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

Hiroaki Narita, Ralf Schmidt

研究成果: Article

抄録

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.

元の言語English
ページ(範囲)1443-1474
ページ数32
ジャーナルJournal of the Mathematical Society of Japan
69
発行部数4
DOI
出版物ステータスPublished - 2017 1 1
外部発表Yes

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

ASJC Scopus subject areas

  • Mathematics(all)

これを引用

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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 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",
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AU - Schmidt, Ralf

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N2 - 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.

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.

KW - Jacquet-Langlands correspondence

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KW - Theta lifts

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