The subconvexity problem for ${\text{GL}}_2$

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of ${\text{GL}}_1$ and ${\text{GL}}_2$ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.