### 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 language | English |
---|---|

Pages (from-to) | 1443-1474 |

Number of pages | 32 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 69 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I : An explicit functorial correspondence.** / Narita, Hiroaki; Schmidt, Ralf.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 69, no. 4, pp. 1443-1474. https://doi.org/10.2969/jmsj/06941443

}

TY - JOUR

T1 - Jacquet-langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms I

T2 - An explicit functorial correspondence

AU - Narita, Hiroaki

AU - Schmidt, Ralf

PY - 2017/1/1

Y1 - 2017/1/1

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

KW - Spinor L-functions

KW - Theta lifts

UR - http://www.scopus.com/inward/record.url?scp=85032787184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032787184&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06941443

DO - 10.2969/jmsj/06941443

M3 - Article

AN - SCOPUS:85032787184

VL - 69

SP - 1443

EP - 1474

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 4

ER -