##
**A local-global compatibility conjecture in the \(p\)-adic Langlands programme for \(\text{GL}_{2}/\mathbb Q\).**
*(English)*
Zbl 1254.11106

This is a very important paper in which the author proposes a local-global compatibility conjecture for the Langlands correspondence for \(\text{GL}_{2}/\mathbb Q\), in a way that incorporates the \(p\)-adic Langlands correspondence, so as to prove that the proposed local-global compatibility conjecture would imply the Fontaine-Mazur conjecture for two dimensional odd representations of \(G_{\mathbb Q}\). In fact the author proves further that his conjecture would imply an overconvergent modularity conjecture (due to Kisin) for trianguline representations. These conjectures also have have consequences towards the construction of multi-variable \(p\)-adic \(L\)-functions attached to families of modular forms.

The web of conjectures outlined in this paper has been largely proved thanks to the works of Colmez, Kisin and the author himself. More precisely, Colmez and Kisin in [P. Colmez, Astérisque 330, 281–509 (2010; Zbl 1218.11107)] established the \(p\)-adic Langlands correspondence, whereas the author in [M. Emerton, “Local-global compatibility in the \(p\)-adic Langlands programme for \(\text{GL}_{2}/\mathbb Q\)”, preprint available at

http://www.math.uchicago.edu/~emerton/preprints.html]

proved the local-global compatibility conjecture. Note also that Kisin has independently proved most cases of the Fontaine-Mazur conjecture for \(\text{GL}_{2}/\mathbb Q\), using a different method.

The author has given an extremely thorough summary of the contents of the paper under review, which the reviewer includes below:

From the text: “In Section 2 we briefly recall the classical Langlands correspondence for \(\text{GL}_{2}/\mathbb Q\) in both the local and global contexts. Given that our ultimate subject is Conjecture 1.1.1, we put a particular emphasis on the realization of the global correspondence in the cohomology of modular curves.

Sections 3, 4, 5, and 6 are devoted to discussing various aspects of the local \(p\)-adic Langlands conjecture.

In Section 3, after presenting some initial motivation and recalling the notion of admissible unitary Banach space representations of \(\text{GL}_2(\mathbb Q_p)\), we present our conjecture regarding the local \(p\)-adic Langlands correspondence (Conjecture 3.3.1).

In Section 4 we introduce the concept of a refinement of a two dimensional \(G_{\mathbb Q_p}\)-representation, and establish some basic properties on the existence and classification of refinements. (For this we rely heavily on the work of P. Colmez [“Série principale unitaire pour \(\text{GL}_{2}(\mathbb{Q}_p)\) et représentations triangulines de dimension 2”, preprint (2005), available at

http://www.math.jussieu.fr/~colmez/publications.html].)

A fundamental point is that a two-dimensional \(G_{\mathbb Q_p}\)-representation admits a refinement if and only if it is trianguline. Furthermore, as we explain, the language of refinements gives a convenient way to express the classification (due to Colmez) of two-dimensional trianguline representations.

In Section 5 we recall what little is known about the classification of topologically irreducible admissible unitary \(\text{GL}_{\mathbb Q_p}\)-representations. One basic technique for studying such a representation \(B\) is to pass to the subspace \(B_{\text{an}}\) of locally analytic vectors, which is dense in \(B\), and is a strongly admissible locally analytic \(\text{GL}_{2}/\mathbb Q_p\)-representation. Unfortunately, not much is known about the classification of such representations in general. However, one is much closer to having a classification of those \(B\) for which either \(B_{\text{lalg}}\) (the space of locally algebraic vectors in \(B\)) or \(J_P(\mathbb Q_p)(B_{\text{an}})\) (the Jacquet module of \(B_{\text{an}}\) with respect to the Borel subgroup of upper triangular matrices in \(\text{GL}_{2/\mathbb Q_p}\)) is non-zero, as we explain in some detail. The discussion relies heavily on work of P. Colmez [“Une correspondance de Langlands \(p\)-adique pour les représentations semi-stables de dimension 2”, preprint (2004), available at http://www.math.jussieu.fr/~colmez/publications.html and [loc. cit.]] and of the author [“Jacquet modules of locally analytic representations of \(p\)-adic reductive groups. I: Construction and first properties”, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 775–839 (2006; Zbl 1117.22008); “II: The relation to parabolic induction”, preprint available at http://www.math.uchicago.edu/~emerton/preprints.html].

As we remarked above, the local correspondence \(V\mapsto B(V)\) has been defined for most trianguline two dimensional \(G_{\mathbb Q_p}\)-representations \(V\). More precisely, it has been defined for irreducible trianguline \(V\) by C. Breuil [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 4, 559–610 (2004; Zbl 1166.11331)] and P. Colmez [loc. cit.], and for Frobenius semisimple potentially crystalline reducible \(V\) by L. Berger and C. Breuil [“Towards a \(p\)-adic Langlands programme”, preprint available at http://www.math.u-psud.fr/~breuil/publications.html] and C. Breuil and the author [“Représentations \(p\)-adiques ordinaires de \(\text{GL}_2 (\mathbb Q_p)\) et compatibilité local-global”, Astérisque 331, 255–315 (2010; Zbl 1251.11043)]. In Section 6 we recall the definition of the correspondence in these various cases, and also discuss the expected structure of \(B(V)\) in the remaining reducible cases. We discuss the extent to which this explicit correspondence is known to satisfy the conditions of Conjecture 3.3.1. We also state a conjecture on the structure of the space of locally analytic vectors \(B(V)B_{\text{an}}\) in \(B(V)\) (generalizing a conjecture of Breuil in the potentially crystalline case).

With our discussion of the local \(p\)-adic Langlands conjecture completed, in Section 7 we return to the subject of Conjecture 1.1.1. We begin by describing in more detail the completions \(\widehat{H}^{-1}(K^p)_E\) and \(\widehat{H}^1_E\) introduced above. After an aside on various notions and results related to systems of Hecke eigenvalues, we discuss the structure of \((\widehat{H}^1_E)_{\text{lalg}}\), and of the Jacquet module \(J_P(\mathbb Q_p)(\widehat{H}^1(K^p)_{E,\text{an}})\). In particular, we state and sketch the proof of Theorem 7.5.8, which establishes the precise relationship between \(J_P(\mathbb Q_p)(\widehat{H}^1(K^p)_{E,\text{an}})\) and the eigensurface of tame level \(K^p\). After another aside, in which we reformulate some of the results of [M. Kisin, “Overconvergent modular forms and the Fontaine-Mazur conjecture”, Invent. Math. 153, No. 2, 373–454 (2003; Zbl 1045.11029)] in the language of refinements, we turn to our main topic: the local-global compatibility conjecture. We state the conjecture in various forms, and deduce some of its consequences (perhaps the most interesting of which are the two stated in Proposition 1.1.2). Finally, we state and prove some results (summarized in Theorems 1.1.3 and 1.1.4 above) that provide some small amount of evidence for the conjecture. ”

The web of conjectures outlined in this paper has been largely proved thanks to the works of Colmez, Kisin and the author himself. More precisely, Colmez and Kisin in [P. Colmez, Astérisque 330, 281–509 (2010; Zbl 1218.11107)] established the \(p\)-adic Langlands correspondence, whereas the author in [M. Emerton, “Local-global compatibility in the \(p\)-adic Langlands programme for \(\text{GL}_{2}/\mathbb Q\)”, preprint available at

http://www.math.uchicago.edu/~emerton/preprints.html]

proved the local-global compatibility conjecture. Note also that Kisin has independently proved most cases of the Fontaine-Mazur conjecture for \(\text{GL}_{2}/\mathbb Q\), using a different method.

The author has given an extremely thorough summary of the contents of the paper under review, which the reviewer includes below:

From the text: “In Section 2 we briefly recall the classical Langlands correspondence for \(\text{GL}_{2}/\mathbb Q\) in both the local and global contexts. Given that our ultimate subject is Conjecture 1.1.1, we put a particular emphasis on the realization of the global correspondence in the cohomology of modular curves.

Sections 3, 4, 5, and 6 are devoted to discussing various aspects of the local \(p\)-adic Langlands conjecture.

In Section 3, after presenting some initial motivation and recalling the notion of admissible unitary Banach space representations of \(\text{GL}_2(\mathbb Q_p)\), we present our conjecture regarding the local \(p\)-adic Langlands correspondence (Conjecture 3.3.1).

In Section 4 we introduce the concept of a refinement of a two dimensional \(G_{\mathbb Q_p}\)-representation, and establish some basic properties on the existence and classification of refinements. (For this we rely heavily on the work of P. Colmez [“Série principale unitaire pour \(\text{GL}_{2}(\mathbb{Q}_p)\) et représentations triangulines de dimension 2”, preprint (2005), available at

http://www.math.jussieu.fr/~colmez/publications.html].)

A fundamental point is that a two-dimensional \(G_{\mathbb Q_p}\)-representation admits a refinement if and only if it is trianguline. Furthermore, as we explain, the language of refinements gives a convenient way to express the classification (due to Colmez) of two-dimensional trianguline representations.

In Section 5 we recall what little is known about the classification of topologically irreducible admissible unitary \(\text{GL}_{\mathbb Q_p}\)-representations. One basic technique for studying such a representation \(B\) is to pass to the subspace \(B_{\text{an}}\) of locally analytic vectors, which is dense in \(B\), and is a strongly admissible locally analytic \(\text{GL}_{2}/\mathbb Q_p\)-representation. Unfortunately, not much is known about the classification of such representations in general. However, one is much closer to having a classification of those \(B\) for which either \(B_{\text{lalg}}\) (the space of locally algebraic vectors in \(B\)) or \(J_P(\mathbb Q_p)(B_{\text{an}})\) (the Jacquet module of \(B_{\text{an}}\) with respect to the Borel subgroup of upper triangular matrices in \(\text{GL}_{2/\mathbb Q_p}\)) is non-zero, as we explain in some detail. The discussion relies heavily on work of P. Colmez [“Une correspondance de Langlands \(p\)-adique pour les représentations semi-stables de dimension 2”, preprint (2004), available at http://www.math.jussieu.fr/~colmez/publications.html and [loc. cit.]] and of the author [“Jacquet modules of locally analytic representations of \(p\)-adic reductive groups. I: Construction and first properties”, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 775–839 (2006; Zbl 1117.22008); “II: The relation to parabolic induction”, preprint available at http://www.math.uchicago.edu/~emerton/preprints.html].

As we remarked above, the local correspondence \(V\mapsto B(V)\) has been defined for most trianguline two dimensional \(G_{\mathbb Q_p}\)-representations \(V\). More precisely, it has been defined for irreducible trianguline \(V\) by C. Breuil [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 4, 559–610 (2004; Zbl 1166.11331)] and P. Colmez [loc. cit.], and for Frobenius semisimple potentially crystalline reducible \(V\) by L. Berger and C. Breuil [“Towards a \(p\)-adic Langlands programme”, preprint available at http://www.math.u-psud.fr/~breuil/publications.html] and C. Breuil and the author [“Représentations \(p\)-adiques ordinaires de \(\text{GL}_2 (\mathbb Q_p)\) et compatibilité local-global”, Astérisque 331, 255–315 (2010; Zbl 1251.11043)]. In Section 6 we recall the definition of the correspondence in these various cases, and also discuss the expected structure of \(B(V)\) in the remaining reducible cases. We discuss the extent to which this explicit correspondence is known to satisfy the conditions of Conjecture 3.3.1. We also state a conjecture on the structure of the space of locally analytic vectors \(B(V)B_{\text{an}}\) in \(B(V)\) (generalizing a conjecture of Breuil in the potentially crystalline case).

With our discussion of the local \(p\)-adic Langlands conjecture completed, in Section 7 we return to the subject of Conjecture 1.1.1. We begin by describing in more detail the completions \(\widehat{H}^{-1}(K^p)_E\) and \(\widehat{H}^1_E\) introduced above. After an aside on various notions and results related to systems of Hecke eigenvalues, we discuss the structure of \((\widehat{H}^1_E)_{\text{lalg}}\), and of the Jacquet module \(J_P(\mathbb Q_p)(\widehat{H}^1(K^p)_{E,\text{an}})\). In particular, we state and sketch the proof of Theorem 7.5.8, which establishes the precise relationship between \(J_P(\mathbb Q_p)(\widehat{H}^1(K^p)_{E,\text{an}})\) and the eigensurface of tame level \(K^p\). After another aside, in which we reformulate some of the results of [M. Kisin, “Overconvergent modular forms and the Fontaine-Mazur conjecture”, Invent. Math. 153, No. 2, 373–454 (2003; Zbl 1045.11029)] in the language of refinements, we turn to our main topic: the local-global compatibility conjecture. We state the conjecture in various forms, and deduce some of its consequences (perhaps the most interesting of which are the two stated in Proposition 1.1.2). Finally, we state and prove some results (summarized in Theorems 1.1.3 and 1.1.4 above) that provide some small amount of evidence for the conjecture. ”

Reviewer: Kâzım Büyükboduk (Istanbul)

### MSC:

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F80 | Galois representations |