Taylor-Wiles Patching

February 20, 2022February 23, 2022 / Anton Hilado / Leave a comment

In Galois Deformation Rings we mentioned the idea of “modularity lifting“, which forms one part of the approach to proving that a Galois representation arises from a modular form, the other part being residual modularity. In that post we also mentioned “R=T” theorems, which are in turn the approach to proving modularity lifting, the “R” standing for the Galois deformation rings that were the main topic of that post, and “T” standing for (a certain localization of) the Hecke algebra. In this post, we shall discuss R=T theorems in a little more detail, and discuss the ideas involved in its proof. We shall focus on the weight $2$ cusp forms (see also Galois Representations Coming From Weight 2 Eigenforms), although many of these ideas can also be generalized to higher weights.

A review of Galois deformation rings and Hecke algebras

Let us recall again the idea behind R=T theorems. We recall from Galois Deformation Rings that if we have a fixed residual representation $\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathbb{F}$ (here $\mathbb{F}$ is some finite field of characteristic $p$ ), we have a Galois deformation ring $R_{\overline{\rho}}$ , with the defining property that maps from $R_{\overline{\rho}}$ into some complete Noetherian local $W(\mathbb{F})$ -algebra $A$ correspond to certain Galois representations over $A$ , namely those which “lift” the residual representation $\overline{\rho}$ . If we compose these maps with maps from $A$ into $\overline{\mathbb{Q}}_{p}$ , we get maps that correspond to certain Galois representations over $\overline{\mathbb{Q}}_{p}$ .

In addition, since we want to match up Galois representations with modular forms (cusp forms of weight $2$ in particular this post), we will want to impose certain conditions on the Galois representations that are parametrized by our deformation ring $R_{\overline{\rho}}$ . For instance, it is known that p-adic Galois representations that arise from a cusp form of weight $2$ and level $\Gamma=\Gamma(N)$ are unramified at all the primes except $p$ and the ones that divide $N$ . There is a way to construct a modification of our deformation ring $R_{\overline{\rho}}$ so that the Galois representations it parametrizes satisfies these conditions (also known as deformation conditions or deformation problems). We shall denote this modified deformation ring simply by $R$ .

On the other hand, maps from the Hecke algebra to some coefficient field (we will choose this to be $\overline{\mathbb{Q}}_{p}$ ; conventionally this is $\mathbb{C}$ , but $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ are isomorphic as fields) correspond to systems of eigenvalues coming from modular forms.

Now the idea is to match up these maps, since then it would be the same as matching Galois representations and modular forms; however, we note that currently our maps from $R_{\overline{\rho}}$ only correspond to Galois representations that come from lifting our fixed Galois representation $\overline{\rho}$ and we have not made any such restriction on the maps from our Hecke algebra, so they don’t quite match up yet.

Galois representations valued in localizations of the Hecke algebra

What we will do to fix this is to come up with a maximal ideal of the Hecke algebra that corresponds to $\overline{\rho}$ , and, instead of considering the entire Hecke algebra, which is too large, we will instead consider the localization of it with respect to this maximal ideal. We have, following the Hodge decomposition (for weights $k>2$ , a generalization of this is given by a theorem of Eichler and Shimura)

$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong S_{2}(\Gamma,\mathbb{C})\oplus \overline{S_{2}(\Gamma,\mathbb{C})}$

where $M_{2}(\Gamma,\mathbb{C})$ (resp. $S_{2}(\Gamma,\mathbb{C})$ ) is the space of modular forms (resp. cusp forms) of weight $2$ and level $\Gamma$ . The advantage of expressing modular forms in this form is that we shall be able to consider them “integrally”. We have that

$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathbb{C}$

Now let $E$ be a finite extension of $\mathbb{Q}_{p}$ , with ring of integers $\mathcal{O}$ , uniformizer $\varpi$ and residue field $\mathbb{F}$ (the same field our residual representation $\overline{\rho}$ takes values in). We can now consider

$\displaystyle H^{1}(Y(\Gamma), \mathcal{O})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathcal{O}$

Let $\Sigma$ be the set consisting of the prime $p$ and the primes dividing the level, which we shall assume to be squarefree (these conditions put us in the minimal case of Tayor-Wiles patching – though the strategy holds more generally, we assume these conditions to simplify our discussion). We have a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ acting on $H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p})$ , and similarly a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acting on $H^{1}(Y(\Gamma), \mathcal{O})$ . Recall that these are the subrings of their respective endomorphism rings generated by the Hecke operators $T_{\ell}$ and $S_{\ell}$ for all $\ell\not\in \Sigma$ (see also Hecke Operators and Galois Representations Coming From Weight 2 Eigenforms). The eigenvalue map

$\displaystyle \lambda_{g}:\mathbb{T}(S(\Gamma,\overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$

which associates to a Hecke operator its eigenvalue on some cusp form $g\in S(\Gamma,\overline{\mathbb{Q}}_{p})$ extends to a map

$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$ .

Now since $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acts on $H^{1}(\Gamma, \mathcal{O})$ we will also have an eigenvalue map

$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathcal{O}$

compatible with the above, in that applying $\lambda_{g}$ followed by embedding the resulting eigenvalue to $\overline{\mathbb{Q}}_{p}$ is the same as composing the map from $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ into $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ first then applying the eigenvalue map. Now we can compose the eigenvalue map to $\mathcal{O})$ with the reduction mod $\varpi$ so that we get $\displaystyle \overline{\lambda}_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathbb{F}$ .

Now let $\mathfrak{m}$ be the kernel of $\overline{\lambda}_{g}$ . This is a maximal ideal of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ . In fact, we can associate to $\lambda_{g}$ a residual representation $\overline{\rho}_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{F})$ , such that the characteristic polynomial of the $\mathrm{Frob}_{\ell}$ is given by $X^{2}-\lambda_{g}(T_{\ell})X+\ell \lambda_{g}(S_{\ell})$ .

Now let $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ be the completion of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ with respect to $\mathfrak{m}$ . It turns out that there is a Galois representation $\rho_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{T}(\Gamma)_{\mathfrak{m}})$ which lifts $\overline{\rho}_{\mathfrak{m}}$ . Furthermore, $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ is a complete Noetherian local $\mathcal{O}$ -algebra!

Putting all of these together, what this all means is that if $\overline{\rho}=\overline{\rho}_{\mathfrak{m}}$ , there is a map $R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$ . Furthermore, this map is surjective. Again, the fact that we have this surjective map reflects that fact that we can obtain Galois representations (of a certain form) from modular forms. Showing that this is an isomorphism amounts to showing that Galois representations of this form always come from modular forms.

Taylor-Wiles patching: Rough idea behind the approach

So now, to prove our “R=T” theorem, we need to show that this map is actually an isomorphism.

Let $M=H^{1}(Y(\Gamma),\mathcal{O})$ . The idea is that $R$ will have an action on $M$ , which will factor through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ . If we can show that $M$ is free as an $R$ -module, then since this action factors through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ via a surjection, then the map from $R$ to $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ must be an isomorphism.

This, by itself, is still too difficult. So what we will do is build an auxiliary module, sometimes called the patched module and denoted $M_{\infty}$ , which is going to be a module over an auxiliary ring we shall denote by $R_{\infty}$ , from which $M$ and $R$ can be obtained as quotients by a certain ideal. The advantage is that we can bring another ring in play, the power series ring $\mathcal{O}[[x_{1},\ldots,x_{q}]]$ , which maps to $R_{\infty}$ (in fact, two copies of it will map to $R_{\infty}$ , which is important), and we will use what we know about power series rings to show that $M_{\infty}$ is free over $R_{\infty}$ , which will in turn show that $M$ is free over $R$ .

In turn, $M_{\infty}$ and $R_{\infty}$ will be built as inverse limits of modules and rings $R_{Q_{n}}$ and $M_{Q_{n}}$ . The subscript $Q_{n}$ refers to a set of primes , called “Taylor-Wiles primes” at which we shall also allow ramification (recall that initially we have imposed the condition that our Galois representations be unramified at all places outside of $p$ and the primes that divide the level $N$ ). As we shall see, these Taylor-Wiles primes will be specially selected so that we will be able to construct $M_{\infty}$ and $R_{\infty}$ with the properties that we will need. This passage to the limit in order to make use of what we know about power series is inspired by Iwasawa theory (see also Iwasawa theory, p-adic L-functions, and p-adic modular forms).

Taylor-Wiles primes

A Taylor-Wiles prime of level $n$ is defined to be a prime $v$ such that the norm $q_{v}$ is congruent to $1$ mod $p^{n}$ , and such that $\overline{\rho}(\mathrm{Frob}_{v})$ has distinct $\mathbb{F}$ -rational eigenvalues. For our purposes we will need, for every positive integer $n$ , a set $Q_{n}$ of Taylor-Wiles primes of cardinality equal to the dimension of the dual Selmer group of $R$ (which we shall denote by $q$ ), and such that the dual Selmer group of $R_{Q_{n}}$ is trivial. It is known that we can always find such a set $Q_{n}$ for every positive integer $n$ .

Let us first look at how this affects the “Galois side”, i.e. $R_{Q_{n}}$ . There is a surjection $R_{Q_{n}}\twoheadrightarrow R$ , but the important property of this, that is due to how the Taylor-Wiles primes were selected, is that the dimensions of their tangent spaces (which is going to be equal to the dimension of the Selmer group as discussed in More on Galois Deformation Rings) are the same.

Now it so happens that, when we are considering $2$ -dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ , the dimensions of the Selmer group and the dual Selmer group will be the same. This is what is known as the numerical coincidence, and is quite special to our case. In general, for instance when instead of $\mathbb{Q}$ we have a more general number field $F$ , this numerical coincidence may not hold (we will briefly discuss this situation at the end of this post). The numerical coincidence, as well as the fact that the dimension of the tangent spaces of $R$ and $R_{Q_{n}}$ remain the same, are both consequences of the Wiles-Greenberg formula, which relates the Selmer group and the dual Selmer group.

Now let us look at the “automorphic side”, i.e. $M_{Q_{n}}$ . We call this the automorphic side because they are localizations of spaces of modular forms (which are automorphic forms). We first need to come up with a new kind of level structure.

Letting $Q_{n}$ be some set of Taylor-Wiles primes, we define $\Gamma_{0}(Q_{n})=\Gamma\cap\Gamma_{0}(\prod_{v\in Q_{n}}v)$ and we further define $\Gamma_{Q_{n}}$ to be such that the quotient $\Gamma_{0}(Q_{n})/\Gamma_{Q_{n}}$ is isomorphic to the group $\Delta_{Q_{n}}$ , defined to be the product over $v\in Q_{n}$ of the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$ .

We define a new Hecke algebra $\mathbb{T}_{Q_{n}}$ obtained from $\mathbb{T}$ by adjoining new Hecke operators $U_{v}$ for every prime $v$ in $Q_{n}$ . We define a maximal ideal $\mathfrak{m}_{Q_{n}}$ of $\mathbb{T}_{Q_{n}}$ generated by the elements of $\mathfrak{m}$ and $U_{v}-\alpha_{v}$ again for every prime $v$ in $Q_{n}$ .

We now define $M_{Q_{n}}$ to be $H^{1}(Y(\Gamma_{Q}),\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$ . This has an action of $\Delta_{Q_{n}}$ and is therefore a $\mathcal{O}[\Delta_{Q_{n}}]$ -module. In fact, $M_{Q_{n}}$ is a free $\mathcal{O}[\Delta_{Q_{n}}]$ -module. This will become important later. Another important property of $M_{Q_{n}}$ is that its $\Delta_{Q_{n}}$ -coinvariants are isomorphic to $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$ .

Now $R_{Q_{n}}$ also has the structure of a $\mathcal{O}[\Delta_{Q_{n}}]$ -algebra. If we take $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(R_{Q_{n}})$ and restrict it to $\mathrm{Gal}(\overline{\mathbb{Q}}_{v}/\mathbb{Q}_{v})$ (for $v$ in $Q_{n}$ ), we get that the resulting local representation is of the form $\eta_{1}\oplus\eta_{2}$ , where $\eta_{1}$ and $\eta_{2}$ are characters. Using local class field theory (see also The Local Langlands Correspondence for General Linear Groups), we obtain a map $\mathbb{Z}_{v}^{\times}\to R_{Q_{n}}^{\times}$ . This map factors through the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$ . Thus given $Q_{n}$ we have a map $\Delta_{Q_{n}}\to R_{Q_{n}}$ .

Now it so happens that the action of $\Delta_{Q_{n}}$ on $M_{Q_{n}}$ factors through the map to $R_{Q_{n}}$ . So therefore we have

$\displaystyle \mathcal{O}[\Delta_{Q_{n}}]\to R_{Q_{n}}\to\mathbb{T}_{Q_{n}}\curvearrowright M_{Q_{n}}$

Taylor-Wiles patching: The patching construction

Now we will perform the patching construction, which means taking the inverse limit over $n$ . First we must show that this is even possible, i.e. that we have an inverse system. We can formalize this via the notion of a patching datum.

We let $S_{\infty}$ denote $\mathcal{O}[[(\mathbb{Z}_{p})^{q}]]\cong \mathcal{O}[[x_{1},\ldots,x_{q}]]$ and let $\mathfrak{a}$ denote the ideal $(x_{1},\ldots,x_{q})$ . Let us also define $R_{\infty}$ to be another power series ring $\mathcal{O}[[y_{1},\ldots,y_{q}]]$ but in a different set of variables of the same number. In the non-minimal case they might look quite different, but in either case there will be a map from $S_{\infty}$ to $R_{\infty}$ ; this may be thought of as the limiting case of the map from $\mathcal{O}[\Delta_{Q_{n}}]$ to $R_{Q_{n}}$ discussed earlier.

Now let $n$ be a positive integer. Let $\mathfrak{a}_{n}$ be the kernel of the surjection $S_{\infty}\twoheadrightarrow \mathcal{O}[(\mathbb{Z}/p^{n}\mathbb{Z})^{q}]$ , let $S_{n}$ be $S_{\infty}/(\varpi^{n},\mathfrak{a}_{n})$ , and $\mathfrak{d}_{n}$ be the ideal $(\varpi^{n},\mathrm{Ann}_{R}(M)^{n})$ . Abstractly, a patching datum of level $n$ is a triple $(f_{n},X_{n},g_{n})$ where

$f_{n}:R_{\infty}\twoheadrightarrow R/\mathfrak{d}_{n}$ is a surjection of complete Noetherian local $\mathcal{O}$ algebras
is a -module, finite free over , such that
- $\mathrm{im}(S_{N}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(R_{\infty}\to\mathrm{End}_{\mathcal{O}}X)$
- $\mathrm{im}(\mathfrak{a}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(\mathrm{ker}(f)\to\mathrm{End}_{\mathcal{O}}X)$
$g_{n}:X/\mathfrak{a}\xrightarrow M/(\varpi^{n})$ is an isomorphism of $R_{\infty}$ -modules

We say that two patching data $(f_{n},X_{n},g_{n})$ and $(f_{n}',X_{n}',g_{n}')$ of level $n$ are isomorphic if $f_{n}=f_{n}'$ and there exists an isomorphism $X_{n}\cong X_{n}'$ compatible with $g_{n}$ and $g_{n}'$ . We note the important fact that there are only finitely many isomorphism classes of patching data for any level $n$ .

Now we will specialize this abstract construction to help us prove our R=T theorem. We choose

$f_{n}:R_{\infty}\twoheadrightarrow R_{Q_{n}}\twoheadrightarrow R\twoheadrightarrow R/\mathfrak{d}_{n}$
$X_{n}=M_{Q_{n}}\otimes_{S_{\infty}} S_{n}$
$g_{n}$ is induced by the isomorphism between the $\Delta_{Q_{n}}$ -coinvariants of $H^{1}(Y_{Q_{n}},\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$ and $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$

If we have a patching datum $D_{m}=(f_{m},X_{m},g_{m})$ of level $m$ , we may form $D_{m}\mod n=D_{m,n}=(f\mod \mathfrak{d}_{n},X_{m}\otimes_{S_{m}} S_{n},g_{m}\otimes_{S_{m}}S_{n})$ which is a patching datum of level $n$ .

Now recall that for any fixed $n$ , we can only have a finite number of isomorphism classes of patching datum of level $n$ . This means we can find a subsequence $(m_{n})_{n\geq 1}$ of $(m)_{m\geq 1}$ such that $D_{m_{n+1},n+1}\mod n\cong D_{m_{n},n}$ .

We can now take inverse limits. Let $M_{\infty}=\varprojlim_{n}X_{m_{n}}$ , let the surjection $R_{\infty}\twoheadrightarrow R$ be given by $\varprojlim_{n}f_{m_{n},n}$ , and let the surjection $M_{\infty}\twoheadrightarrow M$ be given by $\varprojlim_{n}g_{m_{n},n}$ . We have

$\displaystyle \mathcal{O}[[x_{1},\ldots,x_{g}]]\to R_{\infty}\to\mathbb{T}_{\infty}\curvearrowright M_{\infty}$

Just as $M_{Q_{n}}$ is free as a module over $\mathcal{O}[\Delta_{Q_{n}}]$ , we have that $M_{\infty}$ is free as a module over $S_{\infty}$ . We will now use some commutative algebra to show that $M_{\infty}$ is a free $R_{\infty}$ -module. The depth of a module $M'$ over a local ring $R'$ with maximal ideal $\mathfrak{m'}$ is defined to be the minimum $i$ such that $\mathrm{Ext}^{i}(R'/\mathfrak{m}',M')$ is nonzero. The depth of a module is always bounded above by its dimension.

Now the dimension of $R_{\infty}$ is $1+q$ (we know this since we defined it as a power series $\mathcal{O}[[y_{1},\ldots,y_{q}]]$ ). This bounds $\mathrm{dim}_{R_{\infty}}(M_{\infty})$ , and by the above fact regarding the depth of a module, $\mathrm{dim}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{R_{\infty}}(M_{\infty})$ . Since the action of $S_{\infty}$ on $M_{\infty}$ factors through the action of $R_{\infty}$ , $\mathrm{depth}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{S_{\infty}}(M_{\infty})$ . Finally, since $M_{\infty}$ is a free $S_{\infty}$ -module, we have that $\mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$ . In summary,

$\displaystyle 1+q=\mathrm{dim}(R_{\infty})\geq \mathrm{dim}_{R_{\infty}}(M_{\infty})\geq\mathrm{depth}_{R_{\infty}}(M_{\infty})\geq \mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$

and we can see that all of the inequalities are equalities, and all the quantities are equal to $1+q$ . The Auslander-Buchsbaum formula from commutative algebra tells us that

$\displaystyle \mathrm{proj.dim}_{R_{\infty}}(M_{\infty})=\mathrm{depth}(R_{\infty})-\mathrm{depth}_{R_{\infty}}(M_{\infty})$

and since both terms on the right-hand side are equal to $1+q$ , the right-hand side is zero. Therefore the projective dimension of $M_{\infty}$ relative to $R_{\infty}$ is zero, which means that $M_{\infty}$ is a projective module over $R_{\infty}$ . Since $R_{\infty}$ is local, this is the same as saying that $M_{\infty}$ is a free $R_{\infty}$ -module.

We have that $M\cong M_{\infty}/\mathfrak{a}M_{\infty}$ is a free module over $R_{\infty}/\mathfrak{a}R_{\infty}$ . Since this action factors through maps $R_{\infty}/\mathfrak{a}R_{\infty}\to R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$ which are all surjections, they have to be isomorphisms, and we have that $M$ is a free $R$ -module, and therefore $R\cong\mathbb{T}(\Gamma)_{\mathfrak{m}}$ . This proves our R=T theorem.

Generalizations and other applications of Taylor-Wiles patching

We have discussed only the “minimal case” of Taylor-Wiles patching, but one can make use of the same ideas for the non-minimal case, and one may also apply Taylor-Wiles patching to show the modularity of $2$ -dimensional representations of $\mathrm{Gal}(\overline{F}/F)$ for $F$ a totally real field (in this case on the automorphic side we would have Hilbert modular forms).

However, when $F$ is a more general number field the situation is much more complicated, because one of the facts that we have used, which is vital to Taylor-Wiles patching, now fails. This is the fact that the dimension of the dual Selmer group (which is the cardinality of our sets of Taylor-Wiles primes) and the dimension of the Selmer group (which is also the dimension of the tangent space of the Galois deformation ring $R$ ) are equal (again this is what is known as the “numerical coincidence”). This is the important property that can fail for more general number fields. Here the dimensions of the dual Selmer group and the Selmer group may differ by some nonzero quantity $\delta$ .

Moreover, in our discussion we made use of the fact that the cohomology was concentrated in a single degree. For more general number fields this is no longer true. Instead we will have some interval for which the cohomology is nonzero. However, it so happens (for certain “nice” cases) that the length of this interval is equal to $\delta+1$ . This is a hint that the two complications are related, and in fact can be played off each other so that they “cancel each other out” in a sense. Instead of patching modules, in this case one patches complexes instead. These ideas were developed in the work of Frank Calegari and David Geraghty.

The method of Taylor-Wiles patching is also being put forward as an approach to the p-adic local Langlands correspondence (which is also closely related to modularity as we have seen in Completed Cohomology and Local-Global Compatibility), via the work of Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin. This is also closely related to the ideas discussed at in Moduli Stacks of (phi, Gamma)-modules (where we used the same notation $M_{\infty}$ for the patched module). Namely, we expect a coherent sheaf $\mathcal{M}$ on the moduli stack of $\varphi,\Gamma$ -modules which, “locally” coincides or is at least closely related to the patched module $M_{\infty}$ . This has applications not only to the p-adic local Langlands correspondence as mentioned above, but also to the closely-related Breuil-Mezard conjecture. We will discuss these ideas and more in future posts.

References:

Modularity Lifting (Course Notes) by Patrick Allen

Modularity Lifting Theorems by Toby Gee

Beyond the Taylor-Wiles Method by Jack Thorne

Motives and L-functions by Frank Calegari

Overview of the Taylor-Wiles Method by Andrew Snowden (lecture notes from the Stanford Modularity Lifting Seminar)

Reciprocity in the Langlands Program Since Fermat’s Last Theorem by Frank Calegari

Modularity Lifting Beyond the Taylor-Wiles Method by Frank Calegari and David Geraghty

Patching and the p-adic local Langlands Correspondence by Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin

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

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