The authors prove H. Yoshida’s conjecture [Invent. Math. 60, 193- 248 (1980; Zbl 0453.10022), J. Reine Angew. Math. 352, 184-219 (1984; Zbl 0532.10018)] on the nonvanishing of the theta-lift from O(4) to Sp(2). Here O(4) is the orthogonal group attached to a definite quaternion algebra D over \({\mathbb{Q}}\). They can show that this lifting is “almost injective”. More precisely they are able to describe the kernel by a condition on a special value of the standard L-function attached to a form on O(4).
In the first part the pullback-machinery for Eisenstein series is used. The authors give an Euler product expansion for Hecke operators of level N. The residues of the meromorphic continuation of the Eisenstein series can be described by means of theta series. This leads to a criterion when an eigenform turns out to be a linear combination of theta series.
In the second part the authors derive the injectivity properties of the lifting. In particular they show that the Yoshida lift of a pair of new forms on \(D_ A^{\times}\) is an eigenform under certain Hecke operators.