# Taylor-Wiles Patching

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
• $X_{n}$ is a $R_{\infty}\otimes_{\mathcal{O}} S_{n}$-module, finite free over $S_{n}$, 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

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 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

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

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

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

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

# Moduli Stacks of (phi, Gamma)-modules

In Moduli Stacks of Galois Representations we discussed the moduli stack of representations of the absolute Galois group $G_{K}:=\mathrm{Gal}(\overline{K}/K)$ when $K$ is a finite extension of $\mathbb{Q}_{p}$, for representations valued in some $\mathbb{Z}_{\ell}$-algebra, where $\ell$ is a prime number different from $p$. When $\ell=p$ however, it turns out using the same strategy as for $\ell\neq p$ can lead to some moduli stacks that are difficult to study.

Instead, we are going to use the theory of $(\varphi,\Gamma)$-modules as an intermediary. Let $k$ be the residue field of $K$, and let $\mathbf{A}_{K}$ be the p-adic completion of the Laurent series field $W(k)((T))$ (where $W(k)$ here denotes the ring of Witt vectors over $k$). For future reference, let us now also state that we will use $\mathbf{A}_{K}^{+}$ to denote $W(k)\otimes_{\mathbb{Z}_{p}} A[[T]]$. An etale $(\varphi,\Gamma)$-module over $\mathbf{A}_{K}$ is a finite $\mathbf{A}_{K}$-module equipped with commuting semilinear actions of $\varphi$ (an endomorphism coming from the Frobenius of $W(k)$) and $\Gamma$ (the subgroup of $\mathrm{Gal}(K(\zeta_{p^{\infty}})/K)$ isomorphic to $\mathbb{Z}_{p}$, see also Iwasawa theory, p-adic L-functions, and p-adic modular forms). One of the important facts about etale $(\varphi,\Gamma)$-modules is the following:

The category of etale $(\varphi,\Gamma)$-modules is equivalent to the category of continuous $G_{K}$-modules over finite $\mathbb{Z}_{p}$-modules.

This equivalence is given more explicit as follows. Let $\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}$ be the p-adic completion of the ring of integers of the maximal unramified extension of $\mathbf{A}_{K}[1/p]$ in $W(\mathcal{O}_{\mathbb{C}_{p}}^{\flat})$. Then to obtain a $G_{K}$-module $V$ from a $\varphi,\Gamma$-module $M$, we take

$\displaystyle V=(\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}\otimes_{\mathbf{A}_{K}} M)^{\varphi=1}$

and to obtain the $(\varphi,\Gamma)$-module $M$ from the $G_{K}$– module $V$, we take

$\displaystyle M=(\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}\otimes_{\mathbb{Z}_{p}} V)^{G_{K_{\mathrm{cyc}}}}$

where $K_{\mathrm{cyc}}$ is $K$ adjoin all the $p$-power roots of unity. Note that if we wanted bonafide Galois representations instead of $G_{K}$-modules we can invert $p$, and our Galois representations will be over $\mathbb{Q}_{p}$. They will be equivalent to $(\varphi,\Gamma)$-modules over $\mathbf{A}_{K}[1/p]$.

More generally we can consider etale $(\varphi,\Gamma)$-modules with coefficients in $A$, where $A$ is some $\mathbb{Z}_{p}$-algebra. This means they are finite $\mathbf{A}_{K,A}$-modules, where $\mathbf{A}_{K,A}$ is the p-adic completion of $W(k)\otimes_{\mathbb{Z}_{p}}A((T))$, instead of $\mathbf{A}_{K}$-modules (again for future reference, we note that $\mathbf{A}_{K,A}^{+}$ will be used for $W(k)\otimes_{\mathbb{Z}_{p}}A[[T]]$). The category of etale $(\varphi,\Gamma)$-modules with coefficients in $A$ is equivalent to the category of continuous $G_{K}$-representations over finite $A$-modules.

We define the stack $\mathcal{X}_{d}$ by letting $\mathcal{X}_{d}(A)$ be the groupoid of etale $(\varphi,\Gamma)$-modules with coefficients in $A$ for $A$ a p-adically complete $\mathbb{Z}_{p}$-algebra and are projective of rank $d$. That this is indeed a stack follows from the notion of a finitely generated projective module $\mathbf{A}_{A}$-module being local for the fpqc topology. But $\mathcal{X}_{d}$ has more structure than just being a stack. It is an “ind-algebraic” stack, a notion which we shall explain in the next few paragraphs. As the approach we outline in this post was originally developed by Matthew Emerton and Toby Gee, the stack $\mathcal{X}_{d}$ is also known as the Emerton-Gee stack.

### The ind-algebraicity of the moduli stack $\mathcal{X}_{d}$

As stated earlier, the moduli stack $\mathcal{X}_{d}$ is an ind-algebraic stack. This means it can be written as the 2-colimit $\varinjlim_{i\in I} \mathcal{X}_{d,i}$ in the 2-category of stacks of 2-directed system $\lbrace\mathcal{X}_{d,i}\rbrace_{i\in I}$ of algebraic stacks $\mathcal{X}_{d,i}$. (Recall from Algebraic Spaces and Stacks that an algebraic stack is an fppf stack $\mathcal{Y}$ such that there exists a map from an affine scheme $U$ to $\mathcal{Y}$ and this map is representable by algebraic spaces, surjective, and smooth).

To understand why $\mathcal{X}_{d}$ is an ind-algebraic stack, we need to understand it as the scheme-theoretic image of a certain map of certain moduli stacks. The idea is that the target stacks (which is going to be the moduli stack $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ of $\varphi$-modules with a semilinear action of the “discretization” of $\Gamma$, more on this later) is ind-algebraic and we can deduce the ind-algebraicity of $\mathcal{X}_{d}$ from this.

First let us explain what “scheme-theoretic image” means. Let us suppose that we have a limit-preserving fppf sheaf $\mathcal{F}$ whose diagonal is representable by algebraic spaces and a proper morphism $\mathcal{X}\to\mathcal{F}$ where $\mathcal{X}$ is an algebraic stack. If $\mathcal{F}$ admits versal rings satisfying certain properties at all finite-type points then there exists an algebraic closed stack $\mathcal{Z}\hookrightarrow \mathcal{X}$ such that $\mathcal{X}\to\mathcal{F}$ factors through $\mathcal{X}\to\mathcal{Z}$ and this map is scheme-theoretically dominant.

We will need to discuss moduli stacks $\mathcal{R}_{d}$ of rank $d$ $\varphi$-modules, moduli stacks $\mathcal{C}_{d,h}$ of rank $d$ $\varphi$-modules of $F$-height at most $h$, and moduli stacks $W_{d,h}$ of rank $d$ weak Wach modules of $T$-height at most $h$.

We define $\mathcal{R}_{d}^{a}$ to be the stack over $\mathbb{Z}/p^{a}\mathbb{Z}$ such that for any $\mathbb{Z}/p^{a}\mathbb{Z}$-algebra $A$, $\mathcal{R}_{d}^{a}(A)$ is the groupoid of all $\varphi$-modules which are projective and of rank $d$. We have that $\mathcal{R}_{d}^{a}$ is also a stack over $\mathbb{Z}_{p}$, and we can define $\mathcal{R}_{d}=\varinjlim_{a}\mathcal{R}_{d}^{a}$, which is a stack over $\mathbb{Z}_{p}$ which we may think of as the moduli space of $\varphi$-modules which are projective and of rank $d$.

Let $F$ be a polynomial in $W(k)[T]$ which is congruent to a power of $T$ modulo $p$ and let $h$ be a nonnegative integer. A $\varphi$-module of $F$-height at most $h$ over $\mathbf{A}_{K,A}^{+}$ is a finitely generated $T$-torsion free $\mathbf{A}_{K,A}^{+}$-module $\mathfrak{M}$ together with a $\varphi$-semilinear map $\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$ such that the map $1\otimes \varphi_{\mathfrak{M}}:\varphi^{*}\mathfrak{M}\to\mathfrak{M}$ is injective, and whose cokernel is annihilated by $F^{h}$. We let $\mathcal{C}_{d,h}$ be the stack such that $\mathcal{C}_{d,h}(A)$ is the groupoid of $\varphi$-module of $F$-height at most $h$ over $\mathbf{A}_{K,A}^{+}$ which are projective of rank $d$.

In the special case that the polynomial $F$ is the minimal polynomial of the uniformizer of $K$, a $\varphi$-module of $F$-height at most $h$ over $\mathbf{A}_{K,A}^{+}$ is also called a Breuil-Kisin module of height at most $h$. We will encounter Breuil-Kisin modules again later.

We have the following important properties of the stacks $\mathcal{C}_{d,h}^{a}$ and $\mathcal{R}_{d}^{a}$:

• The moduli stack $\mathcal{C}_{d,h}^{a}$ is an algebraic stack of finite presentation over $\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$, with affine diagonal.
• The moduli stack $\mathcal{R}_{d}^{a}$ is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
• The morphism $\mathcal{C}_{d,h}^{a}\to \mathcal{R}_{d}^{a}$ is representable by algebraic spaces, proper, and of finite presentation.
• The diagonal morphism $\Delta:\mathcal{R}_{d}^{a}\to\mathcal{R}_{d}^{a}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathcal{R}_{d}^{a}$ is representable by algebraic spaces, affine, and of finite presentation.

These properties were shown by Emerton and Gee following a strategy originally employed by George Pappas and Michael Rapoport involving relating these stacks to the affine Grassmannian. After taking limits over $a$, we then have the following:

• The moduli stack $\mathcal{C}_{d,h}$ is an p-adic formal algebraic stack of finite presentation over $\mathrm{Spf}(\mathbb{Z}_{p})$, with affine diagonal.
• The moduli stack $\mathcal{R}_{d}$ is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
• The morphism $\mathcal{C}_{d,h}\to \mathcal{R}_{d}$ is representable by algebraic spaces, proper, and of finite presentation.
• The diagonal morphism $\Delta:\mathcal{R}_{d}\to\mathcal{R}_{d}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathcal{R}_{d}$ is representable by algebraic spaces, affine, and of finite presentation.

In the above, a formal algebraic stack is defined similarly to an algebraic stack except our atlas, instead of being a scheme, is a disjoint union of formal schemes (see also Formal Schemes), and we say that a formal algebraic stack over $\mathrm{Spec}(\mathbb{Z}_{p})$ is a p-adic formal algebraic stack if it admits a morphism to $\mathrm{Spf}(\mathbb{Z}_{p})$ that is representable by an algebraic stack.

Now let $\gamma$ be a topological generator of $\Gamma$, and let $\Gamma_{\mathrm{disc}}=\langle\gamma\rangle$, so that $\Gamma_{\mathrm{disc}}\cong\mathbb{Z}$. Let

$\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}:=\mathcal{R}_{d}\times_{\Delta,\mathcal{R}_{d}\times\mathcal{R}_{d},\Gamma_{\gamma}}\mathcal{R}_{d}$

be the moduli stack of projective etale $\varphi$-modules of rank $d$ together with a semilinear action of $\Gamma_{\mathrm{disc}}$ (in the above $\Delta$ is the diagonal and $\Gamma_{\gamma}$ is the graph of $\gamma$). The stack $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ is an ind-algebraic stack, which follows from the properties stated earlier. Now the stack $\mathcal{X}_{d}$ maps into $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$, however since it may not be a closed substack this is not yet enough to prove the ind-algebraicity of $\mathcal{X}_{d}$. So we need to exhibit it as the scheme-theoretic image of an appropriate map into $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$, and this is where the weak Wach modules come in.

A rank $d$ projective weak Wach module of $T$-height at most $h$ and level at most $s$ over $\mathbf{A}_{K,A}^{+}$ is a rank $d$ projective $\varphi$-module $\mathfrak{M}$ of $T$-height at most $h$ over $\mathbf{A}_{K,A}^{+}$, such that $\mathfrak{M}[1/T]$ has a semilinear action of $\Gamma_{\mathrm{disc}}$ satsifying $(\gamma^{p^{s}}-1)\mathfrak{M}\subseteq T\mathfrak{M}$.

Let $\mathcal{W}_{d,h,s}$ be the moduli stack of rank $d$ projective weak Wach modules, of $T$-height at most $h$, and level at most $s$. This is a p-adic formal algebraic stack of finite presentation over $\mathbb{Z}_{p}$. To show this we make the following steps.

We consider the fiber product $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}\times_{\mathcal{R}_{d}}\mathcal{C}_{d,h}$. This is the moduli stack of rank $d$ projective $\varphi$-modules $\mathfrak{M}$ over $\mathbf{A}_{K,A}^{+}$ of $T$-height at most $h$, equipped with a semilinear action of $\Gamma_{\mathrm{disc}}$ on $\mathfrak{M}[1/T]$. It is a p-adic formal algebraic stack of finite presentation over $\mathrm{Spf}(\mathbb{Z}_{p})$.

Now consider $\mathcal{W}_{d,h}$, the moduli stack of rank $d$ projective weak Wach modules of height at most $h$. We have an isomorphism $\varinjlim_{s}\mathcal{W}_{d,h,s}\xrightarrow{\sim}\mathcal{W}_{d,h}$, and $\mathcal{W}_{d,h}$ has a closed immersion of finite presentation into the fiber product $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}\times_{\mathcal{R}_{d}}\mathcal{C}_{d,h}$.

We let $\mathcal{W}_{d,h,s}^{a}:=\mathcal{W}_{d,h,s}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$. This is a closed substack of $\mathcal{W}_{d,h,s}$. We define $\mathcal{X}_{d,h,s}^{a}$ to be the scheme-theoretic image of the composition $\mathcal{W}_{d,h,s}^{a}\hookrightarrow\mathcal{W}_{d,h,s}\to\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$. The stack $\mathcal{X}_{d,h,s}^{a}$ is a closed substack of $\mathcal{X}_{d}$, and in fact we will see that $\varinjlim\mathcal{X}_{d,h,s}^{a}$ is isomorphic to $\mathcal{X}_{d}$.

Let us explain very briefly how the last statement works. The existence of a morphism from $\mathcal{X}_{d,h,s}^{a}$ to $\mathcal{X}_{d}$ (which factors through $\mathcal{X}_{d}^{a})$ basically comes down to being able to extend the action of $\Gamma_{\mathrm{disc}}$ to a continuous action of $\Gamma$.

Now to show that the morphism $\varinjlim\mathcal{X}_{d,h,s}^{a}\to\mathcal{X}$, we have to show that for any $\mathbb{Z}/p^{a}\mathbb{Z}$-algebra $A$ any morphism $\mathrm{Spec}(A)\to\mathcal{X}_{d}$ must factor through $\mathcal{X}_{d,h,s}^{a}$, for some $h$ and some $s$. It is in fact enough to show this for $B$ such that there is a scheme-theoretically dominant map $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ and such that if $M$ is the $(\varphi,\Gamma)$-module corresponding to $\mathrm{Spec}(A)\to\mathcal{X}_{d}$, then $M_{B}$ is free. The freeness of $M_{B}$ allows us to find a $\varphi$-invariant lattice $\mathfrak{M}$ inside it which corresponds to a weak Wach module over $B$. Associating $M_{B}$ to $\mathfrak{M}$ gives us a map $\mathcal{W}_{d,h,s}^{a}\to\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$. Recalling that the scheme-theoretic image of $\mathcal{W}_{d,h,s}^{a}$ in $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ is $\mathcal{X}_{d,h,s}^{a}$, we see that our map $\mathrm{Spec}(B)\to\mathcal{X}_{d}$ factors through $\mathcal{X}_{d,h,s}^{a}$ and thus $\varinjlim\mathcal{X}_{d,h,s}^{a}\to\mathcal{X}$ is an isomorphism. The existence of $B$ satisfying such properties is guaranteed by the work of Emerton and Gee.

### Crystalline moduli stacks

We briefly mentioned in p-adic Hodge Theory: An Overview that Galois representations that come from the etale cohomology of some scheme are expected to have certain properties related to p-adic Hodge theory (this is part of the Fontaine-Mazur conjecture), It will therefore be interesting to us to have a moduli space of Galois representations that satisfy such p-adic Hodge-theoretic properties. Namely, we can investigate the moduli space of crystalline and semistable representations, and there are going to be corresponding substacks $\mathcal{X}_{d}^{\mathrm{crys},\underline{\lambda}}$ and $\mathcal{X}_{d}^{\mathrm{ss},\underline{\lambda}}$ of $\mathcal{X}_{d}$.

Let $A_{\mathrm{inf},A}$ denote $\varprojlim_{a}(\varprojlim_{i}(W_{a}(\mathcal{O}_{\mathbb{C}_{p}}^{\flat})\otimes_{\mathbb{Z}_{p}}A)/v^{i})$, where $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ denotes the tilt of the ring of integers of the p-adic complex numbers (see also Perfectoid Fields) and $v$ is an element of the maximal ideal of $W_{a}\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ whose image in $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ is nonzero. A Breuil-Kisin-Fargues module of height at most $h$ with $A$-coefficients is a finitely generated $A_{\mathrm{inf},A}$-module $\mathfrak{M}^{\mathrm{inf}}$ together with a $\varphi$-semilinear map $\varphi_{\mathfrak{M}^{\mathrm{inf}}}:\mathfrak{M}^{\mathrm{inf}}\to\mathfrak{M}^{\mathrm{inf}}$ such that the map $1\otimes \varphi_{\mathfrak{M}^{\mathrm{inf}}}:\varphi^{*}\mathfrak{M}^{\mathrm{inf}}\to\mathfrak{M}^{\mathrm{inf}}$ is injective, and whose cokernel is annihilated by $E(u)^{h}$, where $E(u)$ is the minimal polynomial of the uniformizer of $K$. A Breuil-Kisin-Fargues $G_{K}$-module of height at most $h$ is a Breuil-Kisin-Fargues module of height at most $h$ together with a semilinear $G_{K}$ action that commutes with $\varphi$.

Let us note that given a Breuil-Kisin module $\mathfrak{M}$, we can obtain a Breuil-Kisin-Fargues module $\mathfrak{M}^{\mathrm{inf}}$ by taking $\mathfrak{M}^{\mathrm{inf}}=A_{\mathrm{inf},A}\otimes_{\mathbf{A}_{K,A}^{+}}\mathfrak{M}$. To be able to take the tensor product we need a map from $\mathbf{A}_{K,A}^{+}$ to $A_{\mathrm{inf},A}$, which in this case is provided by sending the element $T$ in $\mathbf{A}_{K,A}^{+}\cong W(k)\otimes_{\mathbb{Z}_{p}}A[[T]]$ to a compatible system of p-power roots of the uniformizer $\pi$ in $A_{\mathrm{inf},A}$ (we also say that we are in the “Kummer case“, as opposed to the “cyclotomic case” where p-power roots of unity are used; in the literature the symbol $\mathfrak{S}_{A}$ is also used in place of $\mathbf{A}_{K,A}^{+}$, which is reserved for the cyclotomic case; note also that $G_{K_{\mathrm{cyc}}}$ will be replaced by $G_{K_{\infty}}$ in this case, $K_{\infty}$ being $K$ adjoin all p-power roots of $\pi$).

There is a notion of a Breuil-Kisin-Fargues $G_{K}$-module of height at most $h$ admitting all descents. This means that, for every $\pi$ a uniformizer of $K$ and every $\pi^{\flat}$ the p-power roots of $\pi$ in $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$, we can find a Breuil-Kisin module $\mathfrak{M}_{\pi^{\flat}}$ inside the part of the Breuil-Kisin-Fargues module $\mathfrak{M}^{\mathrm{inf}}$ fixed by the absolute Galois group of the field obtained by adjoining all p-power roots of $\pi$ to $K$ (satisfying some conditions related to certain submodules being independent of the choice of $\pi$ and $\pi^{\flat}$). If $\mathfrak{M}^{\mathrm{inf}}$ is a Breuil-Kisin-Fargues $G_{K}$-module and $L$ is a finite extension of $K$, we say that $\mathfrak{M}^{\mathrm{inf}}$ admits all descents over $L$ if the Breuil-Kisin Fargues $G_{L}$-module obtained by restricting the $G_{K}$ action to $G_{L}$ admits all descents.

Let $\mathfrak{M}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$-module of height at most $h$ admitting all descents. We say that $\mathfrak{M}^{\mathrm{inf}}$ is crystalline if, for all $g\in G_{K}$ and for any choice of $\pi$ and $\pi^{\flat}$ we have

$\displaystyle (g-1)(\mathfrak{M}_{\pi^{\flat}})\subset \varphi^{-1}([\varepsilon]-1)[\pi^{\flat}]\mathfrak{M}^{\mathrm{inf}}$.

As the name implies, the importance of the crystalline condition is that it gives rise to crystalline Galois representations (see p-adic Hodge Theory: An Overview). To obtain a Galois representation from a Breuil-Kisin-Fargues $G_{K}$-module $\mathfrak{M}^{\mathrm{inf}}$ of height at most $h$ admitting all descents, first we take $M=W(\mathbb{C}_{p}^{\flat})\otimes_{A_{\mathrm{inf}}}\mathfrak{M}^{\mathrm{inf}}$. Then $M$ is a $(G_{K},\varphi)$-module. Then we can take $M^{\varphi=1}$ to get a $G_{K}$-module, and finally we can tensor with $\mathbb{Q}_{p}$ to get a Galois representation, which we shall denote by $V(M)$. As hinted at earlier, the Galois representation $V(M)$ will be crystalline if and only if $\mathfrak{M}^{\mathrm{inf}}$ is crystalline. Furthermore $V(M)$ will have Hodge-Tate weights in the range $[0,h]$ if and only if $\mathfrak{M}^{\mathrm{inf}}$ has height at most $h$.

Let $\mathcal{C}_{d,\mathrm{crys},h}^{a}$ be the limit-preserving category of groupoids over $\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$ such that $\mathcal{C}_{d,\mathrm{crys},h}^{a}(A)$, for $A$ a finite type $\mathbb{Z}/p^{a}\mathbb{Z}$-algebra, is the groupoid of Breuil-Kisin-Fargues $G_{K}$-modules with $A$-coefficients of height at most $h$, admitting all descents, and crystalline. We let $\mathcal{C}_{\mathrm{d,crys},h}:=\varinjlim_{a}\mathcal{C}_{d,\mathrm{crys},h}^{a}$.

There is a map from $\mathcal{C}_{d,\mathrm{crys},h}$ to $\mathcal{X}_{d}$ given by sending a Breuil-Kisin-Fargues $G_{K}$-module $\mathfrak{M}$ to the $(\varphi,\Gamma)$-module $\mathfrak{M}^{\mathrm{inf}}\otimes_{\mathbf{A}_{\mathrm{inf},A}}W(C^{\flat})_{A}$. We now let $\mathcal{C}_{d,\mathrm{crys},h}^{\mathrm{fl}}$ be the maximal substack of $\mathcal{C}_{d,\mathrm{crys},h}$ which is flat over $\mathrm{Spf}(\mathbb{Z}_{p})$, and define $\mathcal{X}^{\mathrm{crys},h}$ to be the scheme-theoretic image of $\mathcal{C}_{d,\mathrm{crys},h}^{\mathrm{fl}}$ under the map from $\mathcal{C}_{d,\mathrm{crys},h}$ to $\mathcal{X}_{d}$ as described above.

Let us now introduce the notion of Hodge types. A Hodge type is a set of tuples $\underline{\lambda}=\lbrace\lambda_{\sigma,i}\rbrace_{\sigma:K\hookrightarrow\overline{\mathbb{Q}}_{p},1\leq i\leq d}$ of nonnegative integers such that $\lambda_{\sigma,i}\leq\lambda_{\sigma,i+1}$ for all $\sigma$ and all $1\leq i\leq d-1$. A Hodge type is regular if $\lambda_{\sigma,i}<\lambda_{\sigma,i+1}$ for all $\sigma$ and all $1\leq i\leq d-1$.

We also have the notion of an inertial type, which is defined to be a $\overline{\mathbb{Q}}_{p}$-representation of the inertia group $I_{K}$ which extends to a representation of the Weil group $W_{K}$ with open kernel (which implies that it has finite image).

We can associate to a Breuil-Kisin-Fargues $G_{K}$-module a Hodge type and an inertial type as we now discuss. We let $E$ be a finite extension of $\mathbb{Q}_{p}$ large enough so that it contains all the embeddings of $K$ into $\mathbb{C}_{p}$. Let $A^{\circ}$ be a p-adically complete flat $\mathcal{O}_{E}$-algebra topologically of finite type over $\mathcal{O}_{E}$, and let $A=A^{\circ}[1/p]$. Let $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$ module with $A^{\circ}$ coefficients admitting all descents over $L$. We write $\mathfrak{M}_{A^{\circ}}$ for the associated Breuil-Kisin module, and define $\mathrm{Fil}^{i}\varphi^{*}\mathfrak{M}_{A^{\circ}}:=(1\otimes\varphi_{\mathfrak{M}_{A^{\circ}}})^{-1} (E(u)^{i}\varphi^{*}\mathfrak{M}_{A^{\circ}})$. We write

$\displaystyle D_{\mathrm{dR}}(\mathfrak{M}^{\mathrm{inf}})=((\varphi^{*}\mathfrak{M}_{A^{\circ}}/E(u)\varphi^{*}\mathfrak{M}_{A^{\circ}})\otimes_{A^{\circ}}A)^{\mathrm{Gal}(L/K)}$

and

$\displaystyle \mathrm{Fil}^{i}D_{\mathrm{dR}}(\mathfrak{M}^{\mathrm{inf}})=(\mathrm{Fil}^{i}(\varphi^{*}\mathfrak{M}_{A^{\circ}}/E(u)\mathrm{Fil}^{i-1}\varphi^{*}\mathfrak{M}_{A^{\circ}})\otimes_{A^{\circ}}A)^{\mathrm{Gal}(L/K)}$

We have the decomposition $K\otimes_{\mathbb{Q}_{p}}A=\prod_{\sigma:K\hookrightarrow E}A$. We have idempotents $e_{\sigma}$ corresponding to each factor of this decomposition, and we have the decomposition

$\displaystyle D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})=\prod_{\sigma:K\hookrightarrow E} e_{\sigma}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$

of the $K\otimes_{\mathbb{Q}_{p}}A$-module $D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$ into $A$-modules $e_{\sigma}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$.

Now let $\underline{\lambda}$ be a Hodge type. We say that a Breuil-Kisin-Fargues $G_{K}$-module $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ has Hodge type $\underline{\lambda}$ if $e_{\sigma}\mathrm{Fil}^{i}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$ has constant rank equal to $\#\lbrace j\vert\lambda_{\sigma\vert K,j}\geq i\rbrace$.

Now on to inertial types. Let $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$-module admitting all descents over $L$ and let $\mathfrak{M}_{A^{\circ},\pi^{\flat}}$ be the associated Breuil-Kisin module. Consider $\overline{\mathfrak{M}}_{A^{\circ}}=\mathfrak{M}_{A^{\circ},\pi^{\flat}}/[\pi^{\flat}]\mathfrak{M}_{A^{\circ},\pi^{\flat}}$, a submodule of $W(\overline{k})\otimes_{A_{\mathrm{inf}},A}\mathfrak{M}^{\mathrm{inf}}$. Let $\ell$ be the residue field of $L$ and let $L_{0}=W(\ell)[1/p]$. We have a $W(\ell)\otimes_{\mathbb{Z}_{p}}A$-semilinear action of $\mathrm{Gal}(L/K)$ on $\mathfrak{M}_{A^{\circ},\pi^{\flat}}$ induced from the action of $G_{K}$ on $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$, which in turn induces an action of $I_{L/K}$ on the $L_{0}\otimes A$-module $\overline{\mathfrak{M}}_{A^{\circ}}\otimes_{A^{\circ}} A$.

Fix an embedding $\sigma:L_{0}\hookrightarrow E$. As before we have a corresponding idempotent $e_{\sigma}$. Now let $\tau$ be an inertial type. Given a Breuil-Kisin-Fargues $G_{K}$-module we say that it has inertial type $\tau$ if as an $I_{L/K}$-module, $e_{\sigma} \overline{\mathfrak{M}}_{A^{\circ}}\otimes_{A^{\circ}} A$ is isomorphic to the base change of $\tau$ to $A$.

We now define $\mathcal{C}_{d,\mathrm{crys},h}^{L/K,\mathrm{fl},\underline{\lambda},\tau}$ to be the moduli stacks of Breuil-Kisin-Fargues $G_{K}$-modules of rank $d$, height at most $h$, and admitting all descents to $L$, that give rise to Galois representations which become crystalline over $L$ and with associated Hodge type $\underline{\lambda}$ and inertial type $\tau$. We define $\mathcal{X}_{d,\mathrm{crys}}^{\underline{\lambda},\tau}$ to be the scheme-theoretic image of $\mathcal{C}_{d,\mathrm{crys},h}^{L/K,\mathrm{fl},\underline{\lambda},\tau}$ in $\mathcal{X}$.

It is known, via what we know about the corresponding versal rings $R_{d,\mathrm{crys}}^{\underline{\lambda},\tau}$, that the moduli stacks $\mathcal{X}_{d,\mathrm{crys}}^{\underline{\lambda},\tau}\otimes_{\mathrm{Spf}\mathcal{O}}\mathbb{F}$ are equidimensional of dimension equal to the quantity

$\displaystyle \sum_{\sigma}\#\lbrace1\leq i

In particular, if $\underline{\lambda}$ is a regular Hodge type, then this quantity is equal to $[K:\mathbb{Q}_{p}]d(d-1)/2$. This plays a role in the formulation of the geometric Breuil-Mezard conjecture as we shall see later.

### The reduced substack $\mathcal{X}_{d,\mathrm{red}}$

Let us now consider the reduced substack $\mathcal{X}_{d,\mathrm{red}}$. This is an algebraic stack of finite presentation over $\mathbb{F}_{p}$, equidimensional of dimension $[K:\mathbb{Q}_{p}]d(d-1)/2$, and its irreducible components are labeled by Serre weights.

To see more explicitly the geometry of $\mathcal{X}_{d,\mathrm{red}}$ let us focus on the case $K=\mathbb{Q}_{p}$ and $d=1,2$.

For $d=1$ we are looking at characters $G_{K}\to\overline{\mathbb{F}}_{p}^{\times}$. These are of the form $\mathrm{ur}_{a}\overline{\varepsilon}^{i}$. In the picture of $(\varphi,\Gamma)$-modules, these are obtained from the trivial $(\varphi,\Gamma)$-module over $\mathbf{A}_{\mathbb{Q}_{p},\mathbb{F}_{p}}$ by twisting $\varphi$ by $a$ and twisting $\Gamma$ by $\overline{\varepsilon}^{i}$. For each $i$ the representations are therefore parametrized by $\mathbb{G}_{m}$, but we also have automorphisms parametrized by $\mathbb{G}_{m}$.

For $d=2$, the irreducible representations are of the form $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega_{2}^{i}$. These form a $0$-dimensional substack inside $\mathcal{X}_{2}$. The reducible ones which are of the form

$\displaystyle \begin{pmatrix}\mathrm{ur}_{ab}\overline{\varepsilon}^{i}&*\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{j}\end{pmatrix}$

will belong to the irreducible component of $\mathcal{X}_{2,\mathrm{red}}$ labeled by the Serre weight $\mathrm{Sym}^{i-j-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{j}$ (this is unambiguous except in the case where $i-j=1$ or $i-j=p$, in which case the component labeled by $i-j=1$ is one where the representations with $a=1$ are dense, and the component labeled by $i-j=p$ is one where the representations with $a\neq 1$ are dense). Such a representation will correspond to a closed point if it is semisimple.

More generally, given a family of Galois representations, Emerton and Gee outline a way to construct extensions of this family by some irreducible Galois representation.

Suppose we have a family of $d$-dimensional Galois representations $\overline{\rho}_{T}$ parametrized by a reduced finite scheme $T$ (this family corresponds to a map $T\to\mathcal{X}_{\mathrm{red}}$). Let $\overline{\alpha}$ be a fixed Galois representation of dimension $a$.

The theory of the Herr complex then allows us to find a bounded complex of finite rank locally free $\mathcal{O}_{T}$-modules

$C_{T}^{0}\to C_{T}^{1}\to C_{T}^{2}$

whose cohomology computes $H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ (the finite type points of $H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ correspond to the usual Galois cohomology). If $\mathrm{Ext}^{2}(\overline{\alpha},\overline{\rho}_{T})$ is locally free of some rank $r$, then we have a bounded complex of finite rank locally free $\mathcal{O}_{T}$-modules

$C_{T}^{0}\to Z_{T}^{1}$

where $Z_{T}^{1}$ is defined to be the kernel of the map $C_{T}^{1}\to C_{T}^{2}$. There is a surjection $Z_{T}^{1}\twoheadrightarrow H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$.

Let $V$ be the vector bundle over $T$ corresponding to $Z_{T}^{1}$, and let $\overline{\rho}_{V}$ be the pullback of $\overline{\rho}_{T}$ to $V$. Then we can use the surjection $Z_{T}^{1}\twoheadrightarrow H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ to construct a “universal extension” $\mathcal{E}_{V}$ that fits into the following exact sequence:

$\displaystyle 0\to\overline{\rho}_{V}\to\mathcal{E}_{V}\to\overline{\alpha}\to 0$

This universal extension $\mathcal{E}_{V}$ is a family of Galois representations parametrized by $V$, i.e. a map $\mathcal{E}_{V}\to \mathcal{X}_{d+a, \mathrm{red}}$. Being able to construct families of higher-dimensional Galois representations as extensions of lower-dimensional ones helps us study the moduli stacks of Galois representations for any dimension, and is used for instance, to prove the earlier stated facts about the dimension and irreducible components of these moduli stacks.

### The “coarse moduli space” and the Bernstein center

Let us now look at a “coarse moduli space” $X$ associated to $\mathcal{X}_{d}^{\mathrm{det}=\psi}$. This coarse moduli space $X$ is a moduli space of pseudorepresentations. The associated reduced space $X_{\mathrm{red}}$ should be a chain of projective lines, as we shall shortly explain.

A map from Galois representations to pseudorepresentations should factor through semisimplification. If a reducible mod p Galois representation is semisimple then it must be of the form

$\displaystyle \begin{pmatrix}\mathrm{ur}_{a}\overline{\varepsilon}^{i}&0\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{j}\end{pmatrix}$

and from our earlier discussion we can associate to it the Serre weight $\mathrm{Sym}^{i-j-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{j}$. But we can also see this as

$\displaystyle \begin{pmatrix}\mathrm{ur}_{a}\overline{\varepsilon}^{j}&0\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{i}\end{pmatrix}$

and now the associated Serre weight is $\mathrm{Sym}^{j-i-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{i}$. Therefore there are two Serre weights that we can associate to this reducible mod p Galois representation! Now if we fix the determinant of our Galois representation to be, say $\overline{\varepsilon}$, then besides the two Serre weights our Galois representation only depends on the parameter $a$ (because in this case we must have $\mathrm{ur}_{b}=\mathrm{ur}_{a}^{-1}$).

We can consider our two Serre weights now to be the $0$ and $\infty$ of a projective line (these points will also correspond to irreducible representations) and the points of the projective line in between these gives us the values of the parameter $a$ which parametrize the reducible representations. But a “$0$” Serre weight could also be considered as the “$\infty$” Serre weight associated to another family of Galois representations. Therefore we have a chain of projective lines parametrizing our semisimple Galois representations. This is the reduced space $X_{\mathrm{red}}$ of our “coarse moduli space” $X$.

One interesting application of these ideas, which is currently part of ongoing work by Andrea Dotto, Matthew Emerton, and Toby Gee, is that the category of mod p representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ (which we shall denote by $\mathcal{A}$) forms a stack over the Zariski site of $X_{\mathrm{red}}$!

That is, to every Zariski open set of $X_{\mathrm{red}}$, we can associate a category $\mathcal{A}_{U}$ and these categories glue together well and form a stack over the Zariski site of $X_{\mathrm{red}}$. To define these categories $\mathcal{A}_{U}$ we need to use the theory of “blocks” developed by Vytautas Paskunas. Namely, Paskunas showed that the category of locally admissible representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ decomposes into “blocks” labeled by semisimple Galois representations $G_{\mathbb{Q}_{p}}\to\mathrm{GL}_{2}(\overline{\mathbb{F}})$.

We can now construct the category $\mathcal{A}_{U}$ as follows. Let $Y$ be a closed subset of $X_{\mathrm{red}}$. Then we define $\mathcal{A}_{Y}$ to be the full sub category of $\mathcal{A}$ consisting of all representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ whose irreducible subquotients live in blocks labeled by the $\overline{\mathbb{F}}_{p}$-points of $Y$ (since these correspond to semisimple Galois representations, which in turn label the blocks). Then for $U$ an open subset of $X_{\mathrm{red}}$, we define $\mathcal{A}_{U}$ to be the Serre quotient $\mathcal{A}/\mathcal{A}_{Y}$, where $Y=X_{\mathrm{red}}\subset U$.

If $\mathcal{C}$ is an additive category, the Bernstein center of $\mathcal{C}$, denoted $Z(\mathcal{C})$, is defined to be the ring of endomorphisms of the identity functor $\mathcal{C}\to\mathcal{C}$.

It is expected that $\mathcal{A}$, the category of mod p representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, forms a sheaf over $X$, and the Bernstein center $Z(\mathcal{A})$ coincides with the structure sheaf $\mathcal{O}_{X}$ of $X$.

### Relation to p-adic local Langlands and modularity

Let us now discuss how the moduli stack $\mathcal{X}_{d}$ is related to the p-adic local Langlands correspondence and questions of modularity.

We want there to be a sheaf $\mathcal{M}$ on $\mathcal{X}_{d}$ which realizes the p-adic local Langlands correspondence. In the case $K=\mathbb{Q}_{p}$ and $d=2$, we can apply a construction of Colmez to the universal $(\varphi,\Gamma)$-module on $\mathcal{X}_{2}$ and obtain a quasi-coherent sheaf $\mathcal{M}=(D\boxtimes\mathbb{P}^{1})/(D^{\natural}\boxtimes\mathbb{P}^{1})$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$-representations on it. Being a quasi-coherent sheaf it has an action of the structure sheaf $\mathcal{O}_{\mathcal{X}_{2}}$, which is expected to be the same as the action of the Bernstein center of the category of smooth $\mathrm{GL}_{2}(\mathbb{Q}_{p})$-representations on $\mathbb{Z }_{p}$-modules which are locally $p$-power torsion.

In the case when either $K=\mathbb{Q}_{p}$ or $d\neq 2$, there is so far no known satisfactory analogue of Colmez’ construction. However, it is believed that if there is such a sheaf $\mathcal{M}$ it must coincide with a certain patched module construction $M_{\infty}$, which is a module over the deformation ring $R_{\overline{\rho}}$, after pulling back over the map $\mathrm{Spf}(R_{\overline{\rho}})\to\mathcal{X}_{d}$.

The sheaf $\mathcal{M}$ is also expected to play a role in the geometric version of the Breuil-Mezard conjecture, which in its original form concerns the geometry of Galois deformation rings and has applications in modularity and the Fontaine-Mazur conjecture.

Let $\underline{\lambda}$ be a Hodge type and let $\tau$ be an inertial type. Let $\xi_{\sigma}=(\xi_{\sigma,1},\ldots,\xi_{\sigma,d})$, where $\xi_{\sigma,i}=\lambda_{\sigma,i}-(d-i)$. Let $M_{\xi_{\sigma}}$ be the algebraic $\mathcal{O}_{K}$-representation of $\mathrm{GL}_{d}(\mathcal{O}_{K})$ with highest weight $\xi_{\sigma}$, and let $L_{\underline{\lambda}}=M_{\xi\sigma}\otimes_{\mathcal{O}_{K},\sigma}\mathcal{O}_{E}$.

Now to the inertial type $\tau$, there is an “inertial local Langlands correspondence” that associates to $\tau$ a smooth admissible representation $\sigma^{\mathrm{crys}}(\tau)$ of $\mathrm{GL}_{d}(\overline{\mathbb{Q}}_{p})$ over $\overline{\mathbb{Q}_{p}}$. Let $\sigma^{\mathrm{crys},\circ}(\tau)$ be a $\mathrm{GL}_{d}(\mathcal{O}_{K})$-stable $\mathcal{O}_{E}$-lattice in $\sigma^{\mathrm{crys}}(\tau)$, and let $\sigma^{\mathrm{crys}}(\lambda,\tau)=L_{\underline{\lambda}}\otimes_{\mathcal{O}_{E}}\sigma^{\mathrm{crys}}(\lambda,\tau)$. Finally, we let $\overline{\sigma}^{\mathrm{crys}}(\lambda,\tau)$ be the semisimplification of $\sigma^{\mathrm{crys}}(\lambda,\tau)\otimes\mathbb{F}$. We may now view this as an $\mathbb{F}$ representation of $\mathrm{GL}_{d}(k)$, where $k$ is the residue field of $\mathcal{O}_{K}$. Now let $F_{\underline{k}}$ be the irreducible $\mathbb{F}$ representation of $\mathrm{GL}_{d}(k)$ associated to the tuple $\underline{k}$ (these are higher-dimensional versions of the Serre weights discussed in The mod p local Langlands correspondence for GL_2(Q_p)). We have the decomposition

$\displaystyle \sigma^{\mathrm{crys}}(\lambda,\tau)=\bigoplus F_{\underline{k}}^{n_{\underline{k}}^{\mathrm{crys}}(\lambda,\tau)}$

Let $\mathcal{M}(\sigma^{\circ}(\lambda,\tau)):=\mathrm{Hom}_{\mathrm{GL}_{d}(\mathcal{O}_{K})}(\sigma^{\circ}(\lambda,\tau)^{\vee},\mathcal{M})$. Let $\mathcal{Z}(\sigma^{\circ}(\lambda,\tau))$ be the support of $\mathcal{M}(\sigma^{\circ}(\lambda,\tau))$ on $\mathcal{X}_{d}$. It is expected that $\mathcal{Z}(\sigma^{\circ}(\lambda,\tau))=\mathcal{Z}(\mathcal{X}_{d}^{\mathrm{crys},\underline{\lambda},\tau})_{\mathbb{F}}$.

Let $\mathcal{M}(F_{\underline{k}}):=\mathrm{Hom}_{\mathrm{GL}_{d}(\mathcal{O}_{K})}(F_{\underline{k}}^{\vee},\mathcal{M})$ and let $\mathcal{Z}(F_{\underline{k}})$ be the support of $\mathcal{M}(F_{\underline{k}})$ on $\mathcal{X}_{d}$. The geometric Breuil-Mezard conjecture states that

$\displaystyle \mathcal{Z}(\sigma^{\mathrm{crys}}(\lambda,\tau))=\sum_{\underline{k}} n_{\underline{k}}^{\mathrm{crys}}(\lambda,\tau)\mathcal{Z}(F_{\underline{k}})$.

The Breuil-Mezard conjecture is expected to have applications in modularity, i.e. knowing when a Galois representation comes from a modular form. Some progress towards the conjecture has recently been obtained by Daniel Le, Bao Viet Le Hung, Brandon Levin, and Stefano Morra by the use of local models, which are geometric objects of a more group-theoretic origin (related to affine Grassmannians and flag varieties) which can make them easier to study. Their work also has applications to a generalization of the weight part of Serre’s conjecture. We leave this work and other related topics to future posts.

References:

Moduli stacks of (phi, Gamma)-modules: a survey by Matthew Emerton and Toby Gee

Moduli stacks of etale (phi, Gamma)-modules and the existence of crystalline lifts by Matthew Emerton and Toby Gee

Moduli stacks of (phi, Gamma)-modules by Toby Gee (recording of a talk at the Serre weight conjectures and geometry of Shimura varieties workshop at Centre de Recherches Mathematique)

Moduli of Galois representations by David Savitt (recording of a talk at the 2020 Connecticut Summer School in Number Theory)

“Scheme-theoretic images” of morphisms of stacks by Matthew Emerton and Toby Gee

Phi-modules and coefficient spaces by George Pappas and Michael Rapoport

Mod p Bernstein centers of p-adic groups by Andrea Dotto ( recording of a talk at the Serre weight conjectures and geometry of Shimura varieties workshop at Centre de Recherches Mathematiques)

Localizing GL_2(Q_p) representations by Matthew Emerton (recording of a talk at the INdaM program on Serre conjectures and the p-adic Langlands program)

Local models for Galois deformation rings and applications by Daniel Le, Bao Viet Le Hung, Brandon Levin, Stefano Morra

# The mod p local Langlands correspondence for GL_2(Q_p)

In The Local Langlands Correspondence for General Linear Groups, we stated the local Langlands correspondence for $\mathrm{GL}_n(F)$ where $F$ is a finite extension of $\mathbb{Q}_{p}$. We recall that it states that there is a one-to-one correspondence between irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and F-semisimple Weil-Deligne representations of $\mathrm{Gal}(\overline{F}/F)$. Here all the relevant representations are over $\mathbb{C}$.

In this post, we will consider representations over a finite field $\mathbb{F}$ of characteristic $p$. While we may hope that some sort of “mod p local Langlands correspondence” will also hold in this case, at the moment all we know is the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, which we will discuss in this post. It is a sort of stepping stone to the “p-adic local Langlands correspondence” (where the representations are over some extension of $\mathbb{Q}_{p}$), which, as in the mod p case, is only known for the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. The p-adic local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ plays an important role in the proofs of the known cases of the Fontaine-Mazur conjecture, which concerns when a Galois representation comes from the etale cohomology of some variety.

Let us start by discussing some representation theory of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$. This will be somewhat similar to our discussion in The Local Langlands Correspondence for General Linear Groups, as we will see later when we state the classification of the irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, but we will also need some new ingredients.

A Serre weight is an absolutely irreducible representation (absolutely irreducible means it is irreducible over the algebraic closure of $\mathbb{F}$) of $\mathrm{GL}_{2}(\mathbb{F}_{p})$ over $\mathbb{F}$. This is the same as an absolutely irreducible smooth representation of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ over $\mathbb{F}$.

Serre weights are completely classified and can be explicitly described. Let $r\in\lbrace 0,\ldots,p-1\rbrace$ and let $s\in \lbrace 0,\ldots, p-2\rbrace$. Then a Serre weight is always of the form $\mathrm{Sym}^{r}\mathbb{F}^{2}\otimes\mathrm{det}^{s}$.

The name “Serre weight” originates from its relationship to Serre’s modularity conjecture, which is a conjecture about when a residual representation comes from a modular form, and what the level and the weight of the modular form should be. Avner Ash and Glenn Stevens made the observation that a residual representation $\overline{\rho}$ is modular of weight $k$ ($k\geq 2$) and level $\Gamma_{1}(N)$ if and only if $H^{1}(\Gamma_{1}(N),\mathrm{Sym}^{k-2}\mathbb{F}^{2})_{\mathfrak{m}_{\overline{\rho}}}$ (here $\mathfrak{m}_{\overline{\rho}}$ is a certain maximal ideal of the Hecke algebra associated to $\overline{\rho}$) is nonzero.

For convenience, from here on in this post we shall consider Serre weights not just as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ but as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}$, which sends the uniformizer of $\mathbb{Q}_{p}$ to $1$.

Serre weights are important because from them we can obtain induced representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. Let $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ be the representation of $\mathrm{GL} _{2}(\mathbb{Q}_{p})$ coming from compactly supported functions from $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ to $\sigma$ which satisfy $f(kg)=\sigma(k)f(g)$.

The endomorphisms of $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ form the Hecke algebra, which is isomorphic to $\mathbb{F}[T]$. In other words, we can consider $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ as a module over $\mathbb{F}[T]$, and we can take the quotient $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$. This quotient is irreducible, and it is an important class of absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ called the supersingular representations.

The rest of the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ are very similar to what we discussed in The Local Langlands Correspondence for General Linear Groups. Namely, they can be obtained from principal series representations, which are induced representations of characters from the Borel subgroup $B(\mathbb{Q}_{p})$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ (i.e. the upper-triangular matrices in $\mathrm{GL}_{2}(\mathbb{Q}_{p})$).

To recall, the principal representations are $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$, which means they come from functions $f:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}$ such that $f(hg)=\chi_{1}\otimes\chi_{2}(h)f(g)$ for $h\in B(\mathbb{Q}_{p})$ and $g\in\mathrm{GL}_{2}(\mathbb{Q}_{p})$, where $\chi_{1}$ and $\chi_{2}$ are characters of $\mathbb{Q}_{p}^{\times}$, and $\chi_{1}\otimes\chi_{2}$ as a function on $B(\mathbb{Q}_{p})$ means it sends an element $\begin{pmatrix}a& b\\0& d\end{pmatrix}$ of $B(\mathbb{Q}_{p})$ to $\chi_{1}(a)\otimes \chi_{2}(d)$.

In the case that $\chi_{1}\neq\chi_{2}$, the principal series representations will be absolutely irreducible, in which case we obtain another class of absolutely irreducible admissible smooth representations. Otherwise, we may obtain absolutely irreducible representations as a quotient. These will be twists (this means a tensor product by a character) of the Steinberg representation, which is defined as the quotient $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ (here $\mathbf{1}$ is the trivial representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$). This gives a third class of absolutely irreducible admissible representations. Finally we have the characters, which give a fourth class.

In summary, the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$ can be classified into the following four kinds as follows:

• One-dimensional representations (characters) $\delta: \mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
• Principal series representations $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$ for $\chi_{1},\chi_{2}: \mathbb{Q}_{p}^{\times}\to\mathbb{F^{\times}}, \chi_{1}\neq\chi_{2}$
• Twists of Steinberg representations $\delta\otimes\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ for $\delta:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
• Supersingular representations $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ for $\sigma$ a Serre weight

Let us now discuss the other side of the correspondence, the “Galois side”. For ease of notation let us also denote $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ by $G_{\mathbb{Q}_{p}}$.

Let $g$ be an element of the inertia subgroup of $G_{\mathbb{Q}_{p}}$. Serre’s level 2 fundamental character $\omega_{2}$ is given by composing the map

$\displaystyle g\mapsto \frac{g(\sqrt[p^{2-1}]{-p})}{\sqrt[p^{2-1}]{-p}}$

which takes values in $\mathbb{\mu}_{p^{2}-1}$ with the isomorphism $\mu_{p^{2}-1}\xrightarrow{\sim}\mathbb{F}_{p^{2}}^{\times}$.

Let $h$ be a natural number which is mutually prime to $p+1$. We have that $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega_{2}^{h}$ is an irreducible 2-dimensional representation of $G_{\mathbb{Q}_{p}}$. In fact, any absolutely irreducible representation of $G_{\mathbb{Q}_{p}}$ over $\mathbb{F}$ is of this form, possibly tensored with the unramified character $\lambda_{a}$ which takes the inverse of the Frobenius to $a\in\mathbb{F}^{\times}$.

The mod p local Langlands correspondence is now the bijection described explicitly as follows:

To the supersingular representation $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ where $\sigma=\mathrm{Sym}^{r}\mathbb{F}^{2}$, we associate the Galois representation $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega^{r+1}$.

To the representation $\pi$ which is obtained as the extension $0\to\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}\varepsilon^{-1}\to\pi\to \mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{2}\otimes\chi_{1}\varepsilon^{-1}\to 0$ we associate the Galois representation $\overline{\rho}$ which is obtained as an extension $0\to\chi_{1}\to\overline{\rho}\to\chi_{2}\to 0$. Here $\varepsilon$ is the reduction mod p of the p-adic cyclotomic character, and $\chi_{1}$ and $\chi_{2}$ are characters $\mathbb{Q}_{p}^{\times}\to\mathbb{F}^{\times}$ which are not equal to each other nor to the product of the other by the p-adic cyclotomic character or its inverse.

The p-adic local Langlands correspondence, which, as stated earlier concerns representations over some finite extension of $\mathbb{Q}_{p}$ and is important in the Fontaine-Mazur conjecture, needs to be compatible with the mod p local Langlands correspondence as well. Its statement is more involved than the mod p local Langlands correspondence, and its proof involves $(\varphi,\Gamma)$-modules. We reserve further discussion of the p-adic local Langlands correspondence to future posts.

References:

The emerging p-adic Langlands programme by Christophe Breuil

Representations of Galois and of GL_2 in characteristic p by Christophe Breuil

Towards a modulo p Langlands correspondence for GL_2 by Christophe Breuil and Vytautas Paskunas

# Completed Cohomology

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

# Moduli Stacks of Galois Representations

In Galois Deformation Rings we introduced the concept of Galois deformations and Galois deformation rings, which had the property that Galois deformations (which are certain equivalence classes of lifts of a fixed residual representation) correspond to maps from those Galois deformations to the one over the Galois deformation ring. In a way this allows us to consider all the deformations of this residual representation altogether.

In this post, we will consider not only the Galois representations that are lifts of some fixed residual representation, but consider Galois representations without the need to fix a residual representation. These Galois representations are going to be parametrized by the moduli stack of Galois representations, whose geometry we will study.

Before we consider Galois representations, let us first consider the simpler case of representations of a finitely presented group. Let $G$ be such a finitely presented group, with generators $g_{1},\ldots, g_{n}$ and relations $r_{1},\ldots r_{t}$. Let us consider its $d$-dimensional representations over some ring $A$. The first thing we have to do is to give $d\times d$ matrices $M_{1},\ldots,M_{n}$, with coefficients in $A$, corresponding to the generators $g_{1},\ldots g_{n}$. Then we have to quotient out by the relations $r_{1},\ldots,r_{t}$, each viewed as a relation on the matrices $M_{1},\ldots,M_{n}$. Then we may see the functor that assigns to a ring $A$ the set of $d$-dimensional representations of $G$ over $A$ is representable by an affine scheme.

Now the theory of stacks (see also Algebraic Spaces and Stacks) comes in when we take into consideration that two representations that differ only by a change of basis may be considered to be “the same”. So we take the quotient of our affine scheme by this action of $\mathrm{GL}_{d}$, and what we get is a stack.

Let us now go back to Galois representations. Note that the absolute Galois groups we will be interested in are not finitely presented, however, the idea is that we will have to find some clever way of relating these absolute Galois groups to some finitely presented groups so we can make use of what we have just learned.

Let us first discuss the local case, for $\ell\neq p$, i.e. our representations will be on $A$-modules, where $A$ is some $\mathbb{Z}_{\ell}$-algebra. Consider $K$, a finite extension of $\mathbb{Q}_{p}$, and let $\kappa$ be its residue field. As a shorthand let us also denote $\mathrm{Gal}(\overline{K}/K)$ by $G_{K}$. Let us recall (see also Splitting of Primes in Extensions and Weil-Deligne Representations) that we have the exact sequence

$\displaystyle 0\to I_{K}\to G_{K}\to\mathrm{Gal}(\overline{\kappa}/\kappa)\to 0$

Recall that $I_{K}$ is called the inertia group. An extension of $K$ is tamely ramified if its ramification index is prime to $p$. Let $K^{\mathrm{tame}}$ be the maximal tamely ramified extension of $K$ and $K^{\mathrm{unr}}$ be the maximal unramified extension of $K$. Let $G_{K}^{\mathrm{tame}}=\mathrm{Gal}(K^{\mathrm{tame}}/K)$ and let $G_{K}^{\mathrm{unr}}=\mathrm{Gal}(K^{\mathrm{unr}}/K)$. We have an exact sequence

$\displaystyle 0\to I_{K}^{\mathrm{tame}}\to G_{K}^{\mathrm{tame}}\to G_{K}^{\mathrm{unr}}\to 0$

Where $I_{K}^{\mathrm{tame}}$ is called the tame inertia. It is a quotient of the inertia group $I_{K}$ by a subgroup $I_{K}^{\mathrm{wild}}$, called the wild inertia. The tame inertia $I_{K}^{\mathrm{tame}}$ is of the form $\prod_{\ell\neq p}\mathbb{Z}_{\ell}(1)$ and is a pro-cyclic group.

Let $\tau$ be a generator of $I_{K}^{\mathrm{tame}}$ as a pro-cyclic group. Let $\sigma$ be a lift of Frobenius in $G_{K}^{\mathrm{tame}}$. We consider the subgroup of $G_{K}^{\mathrm{tame}}$ given by

$\displaystyle \Gamma=\langle \tau,\sigma\vert\sigma\tau\sigma^{-1}=\tau^{q}\rangle$

where $q$ is the cardinality of the residue field $\kappa$. This subgroup $\Gamma$ is is dense inside $G_{K}^{\mathrm{tame}}$, and $G_{K}^{\mathrm{tame}}$ is its profinite completion.

We have the following exact sequence:

$\displaystyle 0\to I_{K}^{\mathrm{wild}}\to G_{K}\to G_{K}^{\mathrm{tame}}\to 0$

Inside $G_{K}^{\mathrm{tame}}$ we have the subgroup $\Gamma$, and we have another exact sequence as follows:

$\displaystyle 0\to I_{K}^{\mathrm{wild}}\to\mathrm{WD}_{K}\to\Gamma\to 0$

The middle term $\mathrm{WD}_{K}$ is defined to be the limit $\varprojlim_{Q}\mathrm{WD}_{K}/Q$, where $Q$ is an open subgroup of $I_{K}^{\mathrm{wild}}$ which is normal in $G_{K}$, and $\mathrm{WD}_{K}/Q$ is in turn defined to be the extension of the finitely presented group $\Gamma$ by the finite group $I_{K}^{\mathrm{wild}}/Q$, i.e. $\mathrm{WD}_{K}/Q$ is the middle term in the exact sequence

$\displaystyle 0\to I_{K}^{\mathrm{wild}}/Q\to\mathrm{WD}_{K}/Q\to\Gamma\to 0$

Now the idea is that $\mathrm{WD}_{K}/Q$, being an extension of a finitely presented group by a finite group, is finitely presented, and we can use what we have learned about moduli stacks of finitely presented groups at the beginning of this post. At the same time, $\mathrm{WD}_{K}/Q$ is dense inside $G_{K}/Q$, and we have $G_{K}=\varprojlim_{Q} G_{K}/Q$.

Therefore, we let $V_{Q}$ be the moduli stack of representations of the finitely presented group $\mathrm{WD}_{K}/Q$, and our moduli stack of Galois representations will be given by the direct limit $V=\varinjlim V_{Q}$ .

Now all of what we just discussed applies to the $\ell\neq p$ case, but the $\ell=p$ case is much more subtle. To properly construct the moduli stack of Galois representations for the $\ell=p$ case we will need the theory of $(\varphi,\Gamma)$-modules, which will not discuss in this post, though hopefully we will be able to in some future post.

Let us now discuss briefly the global case. Let $K$ be a number field, and let $S$ be a finite set of places of $S$. Let $G_{K,S}$ denote the Galois group of the maximal Galois extension of $K$ unramified outside $S$. We want to consider $d$-dimensional representations of $G_{K,S}$ over a $\mathbb{Z}_{p}/p^{a}\mathbb{Z}_{p}$-algebra $A$, for some $a$. The functor that assigns to such an $A$ this set of representations gives us a stack $\mathfrak{X}$ over the formal scheme $\mathrm{Spf}(\mathbb{Z}_{p})$ (see also Formal Schemes).

Not only can we consider representations, but we can also consider pseudo-representations, which are sort of generalizations of the concept of the trace of a representation. These pseudo-representations also have a corresponding moduli space, which is a formal scheme, denoted by $X$, also over $\mathrm{Spf}(\mathbb{Z}_{p})$. Since we can associate a pseudo-representation to a representation, we have a map $\mathfrak{X}\to X$.

It is a theorem of Chenevier that $X$ is a disjoint union of components $X_{\overline{\rho}}$ indexed by residual pseudo-representations (semi-simple pseudo-representations over a finite field). Similarly, $\mathfrak{X}$ will be a disjoint union of components $\mathfrak{X}_{\overline{\rho}}$, each with a map to the corresponding $X_{\overline{\rho}}$. In the case that $\overline{\rho}$ is irreducible, $X_{\overline{\rho}}$ will be $\mathrm{Spf}(R_{\overline{\rho}})$, while $\mathfrak{X}_{\rho}$ will be $\mathrm{Spf}(R_{\overline{\rho}})/\widehat{\mathbb{G}}_{m}$, where $R_{\overline{\rho}}$ is the universal deformation ring, and $\widehat{\mathbb{G}}_{m}$ is some formal completion of $\widehat{\mathbb{G}}_{m}$.

We end this post by mentioning a conjecture related to the conjectural categorical geometric Langlands correspondence mentioned at the end of The Global Langlands Correspondence for Function Fields over a Finite Field. This is currently part of ongoing work by Matthew Emerton and Xinwen Zhu. There is a “restriction” map

$\displaystyle f:\mathfrak{X}\to\prod_{v\in S}\mathfrak{X}_{v}$

from the global moduli stack $\mathfrak{X}$ to the product of local moduli stacks $\mathfrak{X}_{v}$, for all $v$ in the set $S$ (defined at the start of the discussion of the global case). It is then conjectured that there are coherent sheaves $\mathfrak{A}_{v}$ on each $\mathfrak{X}_{v}$, which come from representations of $\mathrm{GL}_{n}(K_{v})$. We can form the product of these sheaves and pull back to get a sheaf $\mathfrak{A}$ on the global stack $\mathfrak{X}$, and after tensoring with the universal Galois representation on $\mathfrak{X}$, it is conjectured that this gives the compactly supported cohomology of Shimura varieties.

One can also form, more generally, moduli stacks not just of Galois representations but of Langlands parameters. More on these, as well as more in-depth details on these moduli stacks and the conjectures regarding coherent sheaves on these moduli stacks, will hopefully be discussed in future posts.

References:

Moduli stacks of Galois representations by Matthew Emerton on YouTube

Moduli Stacks of (phi, Gamma)-modules: a survey by Matthew Emerton and Toby Gee

Moduli of Langlands parameters by Jan-Francois Dat, David Helm, Robert Kurinczuk, and Gilbert Moss

Moduli of Galois representations by Carl Wang-Erickson

# The Global Langlands Correspondence for Function Fields over a Finite Field

In The Local Langlands Correspondence for General Linear Groups, we introduced some ideas related to what is known as the Langlands program, and discussed in a little more detail the local Langlands correspondence, at least for general linear groups.

In this post, we will discuss the global Langlands correspondence, but we will focus on the case of function fields over a finite field. This will be somewhat easier to state than the case of number fields, and at the same time perhaps give us a bit more geometric intuition. Let us fix a smooth, projective, and irreducible curve $X$, defined over a finite field $\mathbb{F}_{q}$. We let $F$ be its function field. For instance, if $X$ is the projective line $\mathbb{P}^{1}$ over $\mathbb{F}_{q}$, then $F=\mathbb{F}(t)$.

### The case of $\mathrm{GL}_{1}$: Global class field theory for function fields over a finite field

To motivate the global Langlands correspondence for function fields, let us first think of the $\mathrm{GL}_{1}$ case, which is a restatement of (unramified) global class field theory for function fields. Recall that in Some Basics of Class Field Theory global class field theory tells us that for global field $F$, its maximal unramified abelian extension $H$, also called the Hilbert class field of $F$, has the property that $\mathrm{Gal}(H/F)$ is isomorphic to the ideal class group.

We recall that there is an analogy between the absolute Galois group and the etale fundamental group in the case when there is no ramification. Therefore, in the case of function fields, the corresponding statement of unramified global class field theory may be stated as

$\displaystyle \pi_{1}(X,\overline{\eta})^{\mathrm{ab}}\times_{\widehat{\mathbb{Z}}}\mathbb{Z}\xrightarrow{\sim} \mathrm{Pic}(\mathbb{F}_{q})$

where $\pi_{1}(X,\overline{\eta})$ is the etale fundamental group of $X$, a profinite quotient of $\mathrm{Gal}(\overline{F}/F)$ through which its action factors ($\overline{\eta}$ here serves as the basepoint, which is needed to define the etale fundamental group). The Picard scheme $\mathrm{Pic}$ is the scheme such that for any scheme $S$ its $S$ points $\mathrm{Pic}(S)$ correspond to the isomorphism classes of line bundles on $X\times S$. This is analogous to the ideal class group. Taking the fiber product with $\mathbb{Z}$ is analogous to taking the Weil group (see also Weil-Deligne Representations and The Local Langlands Correspondence for General Linear Groups).

The global Langlands correspondence, in the case of $\mathrm{GL}_{1}$, is a restatement of this in terms of maps from each side to some field (we will take this field to be $\overline{\mathbb{Q}}_{\ell}$). It states that there is a bijection between characters $\sigma:\pi_{1}(X,\overline{\eta})\to \overline{\mathbb{Q}}_{\ell}^{\times}$, and $\chi:\mathrm{Pic}(\mathbb{F}_{q})/a^{\mathbb{Z}}\to \overline{\mathbb{Q}}_{\ell}^{\times}$ where $a$ is any element of $\mathrm{Pic}(\mathbb{F}_{q})$ of nonzero degree. Again this is merely a restatement of unramified global class field theory, and nothing has changed in its content. However, this restatement points to us the way in which it may be generalized.