Hybrid subconvexity for class group $L$-functions and uniform sup norm bounds of Eisenstein series

TL;DR

This paper establishes a hybrid subconvexity bound for class group L-functions linked to quadratic fields, utilizing Eisenstein series evaluations at Heegner points, and introduces a new uniform sup norm bound for Eisenstein series.

Contribution

It provides a novel hybrid subconvexity bound for class group L-functions and extends the uniform sup norm bounds for Eisenstein series, advancing understanding of their growth and behavior.

Findings

Proved a hybrid subconvexity bound for class group L-functions.

Established a new uniform sup norm bound for Eisenstein series.

Proposed a uniform version of the Eisenstein series sup norm conjecture.

Abstract

In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the following uniform sup norm bound for Eisenstein series $E(z,1/2+it)\ll_\varepsilon y^{1/2} (|t|+1)^{1/3+\varepsilon}, y\gg 1$, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.