Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I

An explicit functorial correspondence

Hiroaki Narita, Ralf Schmidt

Research output: Contribution to journalArticle

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 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)×B× to GSp(1, 1) or GSp(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over 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.

Original languageEnglish
Pages (from-to)1443-1474
Number of pages32
JournalJournal of the Mathematical Society of Japan
Volume69
Issue number4
DOIs
Publication statusPublished - 2017 Jan 1
Externally publishedYes

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Correspondence
Symplectic Group
Quaternion Algebra
Spinor
L-function
Form
Subgroup
Denote
Invariant

Keywords

  • Jacquet-Langlands correspondence
  • Spinor L-functions
  • Theta lifts

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I: An explicit functorial correspondence",
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 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)×B× to GSp(1, 1) or GSp∗(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over 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.",
keywords = "Jacquet-Langlands correspondence, Spinor L-functions, Theta lifts",
author = "Hiroaki Narita and Ralf Schmidt",
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AB - 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 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)×B× to GSp(1, 1) or GSp∗(2) and a theta lift from GL(2) × GL(2) (or GO(2, 2)) to GSp(2). Here B denotes a definite quaternion algebra over 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.

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