Exceptional theta correspondence $\mathbf{F}_{4}\times\mathbf{PGL}_{2}$ for level one automorphic representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910807280844800 |
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| author | Shan, Yi |
| author_facet | Shan, Yi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exceptional theta correspondence $\mathbf{F}_{4}\times\mathbf{PGL}_{2}$ for level one automorphic representations Shan, Yi Number Theory Representation Theory 11F, 11F27, 11F55, 11F67, 11F70, 20G41 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. |
| title | Exceptional theta correspondence $\mathbf{F}_{4}\times\mathbf{PGL}_{2}$ for level one automorphic representations |
| topic | Number Theory Representation Theory 11F, 11F27, 11F55, 11F67, 11F70, 20G41 |
| url | https://arxiv.org/abs/2501.19101 |