On \(L^{\infty }\) norms of holomorphic cusp forms (Q880057)
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scientific article; zbMATH DE number 5151587
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On \(L^{\infty }\) norms of holomorphic cusp forms |
scientific article; zbMATH DE number 5151587 |
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On \(L^{\infty }\) norms of holomorphic cusp forms (English)
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10 May 2007
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Let \(f\) be an \(L^2\)-normalized holomorphic cusp form of weight \(2k\) on \(\mathrm{SL}_2(\mathbb Z)\), and assume that \(f\) is a Hecke eigenform. The aim of this short note is to prove the following precise \(L^\infty\)-estimate for \(f\): \[ k^{\frac 14-\varepsilon}\ll\|y^kf(z) \|_\infty\ll k^{\frac 14+\varepsilon}. \] The proof is based on the estimate \[ k^{-\varepsilon}\ll L(1,\text{sym}^2(f))\ll k^\varepsilon \] and on Deligne's bound.
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cusp form
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Hecke eigenform
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Deligne's theorem
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