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

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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.