Sup norms of newforms on $GL_2$ with highly ramified central character

Abstract

Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $ϕ$ 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 $χ$ 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 $$\|ϕ\|_\infty \ll N^{\frac14+ε},$$ at least under the Ramanujan Conjecture. In particular, when $χ$ has conductor $N$, this improves upon the previous best known bound $\|ϕ\|_\infty \ll N^{\frac12+ε}$ 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.