Effective sup-norm bounds on average for cusp forms of even weight

2018/01/17 by Friedman, Joshua S., Jorgenson, Jay, Kramer, Jürg
#11F12 #11F72 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1801.05740

Abstract

Let Γ\subsetPSL₂(ℝ) be a Fuchsian subgroup of the first kind acting on the upper half-plane ℍ. Consider the d_2k-dimensional space of cusp forms S_2k^Γ of weight 2k for Γ, and let \f₁,…,f_d_2k\ be an orthonormal basis of S_2k^Γ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity S_2k^Γ(z):=∑_j=1^d_2k\vert f_j(z)\vert² Im(z)^2k as z ranges through ℍ.

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