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

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.