Pointwise bounds for eisenstein series on Γ0(q)\SL2(ℝ)

Abstract

Abstract We construct pointwise bounds in the weight aspect for Eisenstein series on $X_0(q) = \Gamma_0(q)\backslash{} SL_2(\mathbb{R})$, with squarefree level q, using a Sobolev technique. More specifically, we show that for an Eisenstein series E on $X_0(q)$ of weight parameter n and type t, one has for all $x\in X_0(q)$: $|E(x,1/2 + it)| \ll_{\epsilon} q^{\epsilon}(1 + |n|^{1/2 + \epsilon} + |t|^{1/2 + \epsilon})\sqrt{y(x) + y(x)^{-1}}$, where y(x) is the Iwasawa y-coordinate of the point x.