectic p-adic groups American Journal of Mathematics101 (1979), 885–908.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Cl] L. Clozel,Characters of non-connected reductive p-adic groups, Canadian Journal of Mathematics39 (1987), 149–167.zbMATHMathSciNetGoogle Scholar
  4. [De] P. Deligne,Forms modulaires et représentations de GL(2), inModular Functions of One Variable II, Proceedings of the International Summer School, Belgium349, Springer-Verlag, Berlin, 1972, pp. 55–106.CrossRefGoogle Scholar
  5. [Gy] Gyoja,Liftings of irreducible characters of finite reductive groups, Osaka Journal of Mathematics16 (1979), 1–30.zbMATHMathSciNetGoogle Scholar
  6. [HC] Harish-Chandra,Admissible invariant distributions on reductive p-adic groups, Queen’s Papers in Pure and Applied Mathematics48 (1978), 281–347.MathSciNetGoogle Scholar
  7. [HK] M. Harris and S. Kudla,Nonholomorphic discrete series of GSp(2), Duke Mathematical Journal66 (1992), 59–121.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [HPS] R. Howe and I. Piatetski-Shapiro,A counterexample to the “generalized Ramanujan conjecture”, Proceedings of Symposia in Pure Mathematics33, part 1 (1979), 315–322.Google Scholar
  9. [Ko] R. Kottwitz,Rational conjugacy classes in reductive groups, Duke Mathematical Journal49 (1982), 785–806.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [La] R. Langlands,Base change for GL(2), Vol. 96, Annals of Mathematics Studies, Princeton University Press, 1980.Google Scholar
  11. [MVW] C. Moeglin, M.-F. Vigneras and J.-L. Waldspurger,Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics1291, Springer-Verlag, Berlin, 1987.zbMATHGoogle Scholar
  12. [PS] I. Piatetski-Shapiro,L-functions for GSp(4), Olga Taussky-Todd: In Memoriam. Pacific Journal of Mathematics Special Issue (1997), 259–275.Google Scholar
  13. [PSS] I. Piatetski-Shapiro and D. Soudry,L and ∈ factors of GSp(4), Journal of the Fauclty of Science of the University of Tokyo, Section IA28 (1981), 505–530.zbMATHMathSciNetGoogle Scholar
  14. [Re] M. Reeder,On the Iwahori spherical discrete series of p-acid Chevalley groups; formal degrees and L-packets, Annales Scientifiques de l’École Normale Supérieure27 (1994), 463–491.zbMATHMathSciNetGoogle Scholar
  15. [Ro] B. Roberts,The theta correspondence for similitudes, Israel Journal of Mathematics94 (1996), 285–317.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Sr] B. Srinivasan,The characters of finite symplectic group Sp(4,q), Transactions of the American Mathematical Society131 (1968), 488–525.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [T] J. Tate,Number theoretic background, Proceedings of Symposia in Pure Mathematics33, part 2 (1979), 3–26.Google Scholar

Copyright information

© The Hebrew University Magnes Press 2001

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations