Abstract Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

中文翻译：

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GL2 上新形式的最优 sup 范数边界具有最大分支中心特征

摘要 最近，在水平方面对 GL 2 {\mathrm{GL}_{2}} 上的 L 2 {L^{2}} -归一化尖牙自守新形 ϕ 的 sup 范数进行界定的问题备受关注。然而，目前只有当 ϕ 的中心字符 χ 不是太高分叉时，才可以使用强上限。在本文中，我们为一般 χ 在水平方面建立了统一的上限。如果水平 N 是一个正方形，我们的结果简化为 ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon }，至少在拉马努金猜想下是这样。特别是，当 χ 具有导体 N 时，这改进了之前的最佳已知界限 ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{ 2}+\epsilon}} 在这个设置中（由于 [A. Saha, Hybrid sup-norm bounds for Maass newforms of strong level, Algebra Number Theory 11 2017, 1009–1045]) 并匹配由于 [N. Templier，模块化形式的大值，Camb。J. 数学。2 2014, 1, 91–116]，因此我们的结果在这种情况下基本上是最优的。