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On some potentially crystalline representations of $$\text{GL}_2 (\mathbb Q_p)$$. (Sur quelques représentations potentiellement cristallines de $$\text{GL}_2 (\mathbb Q_p)$$.) (French. English summary) Zbl 1243.11063
Berger, Laurent (ed.) et al., Représentations $$p$$-adiques de groupes $$p$$-adiques II: Représentations de $$\text{GL}_2 (\mathbb Q_p)$$ et $$(\varphi, \gamma)$$-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-281-5/pbk). Astérisque 330, 155-211 (2010).
Let $$p$$ be a prime number. Let $$\bar{\mathbb Q}_p$$ denote an algebraic closure of of $$\mathbb Q_p$$. Let $$L$$ be a finite extension of $$\mathbb Q_p$$. Recall that a $$p$$-adic representation of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$ is by definition a finite dimensional $$L$$-vector space $$V$$ equipped with a continuous linear action of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$.

Let $$V$$ be an absolutely irreducible $$p$$-adic representation of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$ of dimension $$2$$, which becomes crystalline over an abelian extension of $$\mathbb Q_p$$. Let $$\text{D}_{\text{cris}}(V)$$ denote the filtered $$\varphi$$-module which was attached by Fontaine to $$V$$. Assume that $$V$$ is $$\varphi$$-semisimple, that is, $$\text{D}_{\text{cris}}(V)$$ is semisimple.

The representation $$V$$ has distinct Hodge-Tate weights, say $$i_1<i_2$$. Let $$\text{Alg}(V)$$ denote the algebraic representation of $$\text{GL}_2(\mathbb Q_p)$$ with highest weight $$(i_1,i_2-1)$$, and let $$\text{Smooth}(V)$$ denote the irreducible smooth representation of $$\text{GL}_2(\mathbb Q_p)$$ associated by the local Hecke correspondence to the Weil representation attached to $$V$$.

Assume here for simplicity that $$\text{Smooth}(V)$$ is not of dimension $$1$$, and define $$\text{B}(V)$$ to be the $$p$$-adic completion of the locally algebraic representation $$\text{Alg}(V)\otimes\text{Smooth}(V)$$ with respect to a $$\text{GL}_2(\mathbb Q_p)$$-stable lattice of finite type under the action of $$\text{GL}_2(\mathbb Q_p)$$. Then $$\text{B}(V)$$ is a $$p$$-adic Banach space equipped with a continuous action of $$\text{GL}_2(\mathbb Q_p)$$, and $V\mapsto \text{B}(V)$ gives a correspondence between (some) $$p$$-adic representations of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$ and (some) $$p$$-adic representations of $$\text{GL}_2(\mathbb Q_p)$$. However it is absolutely not obvious that such a lattice exists, or, equivalently, that $$\text{B}(V)$$ is non-zero.

The author of the article proved in previous works that $$\text{B}(V)$$ is non-zero in the case when $$V$$ is cristalline and $$i_1-i_2<2p$$ (essentially), and also that $$\text{B}(V)$$ is admissible, and (with an additional assumption) that it is topologically irreducible. On the other hand, he has defined $$\text{B}(V)$$ in the case when $$V$$ is absolutely irreducible, semi-stable but not cristalline, and has formulated analoguous conjectures (non-nullity, admissibility, etc.). Next, Colmez has seen that the theory of $$(\varphi,\Gamma)$$-modules due to Fontaine allows to demonstrate these conjectures by building a model of the restriction to the upper Borel subgroup of $$\text{GL}_2(\mathbb Q_p)$$ of the dual representation $$\text{B}(V)*$$ of $$\text{B}(V)$$ by using the $$(\varphi,\Gamma)$$-module of $$V$$.

It shown in the article under reviewing that such a model exists in the case of potentially cristalline representations $$V$$ as above, leading to the following theorem which constitutes the main result: If $$V$$ is an absolutely irreducible $$p$$-adic representation of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$ of dimension $$2$$, which becomes cristalline over an abelian extension of $$\mathbb Q_p$$ and is $$\varphi$$-semsimple, then $$\text{B}(V)$$ is non-zero, topologically irreducible and admissible.
For the entire collection see [Zbl 1192.11001].

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields
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