Theta series of degree 2 of quaternary quadratic forms (Q916689)

Let D be a definite quaternion algebra over \({\mathbb{Q}}\). Starting from a pair \((\phi_ 1,\phi_ 2)\) of automorphic forms on \(D^ x_{{\mathbb{A}}}/D^ x_{{\mathbb{Q}}}\) one can obtain automorphic forms on \(Sp_ 2({\mathbb{A}})\) via a theta lifting. In classical language this lift was studied by \textit{H. Yoshida} [J. Reine Angew. Math. 352, 184-219 (1984; Zbl 0532.10018)], who showed that (under certain assumptions) this lift lies in the Saito-Kurokawa space, if \(\phi_ 1\) is a constant function. The purpose of the paper under review is to give a representation-theoretic proof of this result using the characterization of the Saito-Kurokawa space given by \textit{I. Piatetski-Shapiro} [Invent. Math. 71, 309-338 (1983; Zbl 0515.10024)]. This approach works for any level (not just square-free levels as in Yoshida's paper). Finally the author indicates applications of this result to theta series of spinor genera.