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}) (forv 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

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