The Saito - Kurokawa Lifting

Certain nontempered liftings from PGL (2) × PGL (2) to PGSp (4) are constructed using the theory of (local and global) theta lifts. The resulting representations on PGSp (4) are the SaitoKurokawa representations. The lifting is shown to be functorial under certain reasonable assumptions on the local Langlands correspondence for PGSp (4). Introduction. The classical Saito-Kurokawa lifting associates to each eigenform f ∈ S2k−2( SL (2,Z)) with even k a cuspidal Siegel eigenform F of degree 2 and weight k such that the (finite parts of the) L-functions of f and F are related by the formula L(s, F) = ζ(s − k + 1)ζ(s − k + 2)L(s, f ) (see [5], §6). Within the framework of functoriality of automorphic representations, the Saito-Kurokawa lifting can be explained as follows (see [17], §3). Let A be the ring of adeles of Q. Let π1 be the automorphic representation of PGL (2,A) corresponding to the eigenform f . Let π2 be the anomalous automorphic representation of PGL (2,A) whose archimedean component is the lowest discrete series representation, and each of whose non-archimedean components is the trivial representation. We consider the (conjectural) lifting of PGL (2,A) × PGL (2,A) to PGSp (4,A) coming from the standard embedding of L-groups SL (2,C) × SL (2,C) −→ Sp (4,C). (1) The image of the automorphic representation π1 ⊗ π2 under this lifting turns out to be a (holomorphic) cusp form Π on PGSp (4,A) that corresponds to the Saito-Kurokawa lift F of f . The main purpose of this paper is to prove the following generalization of the Saito-Kurokawa lifting. Let F be any number field and A its ring of adeles. Let π = ⊗πv be a cuspidal automorphic representation of PGL (2,A) and Σ the set of places v of F such that πv is square integrable. In generalization of the above representation π2 we shall define a global representation πS of PGL (2,A) for any finite set of places S. Our basic lifting theorem (Theorem 3.1) states that if S ⊂ Σ Manuscript received April 30, 2003. American Journal of Mathematics 127 (2005), 209–240.