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

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 $E(z,1/2+it){\ll }_{\epsilon }{y}^{1/2}(|t|+1{)}^{1/3+\epsilon },y\gg 1$, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

22 pages, minor changes according to the suggestions of the referee, incorporated the results of Wu and Andersen arXiv:1812.04391 (accepted for publication in Forum Mathematicum)