On the automorphic atoms of length 2 of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$

Abstract

Let $p>3$ be a prime. The aim of this paper is to give a description of the invariant space, under principal and Iwahori congruence subgroups of arbitrary level, of extensions of generic principal series representations appearing in the $p$-modular local Langlands correspondence for ${\mathbf{GL}}_2({\mathbf{Q}}_p)$. As an application we describe Hecke isotypical components of the mod $p$ cohomology of the modular curve over $\mathbf{Q}$ with deeply ramified level at $p$.