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

Cuspidal automorphic representations $\tau$ of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ correspond to global long root $A$-parameters for ${\mathsf{G}}_2$. Using an exceptional theta lift between ${\mathrm{P}\mathrm{U}}_3$ and ${\mathsf{G}}_2$, we construct the associated global $A$-packet and prove the Arthur multiplicity formula for these representations when $\tau$ 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 ${\mathsf{G}}_2$.