Sup-norm bounds for Eisenstein series

TL;DR

This paper establishes bounds for Eisenstein series on congruence quotients, improving the analysis by optimizing the amplifier's structure to better control spectral and level parameters.

Contribution

It introduces a novel approach by supporting the amplifier on primes, combining analytic and geometric methods for sharper bounds.

Findings

Derived new sup-norm bounds for Eisenstein series.

Enhanced the amplifier technique by focusing on primes.

Achieved more efficient counting estimates.

Abstract

The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos--Templier did to obtain a more efficient treatment for the counting problem.