Theta functions, fourth moments of eigenforms, and the sup-norm problem II
| Autor: | Khayutin, Ilya, Nelson, Paul D., Steiner, Raphael S. |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Forum of Mathematics, Pi , Volume 12 , 2024 , e11 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/fmp.2024.9 |
| Popis: | 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 \[ \| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\epsilon} (k V)^{\frac{1}{4}+\epsilon} \] with absolute implied constant. For a cuspidal Maa{\ss} newform $\varphi$ of eigenvalue $\lambda$ on such a surface, we prove that \[ \|\varphi \|_{\infty} \ll_{\lambda,\epsilon} V^{\frac{1}{4}+\epsilon}. \] We establish analogous estimates in the setting of definite quaternion algebras. Comment: 49 pages |
| Databáze: | arXiv |
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