Bounds on sup-norms of half-integral weight modular forms
Volume 165 / 2014
Acta Arithmetica 165 (2014), 385-399
MSC: Primary 11F03; Secondary 11F37.
DOI: 10.4064/aa165-4-7
Abstract
Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $\|y^{\kappa/2}\tilde{f}\|_\infty \ll_{\varepsilon,\kappa} N^{1/2 - 1/18 +\varepsilon}\|y^{\kappa/2}\tilde{f}\|_{L^2}$ for a modular form $\tilde{f}$ of level $4N$ and weight $\kappa$, a half-integer.
