Exceptional theta correspondence ${\mathbf{F}}_4\times {\mathbf{P}\mathbf{G}\mathbf{L}}_2$ for level one automorphic representations

Let ${\mathbf{F}}_4$ be the unique (up to isomorphism) connected semisimple algebraic group over $\mathbb{Q}$ of type ${\mathrm{F}}_4$, with compact real points and split over ${\mathbb{Q}}_p$ for all primes $p$. A conjectural computation by the author in arxiv:2407.05859 predicts the existence of a family of level one automorphic representations of ${\mathbf{F}}_4$, which are expected to be functorial lifts of cuspidal representations of ${\mathbf{P}\mathbf{G}\mathbf{L}}_2$ associated with Hecke eigenforms. In this paper, we study the exceptional theta correspondence for ${\mathbf{F}}_4\times {\mathbf{P}\mathbf{G}\mathbf{L}}_2$, and establish the existence of such a family of automorphic representations for ${\mathbf{F}}_4$. Motivated by the work of Pollack, our main tool is a notion of “exceptional theta series” on ${\mathbf{P}\mathbf{G}\mathbf{L}}_2$, arising from certain automorphic representations of ${\mathbf{F}}_4$. These theta series are holomorphic modular forms on ${\mathbf{S}\mathbf{L}}_2(\mathbb{Z})$, with explicit Fourier expansions, and span the entire space of level one cusp forms.