Base change for the Saito-Kurokawa representations of PGSp(4) (Q1106878)

The Saito-Kurokawa representation of PGSp(4) are (local and automorphic) representations constructed by `lifting' from representations of PGL(2). They can also be directly characterized. The authors consider the behaviour of these representations under a base change K/k which is cyclic of prime degree q. The representations are naturally described in term of packets, and the number of cuspidal elements in a packet depends on whether the L-function of the original representation of PGL(2) vanishes at the centre of the critical strip. The size of the complete packet is known. The object of this investigation is to use these ideas to give examples for the `Langlands' philosophy'. In this case, when q is odd, the number of cuspidal representations increases, whereas when \(q=2\) it can happen that cuspidal representations transform under base change into non-cuspidal representations.