Global $L$-packets for GSp(2) and theta lifts

Abstract

Let $F$ be a totally real number field. We define global $L$-packets for $\mathrm{GSp}(2)$ over $F$ which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of $F$ to the $L$-group of $\mathrm{GSp}(2)$ 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 $\mathrm{GSp}(2)$ over $F$ as predicted by Arthur's conjecture. This can be regarded as the $\mathrm{GSp}(2)$ 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 $\mathrm{GSp}(2)$ over $F$.