On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms

In this paper we study the theta lifting of a weight 2 Bianchi modular form \({\mathcal {F}}\) of level \(\Gamma _0({\mathfrak {n}})\) with \({\mathfrak {n}}\) square-free to a weight 2 holomorphic Siegel modular form. Motivated by Prasanna’s work for the Shintani lifting, we define the local Schwartz function at finite places using a quadratic Hecke character \(\chi \) of square-free conductor \({\mathfrak {f}}\) coprime to level \({\mathfrak {n}}\). Then, at certain 2 by 2 g matrices \(\beta \) related to \({\mathfrak {f}}\), we can express the Fourier coefficient of this theta lifting as a multiple of \(L({\mathcal {F}},\chi ,1)\) by a non-zero constant. If the twisted L-value is known to be non-vanishing, we can deduce the non-vanishing of our theta lifting.