Theta functions, fourth moments of eigenforms, and the sup-norm problem II
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911897211633664 |
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| author | Khayutin, Ilya Nelson, Paul D. Steiner, Raphael S. |
| author_facet | Khayutin, Ilya Nelson, Paul D. Steiner, Raphael S. |
| contents | For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate
\[
\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_ε (k V)^{\frac{1}{4}+ε}
\] with absolute implied constant. For a cuspidal Maaß newform $φ$ of eigenvalue $λ$ on such a surface, we prove that
\[
\|φ\|_{\infty} \ll_{λ,ε} V^{\frac{1}{4}+ε}.
\]
We establish analogous estimates in the setting of definite quaternion algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12351 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Theta functions, fourth moments of eigenforms, and the sup-norm problem II Khayutin, Ilya Nelson, Paul D. Steiner, Raphael S. Number Theory 11F12 (11F27, 11F70, 11F72, 11D45, 11N75, 14G35) For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate \[ \| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_ε (k V)^{\frac{1}{4}+ε} \] with absolute implied constant. For a cuspidal Maaß newform $φ$ of eigenvalue $λ$ on such a surface, we prove that \[ \|φ\|_{\infty} \ll_{λ,ε} V^{\frac{1}{4}+ε}. \] We establish analogous estimates in the setting of definite quaternion algebras. |
| title | Theta functions, fourth moments of eigenforms, and the sup-norm problem II |
| topic | Number Theory 11F12 (11F27, 11F70, 11F72, 11D45, 11N75, 14G35) |
| url | https://arxiv.org/abs/2207.12351 |