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

摘要

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.