Abstract
The author shows that the (partial) standard Langlands L-functions on quarternion groups have at most simple poles at certain positive integers.
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References
Arthur, J., Eisenstein series and the trace formula, Automorphic Forms, Representations and L-Functions, Amer. Math. Soc., Providence, R. I., 1979, 253–274.
Casselman, W., The unramified principal series of p-adic groups, I, the spherical function, Compositio Math., 40(3), 1980, 387–406.
Garrett, P. B., Pullbacks of Eisenstein Series; Applications, Automorphic Forms of Several Variables, 46, Progr. Math., Birkhäuser Boston, Boston, MA, 1984, 114–137.
Gelbart, S., Piatetski-Shapiro, I. and Rallis, S., Explicit Constructions of Automorphic L-Functions, 1254, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1987.
Howe, R., θ-Series and Invariant Theory, Automorphic Forms, Amer. Math. Soc., Providence, R. I., 1979, 275–285.
Kudla, S. S. and Rallis, S., Poles of Eisenstein Series and L-Functions, Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part II (Ramat Aviv, 1989), 3, Israel Math. Conf. Proc., Weizmann, Jerusalem, 1990, 81–110.
Kudla, S. S. and Rallis, S., A regularized Siegel-Weil formula: the first term identity, Ann. of Math., 140(1), 1994, 1–80.
Langlands, R. P., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, 544, Springer-Verlag, Berlin, 1976.
Ürtiş, Ç., Special values of L-functions by a Siegel-Weil-Kudla-Rallis formula, J. Number Theory, 125(1), 2007, 149–181.
Ürtiş, Ç., Poles of Eisenstein series on quaternion groups, J. Number Theory, 130(9), 2010, 2065–2077.
Weil, A., Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math., 113, 1965, 1–87.
Yamana, S., L-Functions and theta correspondence for classical groups, Inventiones Mathematicae, 2013, DOI: 10.1007/S00222-013-0476-x.
Yamana, S., On the Siegel-Weil formula for quaternionic unitary groups, Amer. J. Math., 135(5), 2013, 1383–1432.
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Ürtiş, Ç. Poles of L-functions on quaternion groups. Chin. Ann. Math. Ser. B 35, 519–526 (2014). https://doi.org/10.1007/s11401-014-0849-5
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DOI: https://doi.org/10.1007/s11401-014-0849-5