Towards the Finite Slope Part for GLn

Abstract

Abstract Let $L$ be a finite extension of ${\mathbb{Q}}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of ${\operatorname{Gal}}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally ${\mathbb{Q}}_p$-analytic representation of ${\operatorname{GL}}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [6], we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain “ordinary” $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.