Completed Cohomology and Local-Global Compatibility

January 23, 2022May 24, 2022 / Anton Hilado / 1 Comment

In Completed Cohomology, we mentioned that the p-adic local Langlands correspondence may be found inside the completed cohomology, and that this is used in the proof of the Fontaine-Mazur conjecture. In this post, we elaborate on these ideas. We shall be closely following the Séminaire Bourbaki article Correspondance de Langlands p-adique, compatibilité local-global et applications by Christophe Breuil.

Let us make the previous statement more precise. Let $E$ be a finite extension of $\mathbb{Q}_{p}$ , with ring of integers $\mathcal{O}_{E}$ , uniformizer $\varpi$ , and residue field $k_{E}$ . Let us assume that $\mathcal{O}_{E}$ contains the Hecke eigenvalues of a cuspidal eigenform $f$ of weight $2$ . Consider the etale cohomology $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ of the open modular curve $Y(K)$ (we will define this more precisely later). Then we have that $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ contains $\rho_{f}\otimes_{E}\otimes_{\ell}\pi_{\ell}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})})$ , where $\rho_{f}$ is the p-adic Galois representation associated to $f$ (see also Galois Representations Coming From Weight 2 Eigenforms), and $\pi_{\ell}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})})$ is the smooth representation of $\mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ associated to $\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})}$ by the local Langlands correspondence (see also The Local Langlands Correspondence for General Linear Groups).

For $\ell\neq p$ , if we are given $\pi_{\ell}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})})$ , then we can recover $\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})}$ . Therefore the local Langlands correspondence, at least for $\ell\neq p$ , can be found inside $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ . This is what is known as local-global compatibility.

If $\ell=p$ , however, it is no longer true that we can recover $\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})}$ from $\pi_{\ell}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})})$ . Instead, the “classical” local Langlands correspondence needs to be replaced by the p-adic local Langlands correspondence (which at the moment is only known for the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ ). The p-adic local Langlands correspondence associates to a p-adic local Galois representation $\rho_{p}:\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\to\mathrm{GL}_{2}(E)$ a p-adic Banach space $B(\rho_{p})$ over $E$ equipped with a unitary action of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ . The p-adic local Langlands correspondence is expected to be “compatible” with the classical local Langlands correspondence, in that, if the Galois representation is potentially semistable with distinct Hodge-Tate weights the $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ representation provided by the classical local Langlands correspondence (tensored with an algebraic representation that depends on the Hodge-Tate weights) shows up as the “locally algebraic vectors” of the p-adic Banach space provided by the p-adic local Langlands correspondence (we shall make this more precise later).

In the case of the p-adic local Langlands correspondence we actually have a functor that goes the other way, i.e. from p-adic Banach spaces with a unitary action of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ to Galois representations $\rho_{p}:\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\to\mathrm{GL}_{2}(E)$ . We denote this functor by $V$ (it is also known as Colmez’s Montreal functor). In fact the Montreal functor $V$ not only works for representations over $E$ , but also representations over $k_{E}$ (hence realizing one direction of the mod p local Langlands correspondence, see also The mod p local Langlands correspondence for GL_2(Q_p)) and more generally over $\mathcal{O}_{E}/\varpi^{n}$ . The Montreal functor hence offers a solution to our problem of the classical local Langlands correspondence being unable to recover back the Galois representation from the $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ -representation.

Therefore, we want a form of local-global compatibility that takes into account the p-adic local Langlands correspondence. In the rest of this post, if we simply say “local-global compatibility” this is what we refer to. We will use “classical” local-global compatibility to refer to the version that only involves the classical local Langlands correspondence instead of the p-adic local Langlands correspondence.

A review of completed cohomology and the statement of local-global compatibility

As may be hinted at by the title of this post and the opening paragraph, the key to finding this local-global compatibility is completed cohomology. Let us review the relevant definitions (we work in more generality than we did in Completed Cohomology). Let $\mathbb{A}_{f}$ be the finite adeles of $\mathbb{Q}$ . For any compact subgroup $K$ of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ we let

$\displaystyle Y(K)=\mathrm{GL}_{2}(\mathbb{Q})\backslash(\mathbb{C}-\mathbb{R})\times\mathrm{GL}_{2}(\mathbb{A}_{f})/K$ .

Next let $K^{p}$ be a compact open subgroup of $\mathbb{GL}_{2}(\mathbb{A}_{f}^{p})$ (here the superscript ${}^{p}$ means we omit the factor indexed by $p$ in the restricted product) and let $K_{p}$ be a compact open subgroup of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ . We define

$\displaystyle \widehat{H}^{1}(K^{p})_{\mathcal{O}_{E}}:=\varprojlim_{n}\varinjlim_{K_{p}}H_{\mathrm{et}}^{1}(Y(K^{p}K_{p})\otimes_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E}/\varpi_{E}^{n}\mathcal{O}_{E})$ .

We let $\widehat{H}^{1}(K^{p})_{E}=\widehat{H}^{1}(K^{p})_{\mathcal{O}_{E}}\otimes_{\mathcal{O}_{E}}E$ . This is a p-adic Banach space, with unit ball given by $\widehat{H}^{1}(K^{p})_{\mathcal{O}_{E}}$ . It has a continuous action of $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\times\mathrm{GL}_{2}(\mathbb{Q}_{p})$ which preserves the unit ball. We also let $\widehat{H}_{\mathcal{O}_{E}}^{1}=\varinjlim_{K^{p}}\widehat{H}^{1}(K^{p})_{\mathcal{O}_{E}}$ and $\widehat{H}_{E}^{1}=\varinjlim_{K^{p}}\widehat{H}^{1}(K^{p})_{E}$ . We refer to any of these as the completed cohomology. The appearance of Banach spaces should clue us in that this is precisely what we need to formulate a local-global compatibility that includes the p-adic local Langlands correspondence, since the representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ that shows up there is also a Banach space.

Let $\widehat{H}_{E,\Sigma}^{1}=(\widehat{H}^{1}_{E})^{\prod_{\ell\neq p}\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}$ . We define $(\widehat{H}_{E,\Sigma}^{1})^{\mathrm{alg}}$ to be the subspace of $(\widehat{H}_{E,\Sigma}^{1})$ consisting of vectors $v$ for which there exists a compact open subgroup $K_{p}$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ such that the representation of $K_{p}$ generated by $v$ in $(\widehat{H}_{E,\Sigma}^{1})$ restricted to $K_{p}$ is the direct sum of algebraic representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ restricted to $K_{p}$ .

We will work in a more general setting than just weight $2$ cuspidal eigenforms (whose associated Galois representations can be found in $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ , as discussed earlier). Therefore, in order to take account cuspidal eigenforms of weight $\geq 2$ , we will replace $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ with $\bigoplus_{k\geq 2,n\in\mathbb{Z}} \varinjlim_{K}H_{\mathrm{et}}^{1}(Y(K),\mathcal{F}_{k-2})$ , where $\mathcal{F}_{k-2}$ is the sheaf on the etale site of $Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}}$ that corresponds to the local system on $Y(K)(\mathbb{C})$ given by

$\displaystyle \mathrm{GL}_{2}(\mathbb{Q})\backslash((\mathbb{C}-\mathbb{R})\times\mathrm{GL}_{2}(\mathbb{A}_{f})\times \mathrm{Sym}^{k-2}E^{2})/K)$

Now $\bigoplus_{k\geq 2,n\in\mathbb{Z}} \varinjlim_{K}H_{\mathrm{et}}^{1}(Y(K),\mathcal{F}_{k-2})$ , from which we can obtain the “classical” local-global compatibility, is related to the completed cohomology (from which we want to obtain the local-global compatibility that involves the p-adic local Langlands correspondence) via the following $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{A}_{f})$ -equivariant isomorphism:

$\displaystyle \bigoplus_{k\geq 2,n\in\mathbb{Z}} \varinjlim_{K}H_{\mathrm{et}}^{1}(Y(K),\mathcal{F}_{k-2})\otimes_{E}(\mathrm{Sym}^{k-2}E^{2})^{\vee}\otimes_{E}\varepsilon^{n}\cong(\widehat{H}_{E}^{1})^{\mathrm{alg}}$

where $\varepsilon^{n}$ really is shorthand for the character $\varepsilon^{n}\otimes \varepsilon^{n}\circ\mathrm{det}$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{A}_{f})$ , and in this last expression $\varepsilon$ is the p-adic cyclotomic character.

By taking invariants under the action of $\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell})$ , we also have the following $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\prod_{\ell\in\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell})\times\mathbb{T}_{\Sigma}$ -equivariant isomorphism:

$\displaystyle \bigoplus_{k\geq 2,n\in\mathbb{Z}} \varinjlim_{K_{\Sigma}}H_{\mathrm{et}}^{1}(Y(K_{\Sigma}\prod_{\ell\neq \Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell}))\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{F}_{k-2})\otimes_{E}(\mathrm{Sym}^{k-2}E^{2})^{\vee}\otimes_{E}\varepsilon^{n}\cong(\widehat{H}_{E,\Sigma}^{1})^{\mathrm{alg}}$

Before we give the statement of local-global compatibility let us make one more definition. We first need to revisit the Hecke algebra. Let $K$ be a compact open subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ . We define $\mathbb{T}(K)$ to be the $\mathcal{O}_{E}$ -algebra of $\mathrm{End}_{\mathcal{O}_{E}[\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})]}(H_{\mathrm{et}}^{1}(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E}))$ generated by $T_{\ell}:=K\begin{pmatrix}\ell & 0\\0 & 1\end{pmatrix}K$ and $S_{\ell}:=K\begin{pmatrix}\ell & 0\\0 & \ell\end{pmatrix}K$ . We define

$\displaystyle \mathbb{T}_{\Sigma}=\varprojlim_{K_{\Sigma}}(K_{\Sigma}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell}))$

Now let $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(E)$ be an absolutely irreducible odd continuous p-adic Galois representation, unramified at all but finitely many places. We say that $\rho$ is promodular if there exists a finite set of places $\Sigma$ , containing $p$ and the places at which $\rho$ is ramified, such that the ideal of $\mathbb{T}_{\Sigma}[1/p]$ generated by $T_{\ell}-\mathrm{trace}(\rho(K_{\Sigma})(\mathrm{Frob}_{\ell}))$ and $S_{\ell}-\ell^{-1}\mathrm{det}(\rho(K_{\Sigma})(\mathrm{Frob}_{\ell})$ is a maximal ideal of $\mathbb{T}_{\Sigma}[1/p]$ .

We may now give the statement of local-global compatibility. We start with the “weak” version of the statement. Let $\rho$ be a $2$ -dimensional odd representation of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ which is unramified at all but a finite set of places. Assume that the residual representation $\overline{\rho}$ is absolutely irreducible, and that its restriction to $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is not isomorphic to a Galois representation of the form $\begin{pmatrix}1&*\\0&\overline{\epsilon}\end{pmatrix}$ .

For ease of notation we also let $\rho_{p}$ denote $\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})}$ . Then the weak version of local-global compatibility says that, if $\rho$ is promodular, then there exists a finite set of places $\Sigma$ containing $p$ and the places at which $\rho$ is ramified, such that we have the following nonzero continuous $\mathrm{GL}_{2}(\mathbb{Q}_{p})\times\mathrm{GL}_{2}(\mathbb{A}_{f}^{\Sigma})$ -equivariant morphism:

$\displaystyle B(\rho_{p})\otimes_{E}\otimes_{\ell\not\in\Sigma}^{'}\pi_{\ell}(\rho_{p})\to\mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}(\rho,\widehat{H}_{E}^{1})$

Furthermore, if $\rho_{p}$ is not the direct sum of two characters or the extension of a character by itself, all the morphisms will be closed injections.

The strong version of local-global compatibility is as follows. Assume the hypothesis of the weak version and assume further that the restriction of $\overline{\rho}$ to $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is not isomorphic to a twist of $\begin{pmatrix}1&*\\0&1\end{pmatrix}$ by some character. Then we have a $\mathrm{GL}_{2}(\mathbb{Q}_{p})\times\mathrm{GL}_{2}(\mathbb{A}_{f}^{\Sigma})$ -equivariant homeomorphism

$\displaystyle B(\rho_{p})\otimes_{E}\otimes_{\ell\neq p}^{'}\pi_{\ell}(\rho_{p})\xrightarrow{\sim}\mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}(\rho,\widehat{H}_{E}^{1})$

In this post we will only discuss ideas related to the proof of the weak version of local-global compatibility. It will proceed as follows. First we reduce the problem of showing local-global compatibility to the existence of a map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ . Then to show that this map exists, we construct, using (completions of) Hecke algebra-valued deformations of the relevant residual representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , a module $X_{\mathcal{O}_{E}}$ , and showing that, for any maximal ideal $\mathfrak{p}$ , the submodule of $X_{\mathcal{O}_{E}}$ annihilated by $\mathfrak{p}$ is nonzero. Initially we shall show this only for “crystalline classical maximal ideals”, but these will turn out to be dense in the completion of the Hecke algebra, which will show that the result is true for all maximal ideals.

A Preliminary Reduction

To show local-global compatibility, it is in fact enough for us to show the existence of a $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{Q}_{p})$ -equivariant map

$\displaystyle \rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}.$

Let us briefly discuss why this is true. Consider the smooth induced representation $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}^{{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}}1$ with compact support over $E$ . We have that $\mathrm{End}_{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}^{{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}}1)\cong E[T_{\ell},S_{\ell}]$ . Now let $\pi_{\ell}$ be a smooth representation of $\mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ over $E$ , and let $\lambda_{1}$ , $\lambda_{2}$ be in $E$ . We have

$\displaystyle \mathrm{Hom}_{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}\left(\frac{\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}^{{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}}1}{(T_{\ell}-\lambda_{1},S_{\ell}-\lambda_{2})},\pi_{\ell}\right)=\pi_{\ell}^{\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}[T_{\ell}-\lambda_{1},S_{\ell}-\lambda_{2}]$

Now let $\lambda:\mathbb{T}_{\Sigma}\to E$ be such that $\lambda(T_{\ell})=\mathrm{trace}(\rho(\mathrm{Frob}_{\ell}))$ and $\lambda(S_{\ell})=\ell^{-1}\mathrm{det}(\rho(\mathrm{Frob}_{\ell}))$ , for $\ell\not\in\Sigma$ . It follows from the (classical) local Langlands correspondence that

$\displaystyle \pi_{\ell}(\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}}))=\frac{\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{\ell})}^{{\mathrm{GL}_{2}(\mathbb{Q}_{\ell})}}1}{(T_{\ell}-\lambda_{1},S_{\ell}-\lambda_{2})}$

Let $\widehat{H}_{E,\Sigma}^{1}[\lambda]$ denote the subspace of $\widehat{H}_{E,\Sigma}^{1}$ on which $\mathbb{T}_{\Sigma}$ acts by $\lambda$ . The results that we have just discussed now tell us that the space

$\displaystyle \mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times \mathrm{GL}_{2}(\mathbb{Q}_{p})\times\mathrm{GL}_{2}(\mathbb{A}_{f}^{\Sigma})}(\rho\otimes_{E} B(\rho_{p})\otimes'_{\ell\not\in\Sigma}\pi_{\ell}(\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})}),\widehat{H}_{E}^{1})$

is isomorphic to the space

$\displaystyle \mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times \mathrm{GL}_{2}(\mathbb{Q}_{p})}(\rho\otimes_{E} B(\rho_{p}),\widehat{H}_{E,\Sigma}^{1}[\lambda])$ .

Furthermore, it follows from Eichler-Shimura relations (which relate the action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $\mathbb{T}_{\Sigma}$ on $\widehat{H}_{E,\Sigma}^{1}$ that the previous space is also isomorphic to

$\displaystyle \mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times \mathrm{GL}_{2}(\mathbb{Q}_{p})}(\rho\otimes_{E} B(\rho_{p}),\widehat{H}_{E,\Sigma}^{1})$ .

Furthermore, for each of these isomorphisms, a morphism on one side of the isomorphism is a closed injection if and only if the corresponding morphism is also a closed injection. Therefore, as earlier stated, to show local-global compatibility it will be enough for us to show that a $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{Q}_{p})$ -equivariant map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ exists.

Representations valued in a completion of the Hecke algebra

To show the existence of this map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ , we will construct a module that we shall denote by $X_{\mathcal{O}_{E}}$ . Before we can define this module though, we need to make some definitions involving the Hecke algebra, and representations valued in (completions of) this Hecke algebra.

Let $\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(k_{E})$ be an absolutely irreducible odd continuous residual Galois representation. Let us suppose furthermore that $\overline{\rho}$ is modular.

Let $K_{\Sigma}$ be a compact open subgroup of $\prod_{\ell\in\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ . We let $\mathbb{T}(K_{\Sigma}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell}))_{\overline{\rho}}$ be the completion of $\mathbb{T}_{\Sigma}(K_{\Sigma}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell}))$ with respect to the maximal ideal generated by $\varpi$ , $T_{\ell}-\mathrm{tr}(\overline{\rho}(\mathrm{Frob}_{\ell})$ , and $S_{\ell}-\ell^{-1}\mathrm{det}(\overline{\rho}(\mathrm{Frob}_{\ell}))$ . We define

$\displaystyle \mathbb{T}_{\Sigma,\overline{\rho}}:=\varprojlim_{K_{\Sigma}}\mathbb{T}(K_{\Sigma}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell}))_{\overline{\rho}}.$

Since $\overline{\rho}$ is absolutely irreducible, for every compact open subgroup $K_{\Sigma}$ of $\prod_{\ell\in\Sigma} \mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ such that $\mathbb{T}(K_{\sigma}\prod_{\ell\neq\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell}))_{\overline{\rho}}\neq 0$ the work of Carayol provides us with a unique continuous Galois module $\rho(\Sigma):\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{T}(K_{\sigma}\prod_{\ell\neq\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell}))_{\overline{\rho}})$ unramified outside $\Sigma$ such that $\mathrm{trace}(\rho(K_{\Sigma})(\mathrm{Frob}_{\ell}))=T_{\ell}$ and $\mathrm{det}(\rho(K_{\Sigma})(\mathrm{Frob}_{\ell})=\ell S_{\ell}$ .

We define $\rho_{\Sigma}:=\varprojlim_{K_{\Sigma}}\rho(K_{\Sigma})$ . This is a deformation of $\overline{\rho}$ over the complete Noetherian local $\mathcal{O}_{E}$ -algebra $\mathbb{T}_{\Sigma,\overline{\rho}}$ (see also Galois Deformation Rings). After restriction to $\mathbb{Q}_{p}$ , we may also look at $\rho_{\Sigma}$ as a deformation of $\overline{\rho}_{p}$ .

Now let $\overline{\pi}_{p}$ is the representation associated to $\overline{\rho}_{p}$ by the mod p local Langlands correspondence. We also want to construct a deformation $\pi_{\Sigma}$ of $\overline{\pi}_{p}$ , that is related to $\rho_{\Sigma}$ by the p-adic local Langlands correspondence.

Let $R(\overline{\rho}_{p})$ be the deformation ring that represents the functor which assigns to a complete Noetherian local $\mathcal{O}_{E}$ -algebra the set of deformations of $\overline{\rho}_{p}$ over $A$ . We define $R(\overline{\rho}_{p})^{\mathrm{cris}}$ to be the the quotient of $R(\overline{\rho}_{p})$ by the intersection of all maximal ideals which are kernels of a map $R(\rho_{p})\to E'$ for some extension $E'$ of $E$ such that the representation over $E'$ obtained by base change from the universal representation over $R(\overline{\rho}_{p})$ is crystalline with distinct Hodge-Tate weights (see also p-adic Hodge Theory: An Overview).

Similarly, we have a deformation ring Let $R(\overline{\pi}_{p})$ that represents the functor which assigns to a complete Noetherian local $\mathcal{O}_{E}$ -algebra $A$ the set of deformations of $\overline{\rho}_{p}$ over $A$ . Recall that the p-adic local Langlands correspondence provides us with the Montreal functor $V$ from representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ to representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ , which means we have a map $R(\overline{\rho}_{p})\to R(\overline{\pi}_{p})$ . We let $R(\overline{\pi}_{p})^{\mathrm{det}}$ be the quotient of $R(\overline{\pi}_{p})$ parametrizing deformations $\pi_{p}$ of $\overline{\pi}_{p}$ whose central character corresponds to $\mathrm{det} V(\pi_{p})\varepsilon$ under local class field theory. We define

$R(\overline{\pi}_{p})^{\mathrm{cris}}:=R(\overline{\pi}_{p})\otimes_{R(\overline{\rho}_{p})}R(\overline{\rho}_{p})^{\mathrm{cris}}$

Now it turns out that the surjection $R(\overline{\pi}_{p})^{\mathrm{cris}}\twoheadrightarrow R(\overline{\rho}_{p})^{\mathrm{cris}}$ is actually an isomorphism. A consequence of this is that, if we have a complete Noetherian local $\mathcal{O}_{E}$ -algebra $T$ that is a quotient of $R(\overline{\rho}_{p})^{\mathrm{cris}}$ , any deformation $\rho_{p}$ of $\overline{\rho}_{p}$ over $T$ comes from a deformation $\pi_{p}$ of $\overline{\pi}_{p}$ via the Montreal functor $V$ .

Now all we need to do to construct $\pi_{\Sigma}$ is to find an appropriate complete Noetherian local $\mathcal{O}_{E}$ -algebra $T$ . We recall that $\rho_{\Sigma}$ is a deformation of $\overline{\rho}_{p}$ over $\mathbb{T}_{\Sigma,\overline{\rho}}$ , so we want to find $T$ inside of $\mathbb{T}_{\Sigma},\overline{\rho}$ , apply the discussion in the previous paragraph, and then we can extend scalars to obtain the deformation $\pi_{\Sigma}$ over $\mathbb{T}_{\Sigma,\overline{\rho}}$ . To do this we need to show to discuss crystalline classical maximal ideals, and show that they are Zariski dense inside $\mathbb{T}_{\Sigma,\overline{\rho}}$ (this fact will also be used again to achieve the goal we stated earlier of showing the existence of a map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ ).

We say that a maximal ideal $\mathfrak{p}$ of $\mathbb{T}_{\Sigma}$ is classical if the system of Hecke eigenvalues associated to $\mathbb{T}_{\Sigma}\to\mathbb{T}_{\Sigma}[1/p]/\mathfrak{p}$ comes from a cuspidal eigenform of weight $\geq 2$ .

Let $\mathfrak{p}$ be a classical maximal ideal of $\mathbb{T}_{\Sigma,\overline{\rho}}$ . Then we have a representation

$\displaystyle \rho_{\Sigma}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}} \mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$

which is potentially semistable with distinct Hodge-Tate weights. We say that the classical maximal ideal $\mathfrak{p}$ is crystalline if the associated Galois representation is crystalline.

Let us now outline the argument showing that the crystalline classical maximal ideals are dense in $\mathbb{T}_{\Sigma,\overline{\rho}}$ . This is the same as the statement that the intersection of all crystalline classical maximal ideals is zero. And so our strategy will be to show that any element $t$ in this intersection acts by $0$ on $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ .

Let $K_{\Sigma}^{p}$ be a sufficiently small compact open subgroup of $\prod_{\ell\in\Sigma\setminus p}\mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ . Then the $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ -representation $\widehat{H}^{1}(K_{\Sigma}^{p}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{p}))_{E,\overline{\rho}}$ is a topological direct factor of $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)^{r}$ for some $r>0$ , where $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)$ is the $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ -representation provided by the continuous $E$ -valued functions on $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ .

Now it happens that the polynomial functions of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ are dense inside the continuous functions $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)$ . This implies that the vectors in $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)^{r}$ for which $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ acts algebraically are dense in $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)^{r}$ . Since, by the previous paragraph, $\widehat{H}^{1}(K_{\Sigma}^{p}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{p}))_{E,\overline{\rho}}$ is a topological direct factor of $\mathcal{C}(\mathrm{GL}_{2}(\mathbb{Z}),E)^{r}$ for $K_{\Sigma}^{p}$ sufficiently small, this implies that a similar result holds for $\widehat{H}^{1}(K_{\Sigma}^{p}\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{p}))_{E,\overline{\rho}}$ . Taking limits over $K_{\Sigma}^{p}$ , we obtain that the vectors in $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ for which $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ acts by an algebraic representation of $\mathrm{GL}_{2}$ are dense in $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ .

If $\mathfrak{p}$ is a maximal ideal of $\mathbb{T}_{\Sigma,\overline{\rho}}$ , we write $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}[\mathfrak{p}]$ to denote the submodule of $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ annihilated by $\mathfrak{p}$ . We now have that $(\widehat{H}_{E,\Sigma,\overline{\rho}}^{1})^{\mathrm{alg}}$ is contained in $\oplus_{\mathfrak{p}}\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}[\mathfrak{p}]$ , where the direct sum is over all classical maximal ideals of $\mathbb{T}_{\Sigma,\overline{\rho}}$ . Furthermore, the subrepresentation of $(\widehat{H}_{E,\Sigma,\overline{\rho}}^{1})^{\mathrm{alg}}$ generated by the vectors for which $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ acts by an algebraic representation is contained in $\oplus_{\mathfrak{p}\in\mathcal{C}}\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}[\mathfrak{p}]$ , where the direct sum is now over all crystalline classical maximal ideals of $\mathbb{T}_{\Sigma,\overline{\rho}}$ . Now it turns out that, if $\rho_{f}$ is the Galois representation associated to some cuspidal eigenform $f$ of weight $\geq 2$ , the representation $\pi_{p}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}))}$ contains a vector fixed under the action of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ if and only if $\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}))}$ is crystalline. If $t$ is an element in the intersection of all the crystalline classical maximal ideals, it annihilates $\oplus_{\mathfrak{p}\in\mathcal{C}}\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}[\mathfrak{p}]$ , and therefore also the subrepresentation of $(\widehat{H}_{E,\Sigma,\overline{\rho}}^{1})^{\mathrm{alg}}$ generated by the vectors for which $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ acts by an algebraic representation. But this subrepresentation is dense in $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ and by continuity $t$ acts by zero on $\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}$ . This shows that the intersection of all the crystalline classical maximal ideals is zero and that they are Zariski dense in $\mathbb{T}_{\Sigma,\overline{\rho}}$ .

Since the crystalline classical maximal ideals are dense in in $\mathbb{T}_{\Sigma,\overline{\rho}}$ , we have that the map $R(\overline{\rho}_{p})\to\mathbb{T}_{\Sigma,\overline{\rho}}$ factors through $R(\overline{\rho}_{p})^{\mathrm{cris}}\to\mathbb{T}_{\Sigma,\overline{\rho}}$ . Now we find our complete Noetherian local $\mathcal{O}_{E}$ -algebra $T$ mentioned earlier as the image of the map $R(\overline{\rho}_{p})^{\mathrm{cris}}\to\mathbb{T}_{\Sigma,\overline{\rho}}$ , so that we can obtain a deformation $\pi_{p}$ of $\overline{\pi}_{p}$ that gives rise to $\rho_{p}$ via the Montreal functor $V$ . Then we extend scalars to $\mathbb{T}_{\Sigma,\overline{\rho}}$ to obtain $\pi_{\Sigma}$ .

Existence of the map

Now that we have the $\mathbb{T}(K_{\sigma}\prod_{\ell\neq\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell}))_{\overline{\rho}}$ -valued representations $\rho_{\Sigma}$ and $\pi_{\Sigma}$ , we may now define the module $X_{\mathcal{O}_{E}}$ which as we said will help us prove the existence of a map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ . It is defined as follows:

$\displaystyle X_{\mathcal{O}_{E}}:=\mathrm{Hom}_{\mathbb{T}_{\Sigma,\overline{\rho}}[\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{Q}_{p})]}(\rho_{\Sigma}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}}\pi_{\Sigma},\widehat{H}_{\mathcal{O}_{E}\Sigma,\overline{\rho}}^{1})$

Let $\mathfrak{p}$ be a maximal ideal of $\mathbb{T}_{\Sigma,\overline{\rho}}$ . We let $X_{E}[\mathfrak{p}]$ denote the set of elements of $X_{E}$ that are annihilated by the elements of $\mathfrak{p}$ . Our aim is to show that $X_{E}[\mathfrak{p}]\neq 0$ for all maximal ideals. As we shall show later, applying this to the maximal ideal generated by $\varpi$ , $T_{\ell}-\mathrm{tr}(\overline{\rho}(\mathrm{Frob}_{\ell})$ , and $S_{\ell}-\ell^{-1}\mathrm{det}(\overline{\rho}(\mathrm{Frob}_{\ell}))$ will give us our result. Our approach will be to show first that $X_{E}[\mathfrak{p}]\neq 0$ for “crystalline” maximal ideals, then, using the fact that the crystalline classical maximal ideals are Zariski dense in $\mathbb{T}_{\Sigma,\overline{\rho}}$ , show that this is true for all maximal ideals of $\mathbb{T}_{\Sigma,\overline{\rho}}$ .

Let $\mathfrak{p}$ be a crystalline classical maximal ideal of $\mathbb{T}_{\Sigma,\overline{\rho}}$ . Then $X_{E}[\mathfrak{p}]\neq 0$ . To show this, we choose some field $\widetilde{E}$ that contains $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ . Now recall again that we have

$\displaystyle \bigoplus_{k\geq 2,n\in\mathbb{Z}} \varinjlim_{K_{\Sigma}}H_{\mathrm{et}}^{1}(Y(K_{\Sigma}\prod_{\ell\neq \Sigma}\mathrm{GL}_{2}(\mathbb{Z}_{\ell}))\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{F}_{k-2})\otimes_{\widetilde{E}}\mathrm{Sym}^{k-2}E'^{2})^{\vee}\otimes_{\widetilde{E}}\varepsilon^{n}\cong(\widehat{H}_{\widetilde{E},\Sigma}^{1})^{\mathrm{alg}}$

Now since $\widetilde{E}$ contains $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ , we find that inside $(\widehat{H}_{E',\Sigma}^{1})[\mathfrak{p}]$ there lies a tensor product of $\rho(\mathfrak{p})$ and some locally algebraic representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ . What the crystalline condition on $\mathfrak{p}$ does is it actually provides us with at most one equivalence class of invariant norms on this locally algebraic representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , which must be the one induced by $(\widehat{H}_{\mathcal{O}_{\widetilde{E}},\Sigma}^{1})[\mathfrak{p}]$ on $(\widehat{H}_{\widetilde{E},\Sigma}^{1})[\mathfrak{p}]$ . It turns out that after completion, the representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ on the resulting p-adic Banach space is precisely $B(\rho(\mathfrak{p})\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})$ if $\rho(\mathfrak{p}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})$ is irreducible, and a closed subrepresentation of $B(\rho(\mathfrak{p})\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})$ if $\rho(\mathfrak{p}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})$ is reducible (here $\rho(\mathfrak{p})=\rho_{\Sigma}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}}\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ and $B(\rho(\mathfrak{p})\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})=\pi_{\Sigma}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}}\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ , and these two correspond to each other under the p-adic local Langlands correspondence).

Now we know that $X_{E}[\mathfrak{p}]\neq 0$ if $\mathfrak{p}$ is a crystalline classical maximal ideal. Now we want to extend this to all the maximal ideals of $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ by making use of the fact that the set of crystalline classical maximal ideals is Zariski dense inside $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ .

The idea is that, if $X_{E}[\mathfrak{p}]\neq 0$ for all maximal ideals $\widetilde{\mathfrak{p}}$ that belong to some set $\mathcal{E}$ that is Zariski dense in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ , then $X_{E}[\mathfrak{p}]\neq 0$ for all maximal ideals $\mathfrak{p}$ in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ . Let us consider first the simpler case of a module $M$ of finite type over $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ . We want to show that if $M/\widetilde{\mathfrak{p}}\neq 0$ for all $\widetilde{\mathfrak{p}}\in \mathcal{E}$ then $M/\mathfrak{p}\neq 0$ for all maximal ideals $\mathfrak{p}$ in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ .

Since $\mathfrak{p}$ is maximal, $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ is a field, and $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}$ acts faithfully on $M/\mathfrak{p}M$ . If some element $t\in\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ acts by zero on $M$ it must act by zero on $M/\mathfrak{p}M$ for all $\mathfrak{p}$ . If $M/\mathfrak{p}M\neq 0$ for all $\widetilde{\mathfrak{p}}\in \mathcal{E}$ , then this element $t$ must be in the intersection of all the $\widetilde{\mathfrak{p}}$ in $\mathcal{E}$ , but since $\mathcal{E}$ is Zariski dense in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ , this intersection is zero and $t$ has to be zero.

Suppose for the sake of contradiction that $M/\mathfrak{p}M\neq 0$ for all $\widetilde{\mathfrak{p}}\in \mathcal{E}$ but $M/\mathfrak{p}M=0$ for some maximal ideal $\mathfrak{p}$ in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ . Then Nakayama’s lemma says that there exists some nonzero element $t\in\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ such that $tM=0$ . But this contradicts the above paragraph, so we must have $M/\mathfrak{p}M\neq 0$ for all maximal ideals $\mathfrak{p}$ in $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ .

Now let $K_{\Sigma}^{p}$ be a compact open subgroup of $\prod_{\ell\not\in\Sigma}\mathrm{GL}_{2}(\mathbb{Q}_{\ell})$ , and let $X_{\mathcal{O}_{E}}^{K_{\Sigma}^{p}}$ be defined similarly to $X_{\mathcal{O}_{E}}$ but with $\widehat{H}_{\mathcal{O}_{E}}^{1}(K_{\Sigma}^{p})$ in place of $\widehat{H}_{\mathcal{O}_{E}}^{1}$ . We apply the above argument to $\mathrm{Hom}_{\mathcal{O_{E}}}(X_{\mathcal{O}_{E}}^{K_{\Sigma}^{p}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}} E$ , which is a $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ -module of finite type. Then it is a property of $X_{\mathcal{O}_{E}}$ (which is $\varinjlim_{K_{\Sigma}^{p}}X_{\mathcal{O}_{E}}^{K_{\Sigma}^{p}}$ ) that $X_{\mathcal{O}_{E}}[\mathfrak{p}]\neq 0$ if $X_{\mathcal{O}_{E}}^{K_{\Sigma}^{p}}[\mathfrak{p}]\neq 0$ for sufficiently small $K_{\Sigma}^{p}$ .

Now that we know that $X_{\mathcal{O}_{E}}[\mathfrak{p}]\neq 0$ for all maximal ideals $\mathfrak{p}$ of $\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]$ , we apply this to the particular maximal ideal $\mathfrak{p}_{\rho}$ generated by $T_{\ell}-\mathrm{trace}(\rho(\mathrm{Frob}_{\ell}))$ and $S_{\ell}-\ell^{-1}\mathrm{det}(\rho(\mathrm{Frob}_{\ell}))$ . But we have

$\displaystyle X_{\mathcal{O}_{E}}[\mathfrak{p}_{\rho}]\otimes E=\mathrm{Hom}_{\mathbb{T}_{\Sigma,\overline{\rho}}/\mathfrak{p_{\rho}}[\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\times\mathrm{GL}_{2}(\mathbb{Q}_{p})]}(\rho(\mathfrak{p}_{\rho})\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}/\mathfrak{p}_{\rho}}B(\rho(\mathfrak{p}_{\rho})\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})}),\widehat{H}_{E,\Sigma,\overline{\rho}}^{1}[\mathfrak{p}_{\rho}])$

where again $\rho(\mathfrak{p}_{\rho})=\rho_{\Sigma}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}}\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}_{\rho}$ and $B(\rho(\mathfrak{p}_{\rho})\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})})=\pi_{\Sigma}\otimes_{\mathbb{T}_{\Sigma,\overline{\rho}}}\mathbb{T}_{\Sigma,\overline{\rho}}[1/p]/\mathfrak{p}_{\rho}$ . Since we have just shown that the left-hand side of the above isomorphism is nonzero, then so must the right hand-side, which means there is map $\rho\otimes_{E} B(\rho_{p})\to \widehat{H}_{E,\Sigma}^{1}$ .

Furthermore this map is a closed injection if $\rho_{p}$ is not a direct sum of two characters or an extension of a character by itself. In the case that $\rho_{p}$ is absolutely irreducible, this follows from the fact that $B(\rho_{p})$ is topologically irreducible and admissible. If $\rho_{p}$ is reducible and indecomposable, then $B(\rho_{p})$ is also reducible and indecomposable and one needs to show that a nonzero morphism cannot be factorized by a strict quotient of $B(\rho_{p})$ . We leave further discussion of these to the references.

Application to the Fontaine-Mazur conjecture

Let us now discuss the application of local-global compatibility to (a special case of) the Fontaine-Mazur conjecture, whose statement is as follows.

Let $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(E)$ be an absolutely irreducible odd continuous p-adic Galois representation, unramified at all but finitely many places, and whose restriction to $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is potentially semistable with distinct Hodge-Tate weights. Then the Fontaine-Mazur conjecture states that there exists some cuspidal eigenform $f$ of weight $\geq 2$ such that $\rho$ is the twist of $\rho_{f}$ (the Galois representation associated to $f$ ) by some character.

The Fontaine-Mazur conjecture is also often stated in the following manner. Let $\rho$ be as in the previous paragraph. Then $\rho$ can be obtained as the subquotient of the etale cohomology of some variety. This statement in fact follows from the previous one, because if $\rho_{f}$ is a Galois representation obtained from some cuspidal eigenform $f$ of weight $\geq 2$ , then it may be found as the subquotient of the etale cohomology of what is known as a Kuga-Sato variety.

Now let us discuss how local-global compatibility figures into the proof (due to Matthew Emerton) of a special case of the Fontaine-Mazur conjecture. This special case is when $p>2$ and we have the restriction of the corresponding residual Galois representation $\overline{\rho}$ to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_{p}))$ is absolutely irreducible, and the restriction of $\overline{\rho}$ to $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is not isomorphic to a Galois representation of the form $\begin{pmatrix}1&0\\0&\overline{\epsilon}\end{pmatrix}$ twisted by a character for $p>3$ , or $\begin{pmatrix}1&*\\0&\overline{\epsilon}\end{pmatrix}$ twisted by a character for $p=3$ .

In this case it follows from the work of Böckle, Diamond-Flach-Guo, Khare-Wintenberger, and Kisin that $\rho$ is promodular. Then the local-global compatibility that we have discussed tells us that we have a closed injective map $B(\rho_{p})\hookrightarrow \mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}(\rho,\widehat{H}_{E}^{1})$ . The condition of the restriction $\rho_{p}$ being potentially semistable with distinct Hodge-Tate weights guarantees that $B(\rho_{p})^{\mathrm{alg}}\neq 0$ (here $B(\rho_{p})^{\mathrm{alg}}$ is defined exactly the same as $(\widehat{H}_{E,\Sigma}^{1})^{\mathrm{alg}}$ except with $B(\rho_{p})$ in place of $(\widehat{H}_{E,\Sigma}^{1})$ ). This follows from the compatibility of the p-adic local Langlands correspondence and the “classical” local Langlands correspondence, which says that if $\rho_{p}$ is potentially semistable with distinct Hodge-Tate weights $a<b$ then we have the following isomorphism:

$\displaystyle \mathrm{det}^{a+1}\otimes_{E}\mathrm{Sym}^{b-a-1}E^{2}\otimes_{E}\pi_{p}(\rho_{p})\xrightarrow{\sim}B(\rho_{p})^{\mathrm{alg}}$

The closed injective map $B(\rho_{p})\hookrightarrow \mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}(\rho,\widehat{H}_{E}^{1})$ then tells us that, since $B(\rho_{p})^{\mathrm{alg}}\neq 0$ , we must have $(\widehat{H}_{E,\Sigma}^{1})^{\mathrm{alg}}\neq 0$ as well. But we have the isomorphism

and the Galois representations that show up on the left hand side of this isomorphism are associated to cuspidal eigenforms of weights $k\geq 2$ . This completes our sketch of the proof of the special case of the Fontaine-Mazur conjecture.

We have discussed here the ideas involved in Emerton’s proof of a special case of the Fontaine-Mazur conjecture. There is also another proof due to Mark Kisin that makes use of a different approach, namely, ideas related to the Breuil-Mezard conjecture (a version of which was briefly discussed in Moduli Stacks of (phi, Gamma)-modules) and the method of “patching” (originally developed as part of the approach to proving Fermat’s Last Theorem). This approach will be discussed in future posts on this blog.

References:

Correspondance de Langlands p-adique, compatibilité local-global et applications by Christophe Breuil

Local-global compatibility in the p-adic Langlands programme for GL_2/Q by Matthew Emerton

A local-global compatibility conjecture in the p-adic Langlands programme for GL_2/Q by Matthew Emerton

Completed cohomology and the p-adic Langlands program by Matthew Emerton

The Breuil-Schneider conjecture, a survey by Claus M. Sorensen

Completed Cohomology

November 1, 2021February 22, 2022 / Anton Hilado / 1 Comment

Let $F$ be a number field, and let $G_{F,S}$ be the Galois group over $F$ of the maximal extension of $F$ unramified outside a finite set of primes $S$ . It should follow from the Langlands correspondence that $n$ -dimensional continuous (we shall only be talking about continuous Galois representations in this post, so we omit the word “continuous” from here on) representations $\rho:G_{F,S}\to \mathrm{GL_{n}}(\overline{\mathbb{Q}}_{p})$ should correspond to certain automorphic representations $\pi$ of $\mathrm{GL}_{n}$ unramified outside $S$ (see also Automorphic Forms).

The Fontaine-Mazur-Langlands conjecture further states that such Galois representations $\rho$ that are irreducible and “geometric” (i.e. its restrictions to the primes above $p$ satisfy some conditions related to p-adic Hodge theory, see also p-adic Hodge Theory: An Overview) should match up with “algebraic” (we shall explain this shortly) cuspidal $\pi$ . Furthermore this conjecture expects that certain “Hodge numbers” associated to the Galois representation $\rho$ via p-adic Hodge theory should match up to “Hodge numbers” defined for the automorphic representation $\pi$ via its “infinitesimal character” at the archimedean primes (note that they are defined differently, since they are associated to different kinds of representations; they only share the same name because they are expected to coincide).

Generally, whether $\rho$ is “geometric” or not, its Hodge numbers going to be $p$ -adic numbers, and whether $\pi$ is “algebraic” or not, its Hodge numbers are complex numbers. However, if $\rho$ is geometric, then its Hodge numbers are integers, and if $\pi$ is algebraic, its Hodge numbers are also integers (in fact the definition of “algebraic” here just means that its Hodge numbers are integers), and this allows us to match them up.

To see things in a little more detail, let us consider the case of a $1$ -dimensional representation $\rho:G_{F,S}\to \overline{\mathbb{Q}}_{p}$ . We have seen in Galois Representations that an example of this is given by the p-adic cyclotomic character which we can also view as follows. Let $S=\lbrace p,\infty\rbrace$ . Let $G_{F,S}^{\mathrm{ab}}$ be the abelianization of $G_{F,S}$ . It follows from the Kronecker-Weber theorem that $G_{F,S}^{\mathrm{ab}}$ is isomorphic to $\mathbb{Z}_{p}^{\times}$ , and it is precisely the p-adic cyclotomic character that gives this isomorphism. Since $\mathbb{Z}_{p}^{\times}$ embeds into $\overline{\mathbb{Q}}_{p}^{\times}$ , which is also $\mathrm{GL}_{1}(\overline{\mathbb{Q}_{p}})$ , we have our $1$ -dimensional Galois representation. We can also take a power of the p-adic cyclotomic character to get another $1$ -dimensional Galois representation.

But the p-adic cyclotomic character and its powers are not the only $1$ -dimensional Galois representations. For instance, we have a map from $\mathbb{Z}_{p}^{\times}\to \mathbb{Q}_{p}^{\times}$ given by reducing $\mathbb{Z}_{p}$ mod $p^{r}$ and then composing it with the map $\chi$ that sends this element of $(\mathbb{Z}/p^{r})^{\times}$ to the corresponding $p^{r}$ -th root of unity in $\overline{\mathbb{Q}}_{p}^{\times}$ . This is a finite-order character. We also have another map from $\mathbb{Z}_{p}^{\times}\to \overline{\mathbb{Q}}_{p}^{\times}$ which sends $x$ to $x^{s}$ , for some $s$ in $\overline{\mathbb{Q}}_{p}$ such that $\vert s\vert<\frac{p}{p-1}$ . If we compose the p-adic cyclotomic character with either of these maps, we get another $1$ -dimensional Galois representation. It turns out the Hodge number of the latter representation is given by $s$ .

The $1$ -dimensional Galois representations form a rigid analytic space (see also Rigid Analytic Spaces), and their Hodge numbers form p-adic analytic functions on this space. The geometric representations are the ones that are from a power of the p-adic cyclotomic character composed with a finite-order character, and these form a countable dense subset of this rigid analytic space.

Some form of this phenomena happens more generally for higher dimensional Galois representations – they form a rigid analytic space and the geometric ones are a subset of these.

It is convenient that our Galois representations form a rigid analytic space, and suppose we want to do something similar for our automorphic representations. The problem is that the automorphic representations aren’t really “p-adic”, as we may see from the fact that their Hodge numbers are complex instead of p-adic. This is the problem that p-adically completed cohomology, also simply known as completed cohomology, aims to solve.

Let us look at how we want to find automorphic representations in cohomology. Let $G_{\infty}=\mathrm{GL_{n}}(F\otimes_{\mathbb{Q}}\mathbb{R})$ . If $F$ has $r_{1}$ real embeddings and $r_{2}$ complex embeddings, then $G_{\infty}$ will be isomorphic to $\mathrm{GL}_{n}(\mathbb{R})^{r_{1}}\times\mathrm{GL}_{n}(\mathbb{C})^{r_{2}}$ . Let $K_{\infty}^{\circ}$ be a maximal connected compact subgroup of $G_{\infty}$ . With $r_{1}$ and $r_{2}$ as earlier, $K_{\infty}^{\circ}$ will be isomorphic to $\mathrm{SO}(n)^{r_{1}}\times \mathrm{U}(n)^{r_{2}}$ .

Let $X$ be the quotient $G_{\infty}/\mathbb{R}_{>0}^{\times}K_{\infty}^{\circ}$ . This is an example of a symmetric space – for example, if $F=\mathbb{Q}$ and $n=2$ , $X$ is going to be $\mathbb{C}\setminus \mathbb{R}$ .

The space $X$ has an action of $G_{\infty}$ , and its subgroup $\mathrm{GL}_{n}(\mathcal{O}_{F})$ . Letting $N\geq 1$ , we may therefore take the quotient

$\displaystyle Y(N)=\mathrm{GL}_{n}(\mathcal{O}_{F})\backslash (X\times \mathrm{GL}_{n}(\mathcal{O}_{F}/N\mathcal{O}_{F}))$

For example, if $F=\mathbb{Q}$ and $n=2$ , then $Y(N)$ consists of copies of the (uncompactified) modular curve of level $N$ (the number of copies is equal to the number of primes less than $N$ ).

It is this space $Y(N)$ whose cohomology we are interested in. For instance $H^{i}(Y(N),\mathbb{C})$ is related to automorphic forms by a theorem of Jens Franke. However, it is complex, and not the p-adically varying one that we want. There is an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ , but the important part of this cohomology comes from the cohomology with $\mathbb{Q}$ coefficients, which is unchanged when we do this isomorphism, and therefore does not really add anything.

This is now where we introduce completed cohomology. Let us require that $N$ and $p$ be mutually prime. We define the completed cohomology $\widetilde{H}^{i}$ as follows:

$\displaystyle \widetilde{H}^{i}:=\varprojlim_{s\geq 1}\varinjlim_{r\geq 0}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

The order of the limits here is important (we will see shortly what happens when they are interchanged). By first taking the direct limit we are essentially considering the union of $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ for all $r$ with $\mathbb{Z}/p^{s}\mathbb{Z}$ coefficients. This is a very big abelian group that might not even be finitely generated. Then the inverse limit means we are taking the $p$ -adic completion – having this as the last step guarantees that the result is something that is p-adically complete (hence the name p-adically completed cohomology). So the completed cohomology $\widetilde{H}^{i}$ is a p-adically complete module over $\mathbb{Z}_{p}$ , which again may not be finitely generated. Taking the tensor product of $\widetilde{H}^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ gives us a vector space $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ which moreover is a Banach space.

Let us consider now what happens if the order of the limits were interchanged. Let us denote the result by $H^{i}$ :

$\displaystyle H^{i}:=\varinjlim_{r\geq 0}\varprojlim_{s\geq 1}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

By taking the inverse limit first we are simply considering $H^{i}(Y(Np^{r}),\mathbb{Z}_p$ , and taking the direct limit means we are taking the union of $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ for all $r$ . If we take the tensor product of $H^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ , then what we get is $H_{\mathbb{Q}_{p}}^{i}$ , the union of $H^{i}(Y(Np^{r}),\mathbb{Q}_p$ for all $r$ . Being the cohomology with characteristic zero coefficients, this may once again be related to the automorphic forms, as earlier.

Therefore, $H_{\mathbb{Q}_{p}}^{i}$ , via the Fontaine-Mazur-Langlands conjecture, should be related to the geometric Galois representations. Now it happens that we can actually embed $H_{\mathbb{Q}_{p}}^{i}$ into the completed cohomology $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ , because there is a map from $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ to $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ , and then we can take the direct limit over $r$ followed by the inverse limit over $r$ and then tensor over $\mathbb{Q}_{p}$ as previously.

This embedding of $H_{\mathbb{Q}_{p}}^{i}$ into $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should now bring to mind the picture with the geometric Galois representations which sit inside the rigid analytic space of Galois representations which may not necessarily be geometric, as discussed earlier. It is in fact a conjecture that $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should know about the rigid analytic space of Galois representations.

In the case $F=\mathbb{Q}$ and $n=2$ , the completed cohomology is some space of p-adic modular forms, and there is much that is known via the work of Matthew Emerton, who also showed that the p-adic local Langlands correspondence appears inside the completed cohomology. This has led to a proof of many cases of the Fontaine-Mazur conjecture for $2$ -dimensional odd Galois representations.

We have only provided a rough survey of the motivations behind the theory of completed cohomology in this post. We will discuss further deeper aspects of it, and its relations to the p-adic local Langlands correspondence and the Fontaine-Mazur conjecture in future posts.

References:

Completed cohomology and the p-adic Langlands correspondence by Matthew Emerton on YouTube

Completed cohomology and the p-adic Langlands program by Matthew Emerton

Completed cohomology – a survey by Frank Calegari and Matthew Emerton

Theories and Theorems

Math and Physics for Everyone

completed cohomology