On the global sup-norm of GL(3) cusp forms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866910698195386368 |
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| author | Blomer, Valentin Harcos, Gergely Maga, Péter |
| author_facet | Blomer, Valentin Harcos, Gergely Maga, Péter |
| contents | Let $ϕ$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $ϕ$ in terms of its Laplace eigenvalue $λ_ϕ$. These imply, for $ϕ$ arithmetically normalized and tempered at the archimedean place, the bound $\|ϕ\|_\infty\ll_ελ_ϕ^{39/40+ε}$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the $\mathrm{GL}_3$ Jacquet-Whittaker function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1706_02771 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the global sup-norm of GL(3) cusp forms Blomer, Valentin Harcos, Gergely Maga, Péter Number Theory Classical Analysis and ODEs Spectral Theory Primary 11F72, 11F55, Secondary 33E30, 43A85 Let $ϕ$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $ϕ$ in terms of its Laplace eigenvalue $λ_ϕ$. These imply, for $ϕ$ arithmetically normalized and tempered at the archimedean place, the bound $\|ϕ\|_\infty\ll_ελ_ϕ^{39/40+ε}$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the $\mathrm{GL}_3$ Jacquet-Whittaker function. |
| title | On the global sup-norm of GL(3) cusp forms |
| topic | Number Theory Classical Analysis and ODEs Spectral Theory Primary 11F72, 11F55, Secondary 33E30, 43A85 |
| url | https://arxiv.org/abs/1706.02771 |