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A strengthening of the GL(2) converse theorem

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 February 2011

Andrew R. Booker  and
M. Krishnamurthy
Andrew R. Booker
Affiliation:
School of Mathematics, Bristol University, University Walk, Bristol, BS8 1TW, UK (email: andrew.booker@bristol.ac.uk)
M. Krishnamurthy
Affiliation:
Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA (email: muthu-krishnamurthy@uiowa.edu)
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Abstract

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We generalize the method of A. R. Booker (Poles of Artin L-functions and the strong Artin conjecture, Ann. of Math. (2) 158 (2003), 1089–1098; MR 2031863(2004k:11082)) to prove a version of the converse theorem of Jacquet and Langlands with relaxed conditions on the twists by ramified idèle class characters.

Keywords

converse theoremautomorphic forms

MSC classification

Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 669 - 715
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Booker, A. R., Poles of Artin L-functions and the strong Artin conjecture, Ann. of Math. (2) 158 (2003), 1089–1098; MR 2031863(2004k:11082).CrossRefGoogle Scholar
[2]Cogdell, J. W., Kim, H. H. and Murty, M. R., Lectures on automorphic L-functions, Fields Institute Monographs, vol. 20 (American Mathematical Society, Providence, RI, 2004); MR 2071722(2005h:11104).Google Scholar
[3]Deligne, P., Les constantes des équations fonctionnelles des fonctions L, in Modular functions of one variable, II (Proc. Int. Summer School, University of Antwerp, 1972), Lecture Notes in Mathematics, vol. 349 (Springer, Berlin, 1973), 501–597; MR 0349635(50#2128).Google Scholar
[4]Diaconu, A. and Garrett, P., Integral moments of automorphic L-functions, J. Inst. Math. Jussieu 8 (2009), 335–382; MR 2485795.CrossRefGoogle Scholar
[5]Godement, R., Notes on Jacquet–Langlands theory (Institute for Advanced Study, 1970), available at: http://www.math.ubc.ca/∼cass/research/pdf/godement-ias.pdf.zip.Google Scholar
[6]Hecke, E., Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann. 112 (1936), 664–699; MR 1513069.CrossRefGoogle Scholar
[7]Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114 (Springer, Berlin, 1970); MR 0401654(53#5481).CrossRefGoogle Scholar
[8]Langlands, R. P., On the functional equation of the Artin L-functions (1969), available at http://publications.ias.edu/sites/default/files/a-ps.pdf.Google Scholar
[9]Li, W. C. W., Hecke–Weil–Jacquet–Langlands theorem revisited, in Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois University, Carbondale, IL, 1979), Lecture Notes in Mathematics, vol. 751 (Springer, Berlin, 1979), 206–220; MR 564931(81m:10058).Google Scholar
[10]Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322 (Springer, Berlin, 1999), Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder; MR 1697859(2000m:11104).CrossRefGoogle Scholar
[11]Pjateckij-Šapiro, I. I., On the Weil–Jacquet–Langlands theorem, in Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) (Halsted, New York, 1975), 583–595; MR 0406934(53#10719).Google Scholar
[12]Popa, A. A., Whittaker newforms for Archimedean representations, J. Number Theory 128 (2008), 1637–1645; MR 2419183(2009c:22012).CrossRefGoogle Scholar
[13]Sarnak, P., Notes on Booker’s paper (2002), available at http://www.math.princeton.edu/sarnak/.Google Scholar
[14]Weil, A., Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156; MR 0207658(34#7473).CrossRefGoogle Scholar
[15]Weil, A., Dirichlet series and automorphic forms, Lecture Notes in Mathematics, vol. 189 (Springer, Berlin, 1971).CrossRefGoogle Scholar