Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

Abstract

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.