$p$-adic local Langlands correspondence and weight shifting (Q1731782)

Summary: If \(f\) is a modular form of weight \(\geq 2\), the representation of \(\mathrm {GL}_2(\mathbb Q_p)\) attached to it by the \(p\)-adic local Langlands correspondence encodes the representation attached by the classical correspondence. We introduce weight shifting techniques that allow us to extend the result to weight 1. We also give a full description of the Jordan-Hölder components of the locally analytic representations appearing in the correspondence.