TY - JOUR

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

T2 - An explicit formula for Bessel periods and the non-vanishing of theta lifts

AU - Narita, Hiro Aki

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary 11F67; Secondary 11F27, 11F30, 11F55. Key Words and Phrases. Bessel periods, central L-values, Jacquet–Langlands correspondence, theta lifts. This work was partly supported by Grant-in-Aid for Scientific Research (C) 24540025, Japan Society for the Promotion of Science.
Publisher Copyright:
© 2021 The Mathematical Society of Japan

PY - 2021

Y1 - 2021

N2 - 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 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.

AB - 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 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.

KW - Bessel periods

KW - Central L-values

KW - Jacquet-Langlands correspondence

KW - Theta lifts

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U2 - 10.2969/JMSJ/81168116

DO - 10.2969/JMSJ/81168116

M3 - Article

AN - SCOPUS:85100965774

SN - 0025-5645

VL - 73

SP - 125

EP - 159

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

IS - 1

ER -