The subconvexity problem for $\GL_{2}$

TL;DR

This paper establishes a unified approach to the subconvexity problem for L-functions of automorphic representations on GL(1) and GL(2) over a fixed number field, using normalized period relations.

Contribution

It introduces a novel, soft method leveraging canonical period relations to solve the subconvexity problem uniformly across all aspects.

Findings

Unified solution for GL(1) and GL(2) subconvexity

Method based on normalized period relations

Results hold over fixed number fields

Abstract

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of $\GL_{1}$ and $\GL_{2}$ 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.