Cuspidal automorphic representations $\tau$ 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 $\tau$ 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 $\mathsf{ G}_2$.
Categories: math.NT math.RT
