Extensions for supersingular representations of $GL_2(Q_p)$

By Vytautas Paškūnas. 2007

51 citations recorded by the source corpus.

Let $p>2$ be a prime number. Let $G:=GL_2(Q_p)$ and $\pi$, $\tau$ smooth irreducible representations of $G$ on $\bar{F}_p$-vector spaces with a central character. We show if $\pi$ is supersingular then $Ext^1_G(\tau,\pi)\neq 0$ implies $\tau\cong \pi$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ext^1_G(\tau,\pi)$, when $\pi$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ext^1_G(\tau,\pi)$ for all irreducible representations of $G$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e2c41ab2ff6895246b7e3d1237ac521a0d9dfe50

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. F. Herzig
2. Vytautas Paškūnas

In Geometry and Topology

1. F. Herzig
2. Vytautas Paškūnas
3. C. Breuil
4. T. Gee
5. M. Emerton
6. Peter Scholze

In Mathematical Physics

1. F. Herzig
2. Vytautas Paškūnas
3. C. Breuil
4. T. Gee
5. M. Emerton
6. Peter Scholze

Citations in the corpus

• On the Breuil–Mézard conjecture (2012, background): “5 (ii) in [51], or alternatively see [18, §VII.”
• The image of Colmez’s Montreal functor (2010, methodology): “We also note that one may interpret Ẽ as the ring representing the categorical deformation problem of Q, so that it parameterizes all the deformations of V̌(τ 2 ) which lie in the category RepBGQp (O), as is done in the appendix in [54].”
• ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS II. DERIVED FUNCTORS (2010, unknown): “Furthermore, in [15] Paškūnas completes the computation of the spaces Ext(1)G(V1, V2) for absolutely irreducible V1 and V2 (at least for p > 2), by treating the cases when V1 is supersingular and V2 is supersingular or a twist of the Steinberg (cases which are not treated here, nor in [4] or [6].”
• The classification of irreducible admissible mod p representations of a p-adic GLn (2010, methodology): “(For GL2(Qp) extensions between irreducibles were computed in [Col10], [Eme10b], [BP], [Paš10].”