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

Abstract

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