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January, 2021 Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms II: An explicit formula for Bessel periods and the non-vanishing of theta lifts
Hiro-aki NARITA
J. Math. Soc. Japan 73(1): 125-159 (January, 2021). DOI: 10.2969/jmsj/81168116

Abstract

This paper is a continuation of the first paper. The aim of this second paper is to discuss the non-vanishing of the theta lifts to the indefinite symplectic group $GSp(1,1)$, which have been shown to be involved in the Jacquet–Langlands–Shimizu correspondence with some theta lifts to the $\mathbb{Q}$-split symplectic group $GSp(2)$ of degree two. We study an explicit formula for the square norms of the Bessel periods of the theta lifts to $GSp(1,1)$ in terms of central $L$-values. This study involves two aspects in proving the non-vanishing of the theta lifts. One aspect is to apply the results by Hsieh and Chida–Hsieh on “non-vanishing modulo $p$” of central $L$-values for some Rankin $L$-functions. The other is to relate such non-vanishing with studies on some special values of hypergeometric functions. We also take up the theta lifts to the compact inner form $GSp^*(2)$. We provide examples of the non-vanishing theta lifts to $GSp^*(2)$, which are essentially due to Ibukiyama and Ihara.

Funding Statement

This work was partly supported by Grant-in-Aid for Scientific Research (C) 24540025, Japan Society for the Promotion of Science.

Citation

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Hiro-aki NARITA. "Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms II: An explicit formula for Bessel periods and the non-vanishing of theta lifts." J. Math. Soc. Japan 73 (1) 125 - 159, January, 2021. https://doi.org/10.2969/jmsj/81168116

Information

Received: 26 August 2018; Revised: 5 September 2019; Published: January, 2021
First available in Project Euclid: 12 June 2020

Digital Object Identifier: 10.2969/jmsj/81168116

Subjects:
Primary: 11F67
Secondary: 11F27, 11F30, 11F55

Rights: Copyright © 2021 Mathematical Society of Japan

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Vol.73 • No. 1 • January, 2021
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