Global long root $A$-packets for ${\mathsf{G}}_2$: the dihedral case

Abstract

Cuspidal automorphic representations $\ddot{I}„$ of ${\mathrm{PGL}}_2$ correspond to global long root $A$-parameters for ${\mathsf{G}}_2$. Using an exceptional theta lift between ${\mathrm{PU}}_3$ and ${\mathsf{G}}_2$, we construct the associated global $A$-packet and prove the Arthur multiplicity formula for these representations when $\ddot{I}„$ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on ${\mathsf{G}}_2$.

Removed restrictions at the archimedean places. 40 pages. Comments welcome!