Global L - Packets for GSp ( 2 ) and Theta Lifts

Let F be a totally real number field. We define global L-packets for GSp(2) over F which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of F to the L-group of GSp(2) which are reducible, or irreducible and induced from a totally real quadratic extension of F . We prove that the elements of these global L-packets occur in the space of cusp forms on GSp(2) over F as predicted by Arthur’s conjecture. This can be regarded as the GSp(2) analogue of the dihedral case of the Langlands-Tunnell theorem. To obtain these results we prove a nonvanishing theorem for global theta lifts from the similitude group of a general four dimensional quadratic space over F to GSp(2) over F . 2000 Mathematics Subject Classification: Primary 11F70; Secondary 11F27, 11R39.