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; $$E(z,1/2+it)\ll_\varepsilon y^{1/2} (|t|+1)^{1/3+\varepsilon},\quad y\gg 1,$$ extending work of Blomer and Titchmarsh. Finally we propose a uniform version of the sup norm conjecture for Eisenstein series.Comment: 21 pages, with an improved result ($y^{5/6}$ reduced to $y^{1/2}$ in the uniform sup norm bound) and furthermore the present version avoids the use of Duke's equidistribution of Heegner points making the paper more self-containe