Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T15:25:50.926Z Has data issue: false hasContentIssue false
  • English
  • Français

The L-Functions L(s, Symm(r), π)

Published online by Cambridge University Press:  20 November 2018

C. J. Moreno  and
F. Shahidi
C. J. Moreno
Affiliation:
Department of Mathematics, University of IllinoisUrbana, Illinois 61801, U.S.A.
F. Shahidi
Affiliation:
Department of Mathematics, Purdue UniversityWest Lafayette, Indiana 47907, 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.

The exact form for the gamma factor for the L-function corresponding to the m-th symmetric power of a cuspidal automorphic representation of PGL(2) is given. This information is used to obtain, via a theorem of Landau, bounds for the eigenvalues of Hecke operators.

Keywords

10D4012B40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 405 - 410
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Elliott, P.D.T., Multiplicative functions and Ramanujarn's τ-function, J. Austral. Math. Soc. (Ser. A), 30(1981), pp. 461–468.Google Scholar
2. Elliott, P.D.T., Moreno, C.J., and Shahidi, F., On the absolute value of Ramanujan's τ-function, Math. Ann., 266 (1984), pp. 507–511.Google Scholar
3. Gelbart, S. and Jacquet, H., A relation between automorphic representations of GL(2) and GL(3), Ann. Scient. Ec. Norm. Sup. (Ser. 4), 11 (1973), pp. 471–542.Google Scholar
4. Jacquet, H., Principal L-functions of the linear group, Proc. Symp. Pure Math. (AMS), 33 part 2 (1979), pp. 63–86.Google Scholar
5. Jacquet, H., Piatetskii-Shapiro, I. I., and Shalika, J., Rankin-Selberg convolutions, Amer. J. of Math., 105(2) (1983), pp. 367–464.Google Scholar
6. Landau, E., Über die Anzahl der Gitterpunkte in gewissen Bereichen, Gött. Nachr. (1915), pp. 209–243.Google Scholar
7. Landau, E., Einfuhrung in die elementare und analytische Théorie der algebraischen Zahlen und der Ideale, Chelsea Publ. Co., New York, 1949.Google Scholar
8. Langlands, R.P., Problems in the theory of automorphic forms, Lecture Notes in Math., Springer-Verlag, 170 (1970), pp. 18–61.Google Scholar
9. Langlands, R.P., On the classification of irreducible representations of real algebraic groups (preprint).Google Scholar
10. Proskurin, N.V., Estimates for eigenvalues ofHecke operators in the space of parabolic forms of weight zero, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 82 (1979), pp. 136–143.Google Scholar
11. Rankin, R.A., Contributions to the theory of Ramanujan's functions and other arithmetical functions, I. Proc. Cambridge Phil. Soc, 35 (1939), pp. 351–356.Google Scholar
12. Satake, I., Spherical functions and Ramanujan conjecture, Proc. Symp. Pure Math. (AMS), 9 (1966), pp. 258–264.Google Scholar
13. Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces, Ann. of Math., 100 (1974), pp. 131–170.Google Scholar
14. Selberg, A., Bemerkungen uber eine Dirichletsche Reihe, die mit der Theory der Modulformen nahe verbunden ist, Arch Math. Naturvid., 43 (1940), pp. 47–50.Google Scholar
15. Selberg, A., On the estimationof Fourier coefficients of modular forms, Proc. Symp. Pure Math. (AMS), 8(1965), pp. 1–15.Google Scholar
16. Serre, J.-P., Abelian 1-adic representations and elliptic curves, Benjamin Publ. Co., New York, 1968.Google Scholar
17. Serre, J.-P., Une interprétation des congruences relatives à la fonction T de Ramanujan, Sem. Delange-Pisot-Poitou 1967/1968, No. 14.Google Scholar
18. Serre, J.-P., written communication (July 22, 1983).Google Scholar
19. Shahidi, F., Functional equation satisfied by certain L-functions, Compositio Mathematica, 37 (1978), pp. 171–201.Google Scholar
20. Shahidi, F., On certain L-functions, Amer. J. of Math., 103 (1981), pp. 297–355.Google Scholar
21. Shahidi, F. and Moreno, C.J., The fourth moment of the Ramanujan T-function, Math. Annalen, 266 (1983), pp. 233–239.Google Scholar
22. Tamagawa, T., On the functional equation of the generalized L-function, J. Fac. Sc. Tokyo, 6 (1953), pp. 421–428.Google Scholar
23. Ram Murty, M., On the estimation of eigenvalues of Hecke operators, (to appear).Google Scholar