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 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 (here is some finite field of characteristic ), we have a Galois deformation ring , with the defining property that maps from into some complete Noetherian local -algebra correspond to certain Galois representations over , namely those which “lift” the residual representation . If we compose these maps with maps from into , we get maps that correspond to certain Galois representations over .
In addition, since we want to match up Galois representations with modular forms (cusp forms of weight in particular this post), we will want to impose certain conditions on the Galois representations that are parametrized by our deformation ring . For instance, it is known that p-adic Galois representations that arise from a cusp form of weight and level are unramified at all the primes except and the ones that divide . There is a way to construct a modification of our deformation ring 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 .
On the other hand, maps from the Hecke algebra to some coefficient field (we will choose this to be ; conventionally this is , but and 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 only correspond to Galois representations that come from lifting our fixed Galois representation 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 , 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 , a generalization of this is given by a theorem of Eichler and Shimura)
where (resp. ) is the space of modular forms (resp. cusp forms) of weight and level . The advantage of expressing modular forms in this form is that we shall be able to consider them “integrally”. We have that
Now let be a finite extension of , with ring of integers , uniformizer and residue field (the same field our residual representation takes values in). We can now consider
Let be the set consisting of the prime 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 acting on , and similarly a Hecke algebra acting on . Recall that these are the subrings of their respective endomorphism rings generated by the Hecke operators and for all (see also Hecke Operators and Galois Representations Coming From Weight 2 Eigenforms). The eigenvalue map
which associates to a Hecke operator its eigenvalue on some cusp form extends to a map
Now since acts on we will also have an eigenvalue map
compatible with the above, in that applying followed by embedding the resulting eigenvalue to is the same as composing the map from into first then applying the eigenvalue map. Now we can compose the eigenvalue map to with the reduction mod so that we get .
Now let be the kernel of . This is a maximal ideal of . In fact, we can associate to a residual representation , such that the characteristic polynomial of the is given by .
Now let be the completion of with respect to . It turns out that there is a Galois representation which lifts . Furthermore, is a complete Noetherian local -algebra!
Putting all of these together, what this all means is that if , there is a map . 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 . The idea is that will have an action on , which will factor through . If we can show that is free as an -module, then since this action factors through via a surjection, then the map from to 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 , which is going to be a module over an auxiliary ring we shall denote by , from which and can be obtained as quotients by a certain ideal. The advantage is that we can bring another ring in play, the power series ring , which maps to (in fact, two copies of it will map to , which is important), and we will use what we know about power series rings to show that is free over , which will in turn show that is free over .
In turn, and will be built as inverse limits of modules and rings and . The subscript 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 and the primes that divide the level ). As we shall see, these Taylor-Wiles primes will be specially selected so that we will be able to construct and 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).
A Taylor-Wiles prime of level is defined to be a prime such that the norm is congruent to mod , and such that has distinct -rational eigenvalues. For our purposes we will need, for every positive integer , a set of Taylor-Wiles primes of cardinality equal to the dimension of the dual Selmer group of (which we shall denote by ), and such that the dual Selmer group of is trivial. It is known that we can always find such a set for every positive integer .
Let us first look at how this affects the “Galois side”, i.e. . There is a surjection , 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 -dimensional representations of , 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 we have a more general number field , 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 and 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. . 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 be some set of Taylor-Wiles primes, we define and we further define to be such that the quotient is isomorphic to the group , defined to be the product over of the maximal p-power quotient of .
We define a new Hecke algebra obtained from by adjoining new Hecke operators for every prime in . We define a maximal ideal of generated by the elements of and again for every prime in .
We now define to be . This has an action of and is therefore a -module. In fact, is a free -module. This will become important later. Another important property of is that its -coinvariants are isomorphic to .
Now also has the structure of a -algebra. If we take and restrict it to (for in ), we get that the resulting local representation is of the form , where and are characters. Using local class field theory (see also The Local Langlands Correspondence for General Linear Groups), we obtain a map . This map factors through the maximal p-power quotient of . Thus given we have a map .
Now it so happens that the action of on factors through the map to . So therefore we have
Taylor-Wiles patching: The patching construction
Now we will perform the patching construction, which means taking the inverse limit over . 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 denote and let denote the ideal . Let us also define to be another power series ring 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 to ; this may be thought of as the limiting case of the map from to discussed earlier.
Now let be a positive integer. Let be the kernel of the surjection , let be , and be the ideal . Abstractly, a patching datum of level is a triple where
- is a surjection of complete Noetherian local algebras
- is a -module, finite free over , such that
- is an isomorphism of -modules
We say that two patching data and of level are isomorphic if and there exists an isomorphism compatible with and . We note the important fact that there are only finitely many isomorphism classes of patching data for any level .
Now we will specialize this abstract construction to help us prove our R=T theorem. We choose
- is induced by the isomorphism between the -coinvariants of and
If we have a patching datum of level , we may form which is a patching datum of level .
Now recall that for any fixed , we can only have a finite number of isomorphism classes of patching datum of level . This means we can find a subsequence of such that .
We can now take inverse limits. Let , let the surjection be given by , and let the surjection be given by . We have
Just as is free as a module over , we have that is free as a module over . We will now use some commutative algebra to show that is a free -module. The depth of a module over a local ring with maximal ideal is defined to be the minimum such that is nonzero. The depth of a module is always bounded above by its dimension.
Now the dimension of is (we know this since we defined it as a power series ). This bounds , and by the above fact regarding the depth of a module, bounds . Since the action of on factors through the action of , bounds . Finally, since is a free -module, we have that . In summary,
and we can see that all of the inequalities are equalities, and all the quantities are equal to . The Auslander-Buchsbaum formula from commutative algebra tells us that
and since both terms on the right-hand side are equal to , the right-hand side is zero. Therefore the projective dimension of relative to is zero, which means that is a projective module over . Since is local, this is the same as saying that is a free -module.
We have that is a free module over . Since this action factors through maps which are all surjections, they have to be isomorphisms, and we have that is a free -module, and therefore . 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 -dimensional representations of for a totally real field (in this case on the automorphic side we would have Hilbert modular forms).
However, when 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 ) 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 .
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 . 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 for the patched module). Namely, we expect a coherent sheaf on the moduli stack of -modules which, “locally” coincides or is at least closely related to the patched module . 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.
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