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].

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].

Reviewer: Anne-Marie Aubert (Paris)

##### 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 |