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

Authors
  • Comtat, Félicien1
  • 1 Queen Mary University of London, United Kingdom , (United Kingdom)
Type
Published Article
Journal
Forum Mathematicum
Publisher
De Gruyter
Publication Date
Sep 01, 2020
Volume
33
Issue
1
Pages
1–16
Identifiers
DOI: 10.1515/forum-2020-0080
Source
De Gruyter
Keywords
License
Yellow

Abstract

Recently, the problem of bounding the sup norms of L2{L^{2}}-normalized cuspidal automorphic newforms ϕ on GL2{\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∥∞≪N14+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∥∞≪N12+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.

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