Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Refinement of strong multiplicity one
for automorphic representations of $GL(n)$


Author: C. S. Rajan
Journal: Proc. Amer. Math. Soc. 128 (2000), 691-700
MSC (1991): Primary 11F70; Secondary 11F12, 22E55
Published electronically: October 20, 1999
MathSciNet review: 1707005
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We state a qualitative form of strong multiplicity one for $GL_1$. We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on $GL(n)$, and for the cuspidal representations on $GL(n)$ induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Ramakrishnan. We also show that Ramakrishnan's conjecture follows from a weak form of Ramanujan's conjecture. We state a conjecture concerning the structural aspects of refinements of strong multiplicity one for a pair of general automorphic representations.


References [Enhancements On Off] (What's this?)

  • [AC] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
  • [Ca] P. Cartier, Representations of 𝑝-adic groups: a survey, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
  • [GL] P. Gérardin and J.-P. Labesse, The solution of a base change problem for 𝐺𝐿(2) (following Langlands, Saito, Shintani), Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 115–133. MR 546613
  • [Ha] Günter Harder, Some results on the Eisenstein cohomology of arithmetic subgroups of 𝐺𝐿_{𝑛}, Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989) Lecture Notes in Math., vol. 1447, Springer, Berlin, 1990, pp. 85–153. MR 1082964, 10.1007/BFb0085728
  • [He] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Zweite Mitteilung, Mathematische Werke 14 249-289.
  • [JL] H. Jacquet and R. P. Langlands, Automorphic forms on 𝐺𝐿(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
  • [JS] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, 10.2307/2374103
    H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, 10.2307/2374050
  • [JPSh] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, 10.2307/2374264
  • [L] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [La1] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
  • [La2] A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 189–207. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. MR 546598
  • [La3] R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
  • [Li] Wen Ch’ing Winnie Li, On converse theorems for 𝐺𝐿(2) and 𝐺𝐿(1), Amer. J. Math. 103 (1981), no. 5, 851–885. MR 630770, 10.2307/2374250
  • [LRS] W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for $GL(n)$, preprint.
  • [MoW] C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de 𝐺𝐿(𝑛), Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752
  • [MR] M. Ram Murty and C. S. Rajan, Stronger multiplicity one theorems for forms of general type on 𝐺𝐿₂, Analytic number theory, Vol. 2 (Allerton Park, IL, 1995) Progr. Math., vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 669–683. MR 1409385, 10.1016/0009-2614(96)00501-5
  • [R1] C. S. Rajan, Distribution of values of Hecke characters of infinite order, Acta Arith. 85 (1998), no. 3, 279–291. MR 1627843
  • [R2] C. S. Rajan, On strong multiplicity one for 𝑙-adic representations, Internat. Math. Res. Notices 3 (1998), 161–172. MR 1606395, 10.1155/S1073792898000142
  • [DR1] Dinakar Ramakrishnan, A refinement of the strong multiplicity one theorem for 𝐺𝐿(2). Appendix to: “𝑙-adic representations associated to modular forms over imaginary quadratic fields. II” [Invent. Math. 116 (1994), no. 1-3, 619–643; MR1253207 (95h:11050a)] by R. Taylor, Invent. Math. 116 (1994), no. 1-3, 645–649. MR 1253208, 10.1007/BF01231576
  • [DR2] Dinakar Ramakrishnan, Pure motives and automorphic forms, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 411–446. MR 1265561
  • [DR3] Dinakar Ramakrishnan, On the coefficients of cusp forms, Math. Res. Lett. 4 (1997), no. 2-3, 295–307. MR 1453061, 10.4310/MRL.1997.v4.n2.a10
  • [Sh] Freydoon Shahidi, On certain 𝐿-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, 10.2307/2374219

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11F70, 11F12, 22E55

Retrieve articles in all journals with MSC (1991): 11F70, 11F12, 22E55


Additional Information

C. S. Rajan
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay - 400 005, India
Email: rajan@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05616-6
Received by editor(s): April 28, 1998
Published electronically: October 20, 1999
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1999 American Mathematical Society