Ordinary \(p\)-adic representations of \(\text{GL}_2 (\mathbb Q_p)\) and local-global compatibility. (Représentations \(p\)-adiques ordinaires de \(\text{GL}_2 (\mathbb Q_p)\) et compatibilité local-global.) (French. English summary) Zbl 1251.11043

Breuil, Christophe; Emerton, Matthew

Berger, Laurent (ed.) et al., Représentations \(p\)-adiques de groupes \(p\)-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France (ISBN 978-2-85629-282-2/pbk). Astérisque 331, 255-315 (2010).

Let \(f= q+\sum_{n\geq 2} a_n(f)q^n\) be a newform of weight \(k\geq 2\), of level \(N\), of character \(\chi\), an eigenvector of the Hecke operators \(T_\ell\) for \(\ell\) prime to \(N\) and of \(U_\ell\) for \(\ell\) dividing \(N\).
Let \(\sigma(f)\) denote the \(p\)-adic representation of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) attached to \(f\), and let \(\sigma_p(f)\) be its restriction to a decomposition subgroup in \(p\).
Let \(L\) be a finite extension of \(\mathbb{Q}_p\) on which \(f\) is defined, let \(M\) denote the part of \(N\) which is prime to \(p\). Let \(Y_1(M; p^r)\) be the open modular curve associated to the congruence group \(\Gamma_1(M)\cap\Gamma(p^r)\). Let \(\widehat H^1(K^p_1(M))_L\) denote the tensor product (over \(\mathbb{Z}_p\)) of \(L\) by the \(p\)-adic completion of the \(\mathbb{Z}_p\)-module \(\varinjlim_r H^1_{\text{ét}}(Y_1(M; p^r)_{\overline{\mathbb{Q}}}, \mathbb{Z}_p)\). Consider the \(\sigma(f)\)-isotypic component \(\Pi_p(f):= \operatorname{Hom}_{\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}(\sigma(f),\widehat H^1(K^p_1(M)))_L\). It is a \(p\)-adic Banach space which is canonically equipped with a unitary continuous action of the group \(\text{GL}_2(\mathbb{Q}_p)\). It is expected that “to contain” the \(p\)-adic Hodge theory of the form \(f\).
Assume now that \(\sigma_p(f)\) is a reducible potentially crystalline representation of \(\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\) of Hodge-Tate weight \((1- k, 0)\) for some integer \(k\geq 2\). The authors attach to \(\sigma_p(f)\) a certain \(p\)-adic Banach space \(B(\sigma_p(f))\), and state the following conjecture:
There is a \(\text{GL}_2(\mathbb{Q}_p)\)-equivariant topological isomorphism \(B(\sigma_p(f))\simeq\Pi_p(f)\) of \(p\)-adic Banach spaces.
The main result of the article (Theorem 1.1.2) is a weaker version of the above conjecture, which asserts, in particular, that there is a \(\text{GL}_2(\mathbb{Q}_p)\)-equivariant closed immersion \(B(\sigma_p(f))\hookrightarrow\Pi_p(f)\) of \(p\)-adic Banach spaces.
For the entire collection see [Zbl 1192.11002].

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

11F80	Galois representations
22E50	Representations of Lie and linear algebraic groups over local fields
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

Galois representation; \(p\)-adic Langlands correspondence; étale cohomology; modular form; modular curve; \(p\)-adic Banach space; Jacquet functor