Bounds on sup-norms of half-integral weight modular forms

Eren Mehmet Kıral¹

¹ Brown University 151 Thayer Street Providence, RI 02912, U.S.A.

Acta Arithmetica, Tome 165 (2014) no. 4, pp. 385-399

Résumé

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.

Détail

DOI : 10.4064/aa165-4-7

Keywords: bounding sup norms modular forms terms level has focus much recent study work sup norm half integral weight cusp form bounded terms level prove kappa tilde infty varepsilon kappa varepsilon kappa tilde modular form tilde level weight kappa half integer