Journal of the Mathematical Society of Japan

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

Hiro-aki NARITA

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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.

Article information

J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1443-1474.

First available in Project Euclid: 25 October 2017

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Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F55: Other groups and their modular and automorphic forms (several variables)

Jacquet–Langlands correspondence spinor $L$-functions theta lifts


NARITA, Hiro-aki. Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence. J. Math. Soc. Japan 69 (2017), no. 4, 1443--1474. doi:10.2969/jmsj/06941443.

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