Pointwise bounds for Eisenstein series on $\Gamma_0(q)\setminus SL_2(\mathbb{R})$

TL;DR

This paper establishes pointwise bounds for Eisenstein series on modular curves with squarefree level, using Sobolev techniques to relate bounds to level, weight, and spectral parameters.

Contribution

It introduces a Sobolev-based method to derive explicit pointwise bounds for Eisenstein series on $ ext{X}_0(q)$ with squarefree level, incorporating weight and spectral parameters.

Findings

Bound depends polynomially on level $q$ with $q^{ ext{epsilon}}$ factor.

Bounds involve the weight parameter $n$ and spectral parameter $t$ with $|n|^{1/2 + ext{epsilon}}$ and $|t|^{1/2 + ext{epsilon}}$.

Result applies uniformly across the modular curve with explicit dependence on the Iwasawa $y$-coordinate.

Abstract

We construct pointwise bounds in the weight aspect for Eisenstein series on $X_0(q) = \Gamma_0(q)\setminus SL_2(\mathbb{R})$, 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 $x$.