Journal of the Mathematical Society of Japan

Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence

Hiro-aki NARITA

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Abstract

As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group $GSp(1,1)$ or the definite symplectic group $GSp^*(2)$ over $\mathbb{Q}$ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor $L$-functions as those of paramodular new forms of some specified level on the symplectic group $GSp(2)$ (or $GSp(4)$). This can be viewed as a generalization of the Jacquet–Langlands–Shimizu correspondence to the case of $GSp(2)$ and its inner forms $GSp(1,1)$ and $GSp^*(2)$.

In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from $GL(2)\times B^{\times}$ to $GSp(1,1)$ or $GSp^*(2)$ and a theta lift from $GL(2)\times GL(2)$ (or $GO(2,2)$) to $GSp(2)$. Here $B$ denotes a definite quaternion algebra over $\mathbb{Q}$. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet–Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of $GSp(2)$, which is studied in the appendix by Ralf Schmidt.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1443-1474.

Dates
First available in Project Euclid: 25 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1508918564

Digital Object Identifier
doi:10.2969/jmsj/06941443

Mathematical Reviews number (MathSciNet)
MR3715811

Zentralblatt MATH identifier
06821647

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F55: Other groups and their modular and automorphic forms (several variables)

Keywords
Jacquet–Langlands correspondence spinor $L$-functions theta lifts

Citation

NARITA, Hiro-aki. Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence. J. Math. Soc. Japan 69 (2017), no. 4, 1443--1474. doi:10.2969/jmsj/06941443. https://projecteuclid.org/euclid.jmsj/1508918564


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References

  • J. Arthur, The endoscopic classification of representations, orthogonal and symplectic groups, Colloquium Publications, 61, Amer. Math. Soc., 2013.
  • A. I. Badulescu, Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, With an appendix by Neven Grbac, Invent. Math., 172 (2008), 383–438.
  • A. I. Badulescu and D. Renard, Unitary dual of $GL(n)$ at archimedean places and global Jacquet–Langlands correspondence, Compositio Math., 146 (2010), 1115–1164.
  • A. Borel, Automorphic $L$-functions, Proc. Sympos. Pure Math., 33, part 2, Amer. Math. Soc. Providence, RI, 1979, 27–61.
  • D. Bump, Automorphic forms and representations, Cambridge University Press, 1998.
  • P. Cartier, Representations of $\mathfrak{p}$-adic groups: A survey, Proc. Sympos. Pure Math., 33, part 1, Amer. Math. Soc. Providence, RI, 1979, 111–155.
  • M. Eichler, Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math., 195 (1955), 156–171.
  • M. Eichler, Quadratische Formen und Modulfunktionen, Acta Arith., 4 (1958), 217–239.
  • W. T. Gan and S. Takeda, Theta correspondence for $GSp(4)$, Representation Theory, 15 (2011), 670–718.
  • W. T. Gan and T. Takeda, The local Langlands conjecture for ${\rm GSp}(4)$, Ann. of Math., 173 (2011), 1841–1882.
  • W. T. Gan and W. Tantono, Local Langlands conjecture for $GSp(4)$ II, The case of inner forms, Amer. J. Math., 136 (2014), 761–805.
  • M. Harris and S. Kudla, Arithmetic automorphic forms for the non-holomorphic discrete series of $GSp(2)$, Duke Math. J., 66 (1992), 59–121.
  • M. Harris, D. Soudry and R. Taylor, $l$-adic representations attached to modular forms over imaginary quadratic fields I: lifting to $GSp_4(\mathbb{Q})$, Invent. Math., 112 (1993), 377–411.
  • Harish-Chandra, Discrete series for semisimple Lie groups II, Explicit determination of the characters, Acta Math., 116 (1966), 1–111.
  • E. Hecke, Analytische arithmetik der positiven quadratischen formen, Math. Werke, Vandenhoeck und Ruprecht in G öttingen, 1983, 789–918.
  • R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc., 2 (1989), 535–552.
  • T. Ibukiyama, Paramodular forms and compact twist, Automorphic forms on $GSp(4)$, Proceedings of the 9th Autumn workshop on number theory, (ed. Masaaki Furusawa), 2006, 37–48.
  • T. Ibukiyama, On relations of dimensions of automorphic forms of $Sp(2,\mathbb{R})$ and its compact twist $Sp(2)$ (I), Adv. Stu. Pure Math., 7, Automorphic Forms and Number Theory, North-Holland, Amsterdam, 1985, 7–30.
  • T. Ibukiyama and Y. Ihara, On automorphic forms on the unitary symplectic group $Sp(n)$ and $SL_2(\mathbb{R})$, Math. Ann., 278 (1987), 307–327.
  • H. Jacquet and R. P. Langlands, Automorphic forms on $GL(2)$, Lecture Notes in Math., 114, Springer-Verlag, 1970.
  • A. Knapp, Representation theory of semisimple groups, Princeton University Press, 1986.
  • R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., 170, Springer-Verlag, 1970, 18–61.
  • R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., 31, Amer. Math. Soc. Providence, RI, 1989, 101–170.
  • J. S. Li, A. Paul, E. C. Tan and C. B. Zhu, The explicit duality correspondence of $(Sp(p,q), O^*(2n))$, J. Funct. Anal., 200 (2003), 71–100.
  • R. L öschel, Thetakorrespondenz automorpher Formen, Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu K öln, 1997.
  • C. Moeglin and J. L. Waldspurger, Stabilisation de la formule des traces tordue, 1 and 2, Progress in Math., 316, 317.
  • T. Miyake, Modular forms, Springer-Verlag, Berlin, 2006.
  • T. Moriyama, Entireness of the spinor $L$-functions for certain generic cusp forms on $GSp(2)$, Amer. J. Math., 126 (2004), 899–920.
  • A. Murase, CM-values and central $L$-values of elliptic modular forms (II), Max-Planck-Institut für Mathematik Preprint Series, 30, 2008.
  • A. Murase and H. Narita, Commutation relations of Hecke operators for Arakawa lifting, Tohoku Math. J., 60 (2008), 227–251.
  • A. Murase and H. Narita, Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math., 187 (2012), 317–369.
  • A. Murase and H. Narita, Fourier expansion of Arakawa lifting II: Relation with central $L$-values, Internat. J. Math., 27 (2016), 32 pages.
  • H. Narita, Theta lifting from elliptic cusp forms to automorphic forms on $Sp(1,q)$, Math. Z., 259 (2008), 591–615.
  • H. Narita, Bessel periods of theta lifts to $GSp(1,1)$ and central values of some $L$-functions of convolution type, Automorphic Forms, Research in Number Theory from Oman, Springer Proceedings in Mathematics and Statistics, 115 (2014), 179–191.
  • H. Narita, Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms II: An explicit formula for Bessel periods and non-vanishing of theta lifts, preprint.
  • H. Narita, A. Pitale and R. Schmidt, Irreducibility criteria for local and global representations, Proc. Amer. Math. Soc., 141 (2013), 55–63.
  • T. Przebinda, The oscillator duality correspondence for the pair $O(2,2)$ and $Sp(2,\mathbb{R})$, Mem. Amer. Math. Soc., 79, No.403, 1989.
  • B. Roberts, Global $L$-packets for $GSp(2)$ and theta lifts, Documenta Math., 6 (2001), 247–314.
  • B. Roberts and R. Schmidt, Local new forms for $GSp(4)$, Lecture Notes in Math., 1918, Springer-Verlag, 2007.
  • I. Satake, Theory of spherical functions on reductive algebraic groups over $p$-adic fields, Inst. Hautes Études Sci. Publ. Math., 18 (1963), 5–69.
  • H. Shimizu, Theta series and automorphic forms on $GL_2$, J. Math. Soc. Japan, 24 (1972), 638–683.
  • G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33–65.
  • G. Shimura, Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups, Mathematical Surveys and Monographs, 109, Amer. Math. Soc., 2004.
  • C. M. Sorensen, Level-raising for Saito–Kurokawa forms, Compositio Math., 145 (2009), 915–953.
  • T. Sugano, On holomorphic cusp forms on quaternion unitary groups of degree 2, J. Fac. Sci. Univ. Tokyo, 31 (1985), 521–568.
  • N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, 1, Kluwer Academic Publishers, 1991.
  • A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math., 111 (1964), 143–211.
  • N. Wallach, Real reductive groups I, Academic Press, 1988.