Finiteness properties of the category of mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$

Abstract

We establish Bernstein-centre type of results for the category of mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. We treat all the remaining open cases, which occur when $p$ is $2$ or $3$. Our arguments carry over for all primes $p$. This allows us to remove the restrictions on the residual representation at $p$ in Lue Pan's recent proof of the Fontaine--Mazur conjecture for Hodge--Tate representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with equal Hodge--Tate weights.

arXiv commentminor changes after a referee report. Final version

References in corpus (6)

• Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings
• Blocks for mod $p$ representations of $G{L}_2(Q_p)$
• Représentations galoisiennes automorphes et conséquences arithmétiques des conjectures de Langlands et Arthur
• Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$
• Infinitesimal characters in arithmetic families
• On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{{\mathbb{Q}}_p}$

Cited by in corpus (1)

• On local Galois deformation rings