Sup norms of newforms on $G{L}_2$ with highly ramified central character

Abstract

Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $\varphi$ on ${\text{GL}}_2$ in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character $\chi$ of $\varphi$ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general $\chi$. If the level $N$ is a square, our result reduces to $$\Vert \varphi {\Vert }_{\infty }\ll N^{\frac{1}{4}+\epsilon },$$ at least under the Ramanujan Conjecture. In particular, when $\chi$ has conductor $N$, this improves upon the previous best known bound $\Vert \varphi {\Vert }_{\infty }\ll N^{\frac{1}{2}+\epsilon }$ in this setup (due to Saha [14]) and matches a lower bound due to Templier [17], thus our result is essentially optimal in this case.

Published version with updated references