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On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4)

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 August 2010

Wee Teck Gan  and
Atsushi Ichino
Wee Teck Gan
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA ([email protected])
Atsushi Ichino
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan ([email protected])
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Abstract

We prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

Keywords

periods of automorphic formsL-valuesendoscopythe Gross–Prasad conjecture

MSC classification

Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 235 - 324
Copyright
Copyright © Cambridge University Press 2010

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References

1.Adams, J. and Barbasch, D., Reductive dual pair correspondence for complex groups, J. Funct. Analysis 132 (1995), 1–42.Google Scholar
2.Arthur, J., Unipotent automorphic representations: conjectures, Astérisque 171–172 (1989), 13–71.Google Scholar
3.Ban, D. and Jantzen, C., Degenerate principal series for even-orthogonal groups, Represent. Theory 7 (2003), 440–480.CrossRefGoogle Scholar
4.Blasius, D. and Rogawski, J. D., Motives for Hilbert modular forms, Invent. Math. 114 (1993), 55–87.Google Scholar
5.Böcherer, S., Furusawa, M. and Schulze-Pillot, R., On the global Gross–Prasad conjecture for Yoshida liftings, in Contributions to automorphic forms, geometry, and number theory, pp. 105–130 (Johns Hopkins University Press, Baltimore, MD, 2004).Google Scholar
6.Cognet, M., Représentation de Weil et changement de base quadratique, Bull. Soc. Math. France 113 (1985), 403–457.Google Scholar
7.Cognet, M., Représentation de Weil et changement de base quadratique dans le cas archimédien, II, Bull. Soc. Math. France 114 (1986), 325–354.CrossRefGoogle Scholar
8.Gan, W. T. and Gurevich, N., Restriction of Saito–Kurokawa representations (with an appendix by G. Savin), in Automorphic forms and L-functions, I, Global aspects, Contemporary Mathematics, Volume 488, pp. 95–124 (American Mathematical Society, Providence, RI, 2009).Google Scholar
9.Gan, W. T. and Takeda, S., The local Langlands conjecture for GSp(4), Annals Math., in press.Google Scholar
10.Garrett, P. B., Decomposition of Eisenstein series: Rankin triple products, Annals Math. 125 (1987), 209–235.CrossRefGoogle Scholar
11.Gross, B. H. and Prasad, D., On the decomposition of a representation of SOn when restricted to SOn−1, Can. J. Math. 44 (1992), 974–1002.CrossRefGoogle Scholar
12.Harris, M., L-functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms, J. Am. Math. Soc. 6 (1993), 637–719.Google Scholar
13.Harris, M. and Kudla, S. S., Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2), Duke Math. J. 66 (1992), 59–121.Google Scholar
14.Harris, M., Kudla, S. S. and Sweet, W. J. Jr, Theta dichotomy for unitary groups, J. Am. Math. Soc. 9(1996), 941–1004.CrossRefGoogle Scholar
15.Harris, M., Soudry, D. and Taylor, R., l-adic representations associated to modular forms over imaginary quadratic fields, I, Lifting to GSp4(Q), Invent. Math. 112 (1993), 377–411.Google Scholar
16.He, H., Theta correspondence, I, Semistable range: construction and irreducibility, Commun. Contemp. Math. 2 (2000), 255–283.Google Scholar
17.Hiraga, K. and Saito, H., On L-packets for inner forms of SLn, preprint.Google Scholar
18.Howe, R., Transcending classical invariant theory, J. Am. Math. Soc. 2 (1989), 535–552.CrossRefGoogle Scholar
19.Ichino, A., A regularized Siegel–Weil formula for unitary groups, Math. Z. 247 (2004), 241–277.Google Scholar
20.Ichino, A., Trilinear forms and the central values of triple product L-functions, Duke Math. J. 145 (2008), 281–307.Google Scholar
21.Ichino, A. and Ikeda, T., On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Analysis 19 (2010), 1378–1425.Google Scholar
22.Ikeda, T., On the location of poles of the triple L-functions, Compositio Math. 83 (1992), 187–237.Google Scholar
23.Ikeda, T., On the residue of the Eisenstein series and the Siegel–Weil formula, Compositio Math. 103 (1996), 183–218.Google Scholar
24.Jiang, D., The first term identities for Eisenstein series, J. Number Theory 70 (1998), 67–98.CrossRefGoogle Scholar
25.Jiang, D. and Soudry, D., On the genericity of cuspidal automorphic forms of SO(2n+1), II, Compositio Math. 143 (2007), 721–748.Google Scholar
26.Kim, H. H., On local L-functions and normalized intertwining operators, Can. J. Math. 57 (2005), 535–597.Google Scholar
27.Kim, H. H. and Krishnamurthy, M., Stable base change lift from unitary groups to GLn, Int. Math. Res. Pap. 1(2005), 1–52.Google Scholar
28.Kim, H. H. and Shahidi, F., Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177–197.Google Scholar
29.Kudla, S. S., On the local theta-correspondence, Invent. Math. 83 (1986), 229–255.Google Scholar
30.Kudla, S. S., Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), 361–401.Google Scholar
31.Kudla, S. S. and Rallis, S., On the Weil–Siegel formula, J. Reine Angew. Math. 387 (1988), 1–68.Google Scholar
32.Kudla, S. S. and Rallis, S., Poles of Eisenstein series and L-functions, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II, Israel Mathematical Conference Proceedings, Volume 3, pp. 81–110 (Weizmann, Jerusalem, 1990).Google Scholar
33.Kudla, S. S. and Rallis, S., Ramified degenerate principal series representations for Sp(n), Israel J. Math. 78 (1992), 209–256.CrossRefGoogle Scholar
34.Kudla, S. S. and Rallis, S., A regularized Siegel–Weil formula: the first term identity, Annals Math. 140 (1994), 1–80.Google Scholar
35.Kudla, S. S., Rallis, S. and Soudry, D., On the degree 5 L-function for Sp(2), Invent. Math. 107 (1992), 483–541.Google Scholar
36.Lai, K. F., Tamagawa number of reductive algebraic groups, Compositio Math. 41 (1980), 153–188.Google Scholar
37.Lee, S. T. and Zhu, C.-B., Degenerate principal series and local theta correspondence, II, Israel J. Math. 100 (1997), 29–59.CrossRefGoogle Scholar
38.Lee, S. T. and Zhu, C.-B., Degenerate principal series and local theta correspondence, III, The case of complex groups, J. Alg. 319 (2008), 336–359.Google Scholar
39.Li, J.-S., Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237–255.Google Scholar
40.Li, J.-S., Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), 177–217.Google Scholar
41.Loke, H. Y., Howe quotients of unitary characters and unitary lowest weight modules (with an appendix by S. T. Lee), Represent. Theory 10 (2006), 21–47.Google Scholar
42.Muić, G., On the structure of theta lifts of discrete series for dual pairs (Sp(n), O(V)), Israel J. Math. 164 (2008), 87–124.Google Scholar
43Paul, A., On the Howe correspondence for symplectic-orthogonal dual pairs, J. Funct. Analysis 228 (2005), 270–310.Google Scholar
44.Piatetski-Shapiro, I. I. and Rallis, S., L-functions for the classical groups, in Explicit constructions of automorphic L-functions, Lecture Notes in Mathematics, Volume 1254, pp. 1–52 (Springer, 1987).Google Scholar
45.Piatetski-Shapiro, I. I. and Rallis, S., Rankin triple L functions, Compositio Math. 64 (1987), 31–115.Google Scholar
46.Rallis, S., On the Howe duality conjecture, Compositio Math. 51 (1984), 333–399.Google Scholar
47.Roberts, B., The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317.CrossRefGoogle Scholar
48.Roberts, B., The non-Archimedean theta correspondence for GSp(2) and GO(4), Trans. Am. Math. Soc. 351 (1999), 781–811.Google Scholar
49.Roberts, B., Global L-packets for GSp(2) and theta lifts, Documenta Math. 6 (2001), 247–314.CrossRefGoogle Scholar
50.Roberts, B. and Schmidt, R., Local newforms for GSp(4), Lecture Notes in Mathematics, Volume 1918 (Springer, 2007).Google Scholar
51.Sally, P. J. Jr and Tadić, M., Induced representations and classifications for GSp(2, F) and Sp(2, F), Mém. Soc. Math. France (N.S.) 52 (1993), 75–133.Google Scholar
52.Shimizu, H., Theta series and automorphic forms on GL2, J. Math. Soc. Jpn 24 (1972), 638–683.CrossRefGoogle Scholar
53.Soudry, D., The CAP representations of GSp(4,A), J. Reine Angew. Math. 383 (1988), 87–108.Google Scholar
54.Tan, V., A regularized Siegel–Weil formula on U(2,2) and U(3), Duke Math. J. 94 (1998), 341–378.CrossRefGoogle Scholar
55.Waldspurger, J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Math. 54 (1985), 173–242.Google Scholar
56.Waldspurger, J.-L., Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠ 2, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Mathematical Conference Proceedings, Volume 2, pp. 267– 324 (Weizmann, Jerusalem, 1990).Google Scholar
57.Yoshida, H., Siegel's modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), 193–248Google Scholar