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.

Keywords and Phrases: $L$-packets, Arthur's conjecture, $\GSp(2)$, theta lifts

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