### 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 |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)