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

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.