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October, 2017 Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence
Hiro-aki NARITA
J. Math. Soc. Japan 69(4): 1443-1474 (October, 2017). DOI: 10.2969/jmsj/06941443

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.

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Hiro-aki NARITA. "Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence." J. Math. Soc. Japan 69 (4) 1443 - 1474, October, 2017. https://doi.org/10.2969/jmsj/06941443

Information

Published: October, 2017
First available in Project Euclid: 25 October 2017

zbMATH: 06821647
MathSciNet: MR3715811
Digital Object Identifier: 10.2969/jmsj/06941443

Subjects:
Primary: 11F70
Secondary: 11F27 , 11F46 , 11F55

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

Rights: Copyright © 2017 Mathematical Society of Japan

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Vol.69 • No. 4 • October, 2017
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