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

In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K\mathrm{/}\mathbb{Q}$ (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.