Exceptional theta correspondence $\mathbf{F}_{4}\times\mathbf{PGL}_{2}$ for level one automorphic representations

TL;DR

The paper constructs and analyzes an exceptional theta correspondence between the exceptional group $\mathbf{F}_{4}$ and $\mathbf{PGL}_{2}$, using Albert algebras and the minimal representation of $\mathbf{E}_{7}$ to realize global theta lifts. It introduces exceptional theta series built from $\mathbf{F}_{4}$-automorphic data and their polynomial models, proving that these lifts yield holomorphic level-one cusp forms on $\mathbf{SL}_{2}(\mathbb{Z})$ with explicit Fourier expansions. The authors establish non-vanishing of the global theta lift $\Theta(\pi)$ for level-one cuspidal $\mathbf{PGL}_{2}$ representations associated to Hecke eigenforms, and compute the relevant $L$-factors via a Spin$_9$-period and an exceptional Siegel–Weil formula, confirming local–global compatibility. Consequently, the predicted functorial lifts of cuspidal $\mathbf{PGL}_{2}$ forms to $\mathbf{F}_{4}$ exist, and the weighted exceptional theta series span the full space of level-one cusp forms for appropriate weights. The work extends classical theta-lift paradigms to the exceptional setting, providing explicit modular forms and a new avenue for realizing Langlands-type lifts in the exceptional group context.

Abstract

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 $\mathbf{PGL}_{2}$ associated with Hecke eigenforms. In this paper, we study the exceptional theta correspondence for $\mathbf{F}_{4}\times\mathbf{PGL}_{2}$, 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 $\mathbf{PGL}_{2}$, arising from certain automorphic representations of $\mathbf{F}_{4}$. These theta series are holomorphic modular forms on $\mathbf{SL}_{2}(\mathbb{Z})$, with explicit Fourier expansions, and span the entire space of level one cusp forms.

Theorems & Definitions (108)

• Conjecture 1
• Theorem 2
• Remark 1.2.1
• Theorem 3
• Remark 1.2.2
• Remark 1.3.1
• Theorem 4
• Theorem 5
• Definition 2.1.1
• Proposition 2.1.2