Abstract
In this paper, we would like to formulate a conjecture on a relation between a certain period of automorphic forms on special orthogonal groups and some L-value. Our conjecture can be considered as a refinement of the global Gross–Prasad conjecture.
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A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity one theorems, Ann. of Math., to appear.
Arthur J. (1989) Unipotent automorphic representations: conjectures. Astérisque 171/172: 13–71
S. Böcherer, M. Furusawa, R. Schulze-Pillot, On the global Gross–Prasad conjecture for Yoshida liftings, in “Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, (2004), 105–130.
Böcherer S., Schulze-Pillot R. (1991) Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math. J. 121: 35–96
Casselman W. (1980) The unramified principal series of \({\mathfrak{p}}\)-adic groups. I. The spherical function. Compositio Math. 40: 387–406
Cowling M., Haagerup U., Howe R. (1988) Almost L 2 matrix coefficients. J. Reine Angew. Math. 387: 97–110
P. Deligne, Valeurs de fonctions L et périodes d’intégrales, Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, Part 2, Amer. Math. Soc., Providence, RI (1979), 313–346.
W.T. Gan, A. Ichino, On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4), preprint.
Ginzburg D., Piatetski-Shapiro I.I., Rallis S. (1997) L functions for the orthogonal group. Mem. Amer. Math. Soc. 128: 611
R. Godement, H. Jacquet, Zeta Functions of Simple Algebras, Springer Lecture Notes in Math. 260(1972).
Gross B.H. (1997) On the motive of a reductive group. Invent. Math. 130: 287–313
Gross B.H., Prasad D. (1992) On the decomposition of a representation of SO n when restricted to SOn-1. Canad. J. Math. 44: 974–1002
Gross B.H., Prasad D. (1994) On irreducible representations of SO2n+1 × SO2m . Canad. J. Math. 46: 930–950
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104–204.
Harris M. (1990) Period invariants of Hilbert modular forms. I. Trilinear differential operators and L-functions, Cohomology of Arithmetic Groups and Automorphic Forms. Springer Lecture Notes in Math. 1447: 155–202
Harris M. (1993) L-functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms. J. Amer. Math. Soc. 6: 637–719
Harris M. (1994) Period invariants of Hilbert modular forms. II. Compositio Math 94: 201–226
Harris M., Kudla S.S. (1991) The central critical value of a triple product L-function. Ann. of Math. 133: 605–672
Harris M., Soudry D., Taylor R. (1993) l-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to GSp4(Q). Invent. Math. 112: 377–411
He H. (2003) Unitary representations and theta correspondence for type I classical groups. J. Funct. Anal. 199: 92–121
S. Helgason, Groups and Geometric Analysis, Pure and Applied Mathematics 113, Academic Press Inc., Orlando, FL, (1984).
K. Hiraga, H. Saito, On L-packets for inner forms of SL n , preprint.
Howe R., Piatetski-Shapiro I.I. (1983) Some examples of automorphic forms on Sp4. Duke Math. J. 50: 55–106
Ichino A. (2005) Pullbacks of Saito–Kurokawa lifts. Invent. Math. 162: 551–647
Ichino A. (2008) Trilinear forms and the central values of triple product L-functions. Duke Math. J. 145: 281–307
Ichino A., Ikeda T. (2008) On Maass lifts and the central critical values of triple product L-functions. Amer. J. Math. 130: 75–114
Ikeda T. (2006) Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture. Duke Math. J. 131: 469–497
Ikeda T. (2008) On the lifting of hermitian modular forms. Compositio Math. 144: 1107–1154
Kato S., Murase A., Sugano T. (2003) Whittaker–Shintani functions for orthogonal groups. Tohoku Math. J. 55: 1–64
A.W. Knapp, Representations of GL2(R) and GL2(C), Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33:1 Amer. Math. Soc., Providence, RI (1979), 87–91.
Kohnen W. (1980) Modular forms of half-integral weight on Γ0(4). Math. Ann. 248: 249–266
Kohnen W., Skoruppa N.-P. (1989) A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math. 95: 541–558
Kohnen W., Zagier D. (1981) Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64: 175–198
Kojima H. (1982) An arithmetic of Hermitian modular forms of degree two. Invent. Math. 69: 217–227
Krieg A. (1991) The Maaß spaces on the Hermitian half-space of degree 2. Math. Ann. 289: 663–681
Macdonald I.G. (1980) The volume of a compact Lie group. Invent. Math. 56: 93–95
Raghavan S., Sengupta J. (1991) A Dirichlet series for Hermitian modular forms of degree 2. Acta Arith. 58: 181–201
Roberts B. (2001) Global L-packets for GSp(2) and theta lifts. Doc. Math. 6: 247–314
Shimura G. (1981) On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra. Ann. of Math. 114: 569–607
Shimura G. (1983) Algebraic relation between critical values of zeta functions and inner products. Amer. J. Math. 105: 253–285
Shimura G. (1988) On the critical values of certain Dirichlet series and the periods of automorphic forms. Invent. Math. 94: 245–305
Shimura G. (1999) An exact mass formula for orthogonal groups. Duke Math. J. 97: 1–66
A.J. Silberger, Introduction to Harmonic Analysis on Reductive p-Adic Groups, Mathematical Notes 23, Princeton University Press, Princeton, NJ, 1979.
Sugano T. (1995) Jacobi forms and the theta lifting. Comment. Math. Univ. St. Paul. 44: 1–58
B. Sun, C.-B. Zhu, Multiplicity one theorems: the archimedean case, preprint.
J. Tate, Number theoretic background, Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, Part 2, Amer. Math. Soc., Providence, RI (1979), 3–26.
Waldspurger J.-L. (1985) Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54: 173–242
T.C. Watson, Rankin triple products and quantum chaos, Ann. of Math., to appear.
Yoshida H. (1980) Siegel’s modular forms and the arithmetic of quadratic forms. Invent. Math. 60: 193–248
Yoshida H. (1995) On a conjecture of Shimura concerning periods of Hilbert modular forms. Amer. J. Math. 117: 1019–1038
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Dedicated to Professor Hiroyuki Yoshida on the occasion of his sixtieth birthday
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Ichino, A., Ikeda, T. On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture. Geom. Funct. Anal. 19, 1378–1425 (2010). https://doi.org/10.1007/s00039-009-0040-4
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DOI: https://doi.org/10.1007/s00039-009-0040-4