Analytic properties of spherical cusp forms on GL(n)
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866910698217406464 |
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| author | Blomer, Valentin Harcos, Gergely Maga, Péter |
| author_facet | Blomer, Valentin Harcos, Gergely Maga, Péter |
| contents | Let $ϕ$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $λ_ϕ$. We establish explicit estimates for various quantities related to $ϕ$ that are uniform in $λ_ϕ$. This includes uniforms bounds for spherical Whittaker functions on $\mathrm{GL}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $ϕ$, and uniform bounds for the "essential support" of $ϕ$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_05065 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Analytic properties of spherical cusp forms on GL(n) Blomer, Valentin Harcos, Gergely Maga, Péter Number Theory Classical Analysis and ODEs Spectral Theory Primary 11F72, 11F55, Secondary 11H06, 33E30, 43A85 Let $ϕ$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $λ_ϕ$. We establish explicit estimates for various quantities related to $ϕ$ that are uniform in $λ_ϕ$. This includes uniforms bounds for spherical Whittaker functions on $\mathrm{GL}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $ϕ$, and uniform bounds for the "essential support" of $ϕ$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools. |
| title | Analytic properties of spherical cusp forms on GL(n) |
| topic | Number Theory Classical Analysis and ODEs Spectral Theory Primary 11F72, 11F55, Secondary 11H06, 33E30, 43A85 |
| url | https://arxiv.org/abs/1712.05065 |