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

In this paper, we study hybrid subconvexity bounds for class group L-functions associated to quadratic extensions K/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 uniform sup norm bound for Eisenstein series E(z, 1/2 + it) ≪_ε y¹/²(|t| + 1)¹/3+^ε, y ≫ 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.