Level one automorphic representations of an anisotropic exceptional group over $\mathbb{Q}$ of type $\mathrm{F}_{4}$

Abstract

Up to isomorphism, there is a unique connected semisimple algebraic group over $\mathbb{Q}$ of Lie type $\mathrm{F}_{4}$, with compact real points and split over $\mathbb{Q}_{p}$ for all primes $p$. Let $\mathbf{F}_{4}$ be such a group. In this paper, we study the level one automorphic representations of $\mathbf{F}_{4}$ in the spirit of the work of Chenevier, Renard, and Ta\"ibi. First, we give an explicit formula for the number of these representations having any given archimedean component. For this, we study the automorphism group of the two definite exceptional Jordan algebras of rank $27$ over $\mathbb{Z}$ studied by Gross, as well as the dimension of the invariants of these groups in all irreducible representations of $\mathbf{F}_{4}(\mathbb{R})$. Then, assuming standard conjectures by Arthur and Langlands for $\mathbf{F}_{4}$, we refine this counting by studying the contribution of the representations whose global Arthur parameter has any possible image (or "Sato-Tate group"). This includes a detailed description of all those images, as well as precise statements for the Arthur's multiplicity formula in each case. As a consequence, we obtain a conjectural but explicit formula for the number of algebraic, cuspidal, level one automorphic representation of $\mathrm{GL}_{26}$ over $\mathbb{Q}$ with Sato-Tate group $\mathbf{F}_{4}(\mathbb{R})$ of any given weight (assumed "$\mathrm{F}_{4}$-regular"). The first example of such representations occurs in motivic weight $36$.