Potential modularity—a survey

Kevin Buzzard

Abstract

A Spitalfields Day at the Newton Institute was organised on the subject of the recent theorem that any elliptic curve over any totally real field is potentially modular. This article is a survey of the strategy of the proof, together with some history.

Introduction

Our main goal in this article is to talk about recent theorems of Taylor and his co-workers on modularity and potential modularity of Galois representations, particularly those attached to elliptic curves. However, so as to not bog down the exposition unnecessarily with technical definitions right from the off, we will build up to these results by starting our story with Wiles’ breakthrough paper [Wil95], and working towards the more recent results. We will however assume some familiarity with the general area—for example we will assume the reader is familiar with the notion of an elliptic curve over a number field, and a Galois representation, and what it means for such things to be modular (when such a notion makes sense). Let us stress now that, because of this chronological approach, some theorems stated in this paper will be superseded by others (for example Theorem 1.1 gets superseded by Theorem 1.6 which gets superseded by Theorem 1.7), and similarly some conjectures (for example Serre’s conjecture) will become theorems as the story progresses. The author hopes that this slightly non-standard style nevertheless gives the reader the feeling of seeing how the theory evolved.

We thank Toby Gee for reading through a preliminary draft of this article and making several helpful comments, and we also thank Matthew Emerton and Jan Nekovář for pointing out various other inaccuracies and ambiguities.

1.1 Semistable elliptic curves over Q are modular.

The story, of course, starts with the following well-known result proved in [Wil95] and [TW95].

Theorem 1.1 (Wiles, Taylor–Wiles) Any semi-stable elliptic curve over the rationals is modular.

This result, together with work of Ribet and others on Serre’s conjecture, implies Fermat’s Last Theorem. This meant that the work of Wiles and Taylor captured the imagination of the public. But this article is not about Fermat’s Last Theorem, it is about how the modularity theorem above has been vastly generalised. Perhaps we should note here though that that there is still a long way to go! For example, at the time of writing, it is still an open problem as to whether an arbitrary elliptic curve over an arbitrary totally real field is modular, and over an general number field, where we cannot fall back on the theory of Hilbert modular forms, the situation is even worse (we still do not have a satisfactory theorem attaching elliptic curves to modular forms in this generality, let alone a result in the other direction).

Before we go on to explain the generalisations that this article is mainly concerned with, we take some time to remind the reader of some of the details of the strategy of the Wiles/Taylor–Wiles proof. The ingredients are as follows. For the first, main, ingredient, we need to make some definitions. Let p be a prime number, let O denote the integers in a finite extension of Q_{p}, let ρ : Gal(\overline{Q}/Q) → GL_{2}(O) denote a continuous odd (by which we mean det(ρ(c)) = −1, for c a complex conjugation) irreducible representation, unramified outside a finite set of primes, and let \overline{ρ} denote its reduction modulo the maximal ideal of O. Recall that there is a general theorem due to Deligne and others, which attaches padic Galois representations to modular eigenforms; we say a p-adic Galois representation is modular if it arises in this way, and that a mod p Galois representation is modular if it arises as the semisimplification of the mod p reduction of a modular p-adic Galois representation. Note that we will allow myself the standard abuse of notation here, and talk about “mod p reduction” when we really mean “reduction modulo the maximal ideal of O”.

Note that if a p-adic Galois representation ρ is modular, then its reduction \overline{ρ} (semisimplified if necessary) is trivially modular. Wiles’ insight is that one could sometimes go the other way.

Theorem 1.2 (Modularity lifting theorem) If p > 2, if \overline{ρ} is irreducible and modular, and if furthermore ρ is semistable and has cyclotomic determinant, then ρ is modular.

This is Corollary 3.46 of [DDT97] (the aforementioned paper is an overview of the Wiles/Taylor–Wiles work; the theorem is essentially due to Wiles and Taylor–Wiles). The proof is some hard work, but is now regarded as “standard”—many mathematicians have read and verified the proof. We shall say a few words about the proof later on. Semistability is a slightly technical condition (see op. cit. for more details) but we shall be removing it soon so we do not go into details. Rest assured that if E/Q is a semistable elliptic curve then its Tate module is semistable.

The next ingredient is a very special case of the following conjecture of Serre:

Conjecture 1.3 (Serre, 1987) If k is a finite field and \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(k) is a continuous odd absolutely irreducible representation, then \overline{ρ} is modular.

We will say more about this conjecture and its generalisations later. Note that this conjecture is now a theorem of Khare and Wintenberger, but we are taking a chronological approach so will leave it as a conjecture for now. In the early 1990s this conjecture was wide open, but one special case had been proved in 1981 (although perhaps it was not stated in this form in 1981; see for example Prop. 11 of [Ser87] for the statement we need):

Theorem 1.4 (Langlands, Tunnell) If \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{3}) is continuous, odd, and irreducible, then \overline{ρ} is modular.

The original proof of Theorem 1.4 is a huge amount of delicate analysis: let it not be underestimated! One needs (amongst other things) the full force of the trace formula in a non-compact case to prove this result, and hence a lot of delicate analysis. We freely confess to not having checked the details of this proof ourselves. Note also that the point is not just that the image of \overline{ρ} is solvable, it is that the image is very small (just small enough to be manageable, in fact). Note that because we are in odd characteristic, the notions of irreducibility and absolute irreducibility coincide for an odd representation (complex conjugation has two distinct eigenvalues, both defined over the ground field). Langlands’ book [Lan80] proves much of what is needed; the proof was finished by Tunnell in [Tun81] using the non-solvable cubic base change results of [JPSS81]. This result is of course also now regarded as standard— many mathematicians have read and verified this proof too. The author remarks however that due to the rather different techniques involved in the proofs of the two results, he has the impression that the number of mathematicians who have read and verified all the details of the proofs of both the preceding theorems is rather smaller!

Let us see how much of Theorem 1.1 we can prove so far, given Theorems 1.2 and 1.4. Let E be a semistable elliptic curve, set p = 3 and let ρ : Gal(\overline{Q}/Q) → GL_{2}(Z_{3}) be the 3-adic Tate module of E. Then ρ is continuous, odd, unramified outside a finite set of primes, and semistable, with cyclotomic determinant. Furthermore, if \overline{ρ}, the Galois representation on E[3], is irreducible, then \overline{ρ} is modular by the Langlands–Tunnell theorem 1.4 and so ρ, and hence E, is modular, by the modularity lifting theorem. Of course the problem is that E[3] may be reducible—for example if E has a Q-point of order 3 (or more generally a subgroup of order 3 defined over Q). To deal with this situation, Wiles developed a technique known as the “3–5 trick”.

Lemma 1.5 (The 3–5 trick) If E/Q is a semistable elliptic curve with E[3] reducible, then E[5] is irreducible, and there is another semistable elliptic curve A/Q with E[5] ≅ A[5] and A[3] irreducible.

The 3–5 trick is all we need to finish the proof of Theorem 1.1. For if E/Q is semistable but E[3] is reducible, choose A as in the lemma, and note that A[3] is irreducible, hence A[3] is modular (by the Langlands– Tunnell Theorem 1.4), hence A is modular (by the Modularity Lifting Theorem 1.2), so A[5] is modular, so E[5] is modular, and irreducible, so E is modular by the Modularity Lifting Theorem 1.2 applied to the 5-adic Tate module of E.

Let us say a few words about the proof of the last lemma. The reason that reducibility of E[3] implies irreducibility of E[5] is that reducibility of both would imply that E had a rational subgroup of order 15, and would hence give rise to a point on the modular curve Y_{0}(15), whose compactification X_{0}(15) is a curve of genus 1 with finitely many rational points, and it turns out that the points on this curve are known, and one can check that none of them can come from semistable elliptic curves (and all of them are modular anyway). So what is left is that given E as in the lemma, we need to produce A. Again we use a moduli space trick. We consider the moduli space over Q parametrising elliptic curves B equipped with an isomorphism B[5] ≅ E[5] that preserves the Weil pairing. This moduli space has a natural smooth compactification X over Q, obtained by adding cusps. Over the complexes the resulting compactified curve is isomorphic to the modular curve X(5), which has genus zero. Hence X is a genus zero curve over Q. Moreover, X has a rational point (coming from E) and hence X is itself isomorphic to the projective line over Q (rather than a twist of the projective line). In particular, X has infinitely many rational points. Now using Hilbert’s Irreducibility Theorem, which in this setting can be viewed as some sort of refinement of the Chinese Remainder Theorem, it is possible to find a point on X which is 5-adically very close to E, and 3-adically very far away from E (far enough so that the Galois representation on the 3-torsion of the corresponding elliptic curve is irreducible: this is the crux of the Hilbert Irreducibility Theorem, and this is where we are using more than the naive Chinese Remainder Theorem). Such a point corresponds to the elliptic curve A we seek, and the lemma, and hence Theorem 1.1, is proved.

The reason we have broken up the proof of Theorem 1.1 into these pieces is that we would like to discuss generalisations of Theorem 1.1, and this will entail discussing generalisations of the pieces that we have broken it into.

1.2 Why the semistability assumption?

All semistable elliptic curves were known to be modular by 1995, but of course one very natural question was whether the results could be extended to all elliptic curves. Let us try and highlight the issues involved with trying to extend the proof; we will do this by briefly reminding readers of the strategy of the proof of a modularity lifting theorem such as Theorem 1.2. The strategy is that given a modular \overline{ρ}, one considers two kinds of lifting to characteristic zero. The first is a “universal deformation” ρ^{univ} : Gal(\overline{Q}/Q) → GL_{2}(R^{univ}), where one considers all deformations satisfying certain properties fixed beforehand (in Wiles’ case these properties were typically “unramified outside S and semistable at all the primes in S” for some fixed finite set of primes S), and uses the result of Mazur in [Maz89] that says that there is a universal such deformation, taking values in a ring R^{univ}. The second is a “universal modular deformation” ρ_{T} : Gal(\overline{Q}/Q) → GL_{2}(T) comprising of a lift of \overline{ρ} to a representation taking values in a Hecke algebra over Z_{p} built from modular forms of a certain level, weight and character (or perhaps satisfying some more refined local properties). Theorems about modular forms (typically local-global theorems) tell us that the deformation to GL_{2}(T) has the properties used in the definition of R^{univ}, and there is hence a map

R^{univ} → T.

The game is to prove that this map is an isomorphism; then all deformations will be modular, and in particular ρ, the representation we started with, will be modular. The insight that the map may be an isomorphism seems to be due to Mazur: see Conjecture (*) of [MT90] and the comments preceding it. One underlying miracle is that this procedure can only work if R^{univ} has no p-torsion, something which is not at all evident, but which came out of the Wiles/Taylor–Wiles proof as a consequence.

The original proof that the map R^{univ} → T is an isomorphism breaks up into two steps: the first one, referred to as the minimal case, deals with situations where ρ is “no more ramified than \overline{ρ}”, and is proved by a patching argument via the construction of what is now known as a Taylor–Wiles system: one checks that certain projective limits of Rs and T s (using weaker and weaker deformation conditions, and more and more modular forms) are power series rings, and that the natural map between them is an isomorphism for commutative algebra reasons (for dimension reasons, really), and then one descends back to the case of interest. The second is how to deduce the general case from the minimal case—this is an inductive procedure (which relies on a result on Jacobians of modular curves known as Ihara’s Lemma, the analogue of which still appears to be open for GL_{n}, n > 2; this provided a serious stumbling-block in generalising the theory to higher dimensions for many years). Details of both of these arguments can be found in [DDT97], especially §5 (as well, of course, as in the original sources).

To see why the case of semistable elliptic curves was treated first historically, we need to look more closely at the nitty-gritty of the details behind a deformation problem.

Wiles had a semistable irreducible representation \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(k) with k a finite field of characteristic p. Say \overline{ρ} is unramified outside some finite set S ∋ p of primes, and has cyclotomic determinant. Crucially, Wiles knew what it meant (at least when p > 2) for a deformation ρ : Gal(\overline{Q}/Q) → GL_{2}(A) to be semistable and unramified outside S, where A is now a general Artin local ring with residue field k (or even a projective limit of such rings). For a prime q ∉ S it of course means ρ is unramified at q. For q ≠ p, q ∈ S, it means that the image of an inertia group at q under ρ can be conjugated into the upper triangular unipotent matrices. For q = p one needs more theory. The observation is that an elliptic curve with semistable reduction either has good reduction, or multiplicative reduction. The crucial point is that for a general Artin local A with finite residue field one can make sense of the notion that ρ : Gal(\overline{Q}/Q) → GL_{2}(A) has “good reduction”— one demands that it is the Galois action on the generic fibre of a finite flat group scheme with good reduction, and work of Fontaine [Fon77] and Fontaine–Laffaille [FL82] shows that one can translate this notion into “linear algebra” which is much easier to work with (this is where the assumption p > 2 is needed). Also crucial are the results of Raynaud [Ray74], which show that the category of Galois representations with these properties is very well-behaved. Similarly one can make sense of the notion that ρ has “multiplicative reduction”: one can demand that ρ on a decomposition group at p is upper triangular.

We stress again that the crucial point is that the notions of “good reduction” and “multiplicative reduction” above make sense for an arbitrary Artin local A, and patch together well to give well-behaved local deformation conditions which are locally representable (by which we mean the deformations of the Galois representation \overline{ρ}|G_{Q_{p}} are represented by some universal ring). So we get a nicely-behaved local deformation ring—in particular we get a ring for which we can compute the tangent space 𝔪/𝔪^{2} of its mod p reduction. If this tangent space has dimension at most 1 then the dimension calculations work out in the patching argument and the modularity lifting theorem follows.

If one is prepared to take these observations on board, then it becomes manifestly clear what the one of the main problems will be in proving that an arbitrary elliptic curve over the rationals is modular: we will have to come up with deformation conditions that are small enough to make the dimension calculations work, but big enough to encompass Galois representations that are not semistable. At primes q ≠ p this turned out to be an accessible problem; careful calculations by Fred Diamond in [Dia96] basically resolved these issues completely. Diamond’s main theorem had as a consequence the result that if E/Q had semistable reduction at both 3 and 5 then E was modular. The reason that both 3 and 5 occur is of course because he has to use the 3–5 trick if E[3] is reducible. A year or so later, Diamond, and independently Fujiwara, had another insight: instead of taking limits of Hecke algebras to prove that a deformation ring equalled a Hecke ring, one can instead take limits of modules that these algebras act on naturally. The resulting commutative algebra is more delicate, and one does not get modularity of any more elliptic curves in this way, but the result is of importance because it enables one to apply the machinery in situations where certain “mod p multiplicity one” hypotheses are not known. These multiplicity one hypotheses were known in the situations that Wiles initially dealt with but were not known in certain more general situations; the consequence was that Wiles’ method could now be applied more generally. Diamond’s paper [Dia97] illustrated the point by showing that the methods could now be applied in the case of Shimura curves over Q (where new multiplicity one results could be deduced as a byproduct), and Fujiwara (in [Fuj], an article which remains unpublished, for reasons unknown to this author) illustrated that the method enabled one to generalise Wiles’ methods to the Hilbert modular case, on which more later.

Getting back to elliptic curves over the rationals, the situation in the late 1990s, as we just indicated, was that any elliptic curve with semistable reduction at 3 and 5 was now proven to be modular. To get further, new ideas were needed, because in the 1990s the only source of modular mod p Galois representations were those induced from a character, and those coming from the Langlands–Tunnell theorem. Hence in the 1990s one was forced to ultimately work with the prime p = 3 (the prime p = 2 was another possibility; see for example [Dic01], but here other technical issues arise). Hence, even with the 3–5 trick, it was clear that if one wanted to prove that all elliptic curves over Q were modular using these methods then one was going to have to deal with elliptic curves that have rather nasty non-semistable reduction at 3 (one cannot use the 3–5 trick to get around this because if E has very bad reduction at 3 (e.g. if its conductor is divisible by a large power of 3) then this will be reflected in the 5-torsion, which will also have a large power of 3 in its conductor, so any curve A with A[5] ≅ E[5] will also be badly behaved at 3; one can make certain simplifications this way but one cannot remove the problem entirely). The main problem is then deformationtheoretic: given some elliptic curve E which is highly ramified at some odd prime p, how does one write down a reasonable deformation problem for E[p] at p, which is big enough to see the Tate module of the curve, but is still sufficiently small for the Taylor–Wiles method to work? By this we mean that the tangent space of the mod p local deformation problem at p has to have dimension at most 1. This thorny issue explains the five year gap between the proof of the modularity of all semistable elliptic curves, and the proof for all elliptic curves.

1.3 All elliptic curves over Q are modular.

As explained in the previous section, one of the main obstacles in proving that all elliptic curves over the rationals are modular is that we are forced, by Langlands–Tunnell, to work with p = 3, so elliptic curves with conductor a multiple of a high power of 3 are going to be difficult to deal with. Let us review the situation at hand. Let \overline{ρ} : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(k) be an irreducible Galois representation, where here k is a finite field of characteristic p. Such a \overline{ρ} has a universal deformation to ρ^{univ} : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(R^{univ}). This ring R^{univ} is a quotient of a power series ring W (k)[[x_{1}, x_{2}, …, x_{n}]] in finitely many variables, where here W (k) denotes the Witt vectors of the field k. But this universal deformation ring is too big for our purposes—a general lifting of \overline{ρ} to the integers O of a finite extension of Frac(W (k)) will not look anything like the Tate module of an elliptic curve (it will probably not even be Hodge–Tate, for example). The trick that Wiles used was to not look at such a big ring as R^{univ}, but to look at more stringent deformation problems, such as deforming \overline{ρ} to representations which came from finite flat group schemes over Z_{p}. This more restricted space of deformations is represented by a smaller deformation ring R^{♭}, a quotient of R^{univ}, and it is rings such as R^{♭} that Wiles could work with (the relevant computations in this case were done in Ravi Ramakrishna’s thesis [Ram93]).

In trying to generalise this idea we run into a fundamental problem. The kind of deformation problems that one might want to look at are problems of the form “ρ that become finite and flat when restricted to Gal(\overline{Q}_{p}/K) for K this fixed finite extension of Q_{p}”. However, for K a wildly ramified extension of Q_{p} the linear algebra methods alluded to earlier on become much more complex, and indeed at this point historically there was no theorem classifying finite flat group schemes over the integers of such p-adic fields which was concrete enough to enable people to check that the resulting deformation problems were representable, and represented by rings whose tangent spaces were sufficiently small enough to enable the methods to work.

A great new idea, however, was introduced in the paper [CDT99]. Instead of trying to write down a complicated deformation problem that made sense for all Artin local rings and then to analyse the resulting representing ring, Conrad, Diamond and Taylor construct “deformation rings” in the following manner. First, they consider deformations ρ : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(O) of \overline{ρ} in the case that O is the integers of a finite extension of Q_{p}. In this special setting there is a lot of extra theory available: one can ask if ρ is Hodge–Tate, de Rham, potentially semi-stable, crystalline and so on (these words do not make sense when applied to a general deformation ρ : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(A) of \overline{ρ}, they only make sense when applied to a deformation to GL_{2}(O)), and furthermore if ρ is potentially semi-stable then the associated Fontaine module D_{pst}(ρ) is a 2-dimensional vector space with an action of the inertia subgroup of Gal(\overline{Q}_{p}/Q_{p}) which factors through a finite quotient. This finite image 2-dimensional representation of inertia is called the type of the potentially semi-stable representation ρ, and so we can fix a type τ and then ask that a deformation ρ : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(O) be potentially semistable of a given type.

Again we stress that this notion of being potentially semistable of a given type certainly does not make sense for a deformation of \overline{ρ} to an arbitrary Artin local W (k)-algebra, so in particular this notion is not a deformation problem and we cannot speak of its representability. One of the insights of [CDT99] however, is that we can construct a “universal ring” for this problem anyway! Here is the trick, which is really rather simple. We have \overline{ρ} and its universal formal deformation ρ^{univ} to GL_{2}(R^{univ}). Now let us consider all maps s : R^{univ} → O, where O is as above. Given such a map s, we can compose ρ^{univ} with s to get a map ρ_{s} : Gal(\overline{Q}_{p}/Q_{p}) → GL_{2}(O). Let us say that the kernel of s is of type τ if ρ_{s} is of type τ, and if furthermore ρ_{s} is potentially Barsotti–Tate (that is, comes from a p-divisible group over the integers of a finite extension of Q_{p}) and has determinant equal to the cyclotomic character. A good example of a potentially Barsotti–Tate representation is the representation coming from the Tate module of an elliptic curve with potentially good reduction at p, and such things will give rise to points of type τ for an appropriate choice of τ.

Let R_{τ} denote the quotient of R^{univ} by the intersection of all the prime ideals of R^{univ} which are of type τ (with the convention that R_{τ} = 0 if there are no such prime ideals). Geometrically, what is happening is that the kernel of s is a prime ideal and hence a point in Spec(R^{univ}), and we are considering the closed subscheme of Spec(R^{univ}) obtained as the Zariski-closure of all the prime ideals of type τ. So, whilst R_{τ} does not represent the moduli problem of being “of type τ” (because this is not even a moduli problem, as mentioned above), it is a very natural candidate for a ring to look at if one wants to consider deformations of type τ. It also raises the question as to whether the set of points which are of type τ actually form a closed set in, say, the rigid space generic fibre of R^{univ}. If they were to not form a closed set then R_{τ} would have quotients corresponding to points which were not of type τ, but which were “close” to being of type τ (more precisely, whose reductions modulo p^{n} were also reductions of type τ representations). The paper [CDT99] calls the points in the closure “weakly of type τ” and conjectures that being weakly of type τ is equivalent to being of type τ. This conjecture was proved not long afterwards for tame types by David Savitt in [Sav05].

Now of course, one hopes that for certain types τ, the corresponding rings R_{τ} are small enough for the Taylor–Wiles method to work (subject to the restriction that p > 2 and that \overline{ρ} is absolutely irreducible even when restricted to the absolute Galois group of Q(\sqrt{(−1)^{(p−1)/2}p}), an assumption needed to make the Taylor–Wiles machine work), and big enough to capture some new elliptic curves. Even though the definition of R_{τ} is in some sense a little convoluted, one can still hope to write down a surjection W (k)[[t]] → R_{τ} in some cases (and thus control the tangent space of R_{τ}), for example by writing down a deformation problem which is known to be representable by a ring isomorphic to W (k)[[t]], and showing that it contains all the points of type τ (geometrically, we are writing down a closed subset of Spec(R^{univ}) with sufficiently small tangent space, checking it contains all the points of type τ and concluding that it contains all of Spec(R_{τ})). The problem with such a strategy is that it requires a good understanding of finite flat group schemes over the integers of the p-adic field K corresponding to the kernel of τ. In 1998 the only fields for which enough was known were those extensions K of Q_{p} which were tamely ramified. For such extensions, some explicit calculations were done in [CDT99] at the primes 3 and 5, where certain explicit R_{τ} were checked to have small enough tangent space. There is a general modularity lifting theorem announced in [CDT99] but it includes, in the non-ordinary case, an assumption the statement that R_{τ} is small enough for the method to work, and this is difficult to check in practice, so the result has limited applicability. However the authors did manage to check this assumption in several explicit cases when p ∈ {3, 5}, and deduced Theorem 1.6 If E/Q is an elliptic curve which becomes semistable at 3 over a tamely ramified extension of Q_{3}, then E is modular.

This is the main theorem of [CDT99] (see the second page of loc. cit.). The proof is as follows: if E[3] is irreducible when restricted to the absolute Galois group of Q(\sqrt{−3}) then they verify by an explicit calculation that either E is semistable at 3, or some appropriate R_{τ} is small enough, and in either case this is enough. If E[5] is irreducible when restricted to the absolute Galois group of Q(\sqrt{−5}) then E[5] can be checked to be modular via the 3–5 trick, and E can be proven modular as a consequence, although again the argument relies on computing enough about an explicit R_{τ} to check that it is small enough. Finally Noam Elkies checked for the authors that the number of j-invariants of elliptic curves over Q for which neither assertion holds is finite and worked them out explicitly; each j-invariant was individually checked to be modular.

At around the same time, Christophe Breuil had proven the breakthrough theorem [Bre00], giving a “linear algebra” description of the category of finite flat group schemes over the integers of an arbitrary p-adic field. Armed with this, Conrad, Diamond and Taylor knew that there was a chance that further calculations of the sort done in [CDT99] had a chance of proving the full Taniyama-Shimura conjecture. The main problem was that the rings R_{τ} were expected to be small enough for quite a large class of tame types τ, but were rarely expected to be small enough if τ was wild. After much study, Breuil, Conrad, Diamond and Taylor found a list of triples (p, \overline{ρ}, τ) for which R_{τ} could be proved to be small enough (p was always 3 in this list, and in one extreme case they had to use a mild generalisation of a type called an “extended type” in a case where R_{τ} was just too big; the extended type cut it down enough), and this list and the 3–5 trick was enough to prove

Theorem 1.7 (Breuil, Conrad, Diamond, Taylor (2001)) Any elliptic curve E/Q is modular.

1.4 Kisin’s modularity lifting theorems.

In this section we briefly mention some important work of Kisin that takes the ideas above much further.

As we have just explained, the Breuil–Conrad–Diamond–Taylor strategy for proving a modularity lifting theorem was to write down subtle local conditions at p which were representable by a ring which was “not too big” (that is, its tangent space is at most 1-dimensional). The main problem with this approach was that the rings that this method needs to use in cases where the representation is coming from a curve of large conductor at p are (a) difficult to control, and (b) very rarely small enough in practice. The authors of [BCDT01] only just got away with proving modularity of all elliptic curves because of some coincidences specific to the prime 3, where the rings turned out to be computable using Breuil’s ideas, and just manageable enough for the method to work. These calculations inspired conjectures of Breuil and Mézard ([BM02]) relating an invariant of R_{τ} (the Hilbert–Samuel multiplicity of the mod p reduction of this ring) to a representation-theoretic invariant (which is much easier to compute).

Kisin in the breakthrough paper [Kis09] (note that this paper was published in 2009 but the preprint had been available since 2004) gave a revolutionary new way to approach the problem of proving modularity lifting theorems. Kisin realised that rather than doing the commutative algebra in the world of Z_{p}-algebras, one could instead just carry around the awkward rings R_{τ} introduced in [CDT99], and instead do all the dimension-counting in the world of R_{τ} -algebras (that is, count relative dimensions instead). This insight turns out to seriously reduce the amount of information one needs about R_{τ}; rather than it having to have a 1-dimensional tangent space, it now basically only needs to be an integral domain of Krull dimension 2. In fact one can get away with even less (which is good because R_{τ} is not always an integral domain); one can even argue using only an irreducible component of Spec(R_{τ} [1/p]), as long as one can check that the deformations one is interested in live on this component.

There is one problem inherent in this method, as it stands: the resulting modularity lifting theorems have a form containing a condition which might be tough to verify in practice. For example, they might say something like this: “say \overline{ρ} is modular, coming from a modular form f. Say ρ lifts \overline{ρ}. Assume furthermore that ρ_{f} and ρ both correspond to points on the same component of some Spec(R_{τ} [1/p]). Then ρ is modular.” The problem here is that one now needs either to be able to check which component various deformations of \overline{ρ} are on, or to be able somehow to jump between components (more precisely, one needs to prove theorems of the form “if \overline{ρ} is modular coming from some modular form, then it is modular coming from some modular form whose associated local Galois representation lies on a given component of Spec(R_{τ} [1/p])”. Kisin managed to prove that certain R_{τ} only had one component, and others had two components but that sometimes one could move from one to the other, and as a result of these “component-hopping” tricks ended up proving the following much cleaner theorem ([Kis09]):

Theorem 1.8 (Kisin) Let p > 2 be a prime, let ρ be a 2-dimensional p-adic representation of Gal(\overline{Q}/Q) unramified outside a finite set of primes, with reduction \overline{ρ}, and assume that \overline{ρ} is modular and \overline{ρ}| Gal(\overline{Q}/K) is absolutely irreducible, where K = Q(\sqrt{(−1)^{(p−1)/2}p}). Assume furthermore that ρ is potentially Barsotti–Tate and has determinant equal to a finite order character times the cyclotomic character. Then ρ is modular.

Note that we do not make any assumption on the type of ρ; this is why the theorem is so strong. This result gives another proof of the modularity of all elliptic curves, because one can argue at 3 and 5 as in [CDT99] and [BCDT01] but is spared the hard computations of R_{τ} in [BCDT01]: the point is that Kisin’s machine can often deal with them even if their tangent space has dimension greater than one by doing the commutative algebra in this different and more powerful way. These arguments ultimately led to a proof of the Breuil–Mézard conjectures: see for example Kisin’s recent ICM talk.

The next part of the story in the case of 2-dimensional representations of Gal(\overline{Q}/Q) would be the amazing work of Khare and Wintenberger ([KW09a],[KW09b]), proving

Theorem 1.9 (Khare–Wintenberger) Serre’s conjecture (Conjecture 1.3) is true.

We have to stop somewhere however, so simply refer the interested reader to the very readable papers [KW09a] and [Kha07] for an overview of the proof of this breakthrough result.

The work of Khare and Wintenberger means that nowadays we do not have to rely on the Langlands–Tunnell theorem to “get us going”, and indeed we now get two more proofs of Fermat’s Last theorem and of the modularity of all elliptic curves: firstly, given an elliptic curve E, we can just choose a random large prime, apply Khare–Wintenberger to E[p] and then apply the theorem of Kisin above. Secondly, given an elliptic curve, we can apply the Khare–Wintenberger theorem to E[p] for all p at once, and then use known results about level optimisation in Serre’s conjecture to conclude again that E is modular. In particular we get a proof of FLT that avoids non-Galois cubic base change. However it seems to the author that things like cyclic base change and the Jacquet– Langlands theorem will still be essentially used in this proof, and hence even now it seems that to understand a full proof of FLT one still needs to understand both a huge amount of algebraic geometry and algebra, and also a lot of hard analysis.

1.5 Generalisations to totally real fields.

So far we have restricted our discussion to modularity lifting theorems that applied to representations of the absolute Galois group of Q. It has long been realised that even if one is mainly interested in these sorts of questions over Q, it is definitely worthwhile to prove as much as one can for a general totally real field, because then one can use base change tricks (the proofs of which are in [Lan80] and use a lot of hard analysis) to get more information about the situation over Q. One of the first examples of this phenomenon, historically, was the result in §0.8 of [Car83], where Carayol proves that the conductor of a modular elliptic curve over Q was equal to the level of the newform giving rise to the curve—even though this is a statement about forms over Q, the proof uses Hilbert modular forms over totally real fields.

Some of what we have said above goes through to the totally real setting. Let us summarise the current state of play. We fix a totally real number field F. The role of modular forms in the previous sections is now played by Hilbert modular forms; to a Hilbert modular eigenform there is an associated 2-dimensional p-adic representation of the absolute Galois group of F, and this representation is totally odd, in the sense that the determinant of ρ(c) is −1 for all complex conjugations c (there is more than one conjugacy class of such things if F ≠ Q, corresponding to the embeddings F → R). So formally the situation is quite similar to the case of F = Q. “Under the hood” there are some subtle differences, because there is more than one analogue of the theory of modular curves in this setting (Hilbert modular varieties and Shimura curves, both of which play a role, as do certain 0-dimensional “Shimura varieties”), but we will not go any further into these issues; the point is that one can formulate the notion of modularity, and hence of modularity lifting theorems in this setting. But can one prove anything? One thing we certainly cannot prove, at the time of writing, is

Conjecture 1.10 (“Serre”) Any continuous totally odd irreducible representation \overline{ρ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{p}) is modular.

Serre did not (as far as we know) formulate his conjecture in this generality, but it has become part of the folklore and his name seems now to be attached to it. We mention this conjecture because of its importance in the theory. If one could prove this sort of conjecture then modularity of all elliptic curves over all totally real fields would follow.

Just as in the case of F = Q, the conjecture can be refined—for example one can predict the weight and level of a modular form that should give rise to \overline{ρ}, as Serre did for F = Q in [Ser87]. These refinements, in the case of the weight, can be rather subtle: we refer the reader to [BDJ10] when F is unramified at p, and to work of Michael Schein (for example [Sch08]) and Gee (Conjecture 4.2.1 of [Gee]).

Once these refinements are made, one can ask two types of questions. The first is of the form “given \overline{ρ}, is it modular?”. These sorts of questions seem to be wide open for a general totally real field. The second type is of the form “given \overline{ρ} which is assumed modular, is it modular of the conjectured weight and level?”. This is the sort of question which was answered by Ribet (the level: see [Rib90]) and Edixhoven (the weight: see [Edi92]), following work of many many others, for F = Q. Much progress has also been made in the totally real case. For example see work of Jarvis [Jar99] and Rajaei [Raj01] on the level, and Gee [Gee07] on the weight, so the situation for Serre’s conjecture on Hilbert modular forms now is becoming basically the same as it was in the classical case before Khare–Wintenberger: various forms of the conjecture are known to be equivalent, but all are open.

Given that there is a notion of modularity, one can formulate modularity lifting conjectures in this setting. But what can one prove? The first serious results in this setting were produced by Skinner and Wiles in [SW99] and [SW01], which applied in the setting of ordinary representations (that is, basically, to representations ρ which were upper triangular when restricted to a decomposition group for each prime above p). Here is an example of a modularity lifting theorem that they prove ([SW01] and correction in [Ski]):

Theorem 1.11 (Skinner–Wiles, 2001) Suppose p is an odd prime and F is a totally real field. Suppose ρ : Gal(\overline{F} /F) → GL_{2}(\overline{Q}_{p}) is continuous, irreducible, and unramified outside a finite set of places. Suppose that det(ρ) is a finite order character times some positive integer power of the cyclotomic character, that ρ|D_{𝔭} is upper triangular with an unramified quotient, for all 𝔭|p, and that the two characters on the diagonal are distinct modulo p. Finally, suppose that \overline{ρ}|G_{F (ζ_{p})} is absolutely irreducible, and that \overline{ρ} is modular, coming from an ordinary Hilbert modular form f of parallel weight such that, for all 𝔭|p, the unramified quotients of ρ_{f}|D_{𝔭} and ρ|D_{𝔭} are congruent mod p. Then ρ is modular.

Note that Skinner and Wiles show that if \overline{ρ} has an ordinary modular lift, then many of its ordinary lifts are modular. Their technique is rather more involved than the usual numerical criterion argument—they make crucial use of deformations to characteristic p rings, and in fact do not show that the natural map R → T is an isomorphism, using base change techniques to reduce to a case where they can prove that it is a surjection with nilpotent kernel.

Kisin’s work on the rings R_{τ} of the previous section all generalised to the totally real setting, enabling Kisin to prove some stronger modularity lifting theorems which were not confined to the ordinary case. Kisin’s original work required p to be totally split in F, but Gee proved something in the general case. We state Gee’s theorem below.

Theorem 1.12 (Gee [Gee06],[Gee09]) Suppose p > 2, F is totally real, and ρ is a continuous potentially Barsotti–Tate 2-dimensional padic Galois representation of the absolute Galois group of F, unramified outside a finite set of primes, and with determinant equal to a finite order character times the cyclotomic character. Suppose that its reduction \overline{ρ} is modular, coming from a Hilbert modular form f, and suppose that for all v|p, if ρ is potentially ordinary at v then so is ρ_{f}. Finally suppose \overline{ρ} is irreducible when restricted to the absolute Galois group of F (ζ_{p}), and if p = 5 and the projective image of \overline{ρ} is isomorphic to PGL_{2}(F_{5}) then assume furthermore than [F (ζ_{5}) : F] = 4.

Then ρ is modular.

Note that, in contrast to Kisin’s result Theorem 1.8, in this generality “component hopping” is not as easy, and the assumption in this theorem that if ρ is potentially ordinary then ρ_{f} is too, are precisely assumptions ensuring that ρ and ρ_{f} are giving points on the same components of the relevant spaces Spec(R_{τ} [1/p]).

It is also worth remarking here that the Langlands–Tunnell theorem, Theorem 1.4, is true for totally odd irreducible representations of any totally real field to GL_{2}(F_{3}), so we can start to put together what we have to prove some modularity theorems for elliptic curves. Note that Gee’s result above has the delicate assumption that not only is \overline{ρ} modular, but it is modular coming from a Hilbert modular form whose behaviour at primes dividing p is similar to that of ρ. However Kisin’s “component hopping” can be done if p is totally split in F, and Kisin can, using basically the same methods, generalise his Theorem 1.8 to the totally real case if p is totally split, giving the following powerful modularity result:

Theorem 1.13 (Kisin [Kis07]) Let F be a totally real field in which a prime p > 2 is totally split, let ρ be a continuous irreducible 2dimensional representation of Gal(\overline{F} /F), unramified outside a finite set of primes, and potentially Barsotti–Tate at the primes above p. Suppose that det(ρ) is a finite order character times the cyclotomic character, that \overline{ρ} is modular coming from a Hilbert modular form of parallel weight 2, and that \overline{ρ}|G_{F (ζ_{p})} is absolutely irreducible. Then ρ is modular.

As a consequence, if F is a totally real field in which p = 3 is totally split, and if E/F is an elliptic curve with E[3]|F_{F (ζ_{3})} absolutely irreducible, then E is modular.

Of course all elliptic curves over F are conjectured to be modular, but this conjecture still remains inaccessible. If one were to attempt to mimic the strategy of proof in the case F = Q then one problem would be that for a general totally real field, there may be infinitely many elliptic curves with subgroups of order 15 defined over F, and how can one deal with such curves? There are infinitely many, so one cannot knock them off one by one as Elkies did. Their mod 3 and mod 5 Galois representations are globally reducible, and the best modularity lifting theorems we have in this situation are in [SW99], where various hypotheses on F are needed (for example F/Q has to be abelian in Theorem A of [SW99]). On the other hand, because Serre’s conjecture is still open for totally real fields one cannot use the p-torsion for any prime p ≥ 7 either, in general. It is not clear how to proceed in this situation!

1.6 Potential modularity pre-Kisin and the 𝔭–λ trick.

We have been daydreaming in the previous section about the possibilities of proving that a general elliptic curve E over a general totally real field F is modular, and observing that we are not ready to prove this result yet. Modularity is a wonderful thing to know for an elliptic curve; for example, the Birch–Swinnerton-Dyer conjecture is a statement about the behaviour of the L-function of an elliptic curve at the point s = 1, but the L-function of an elliptic curve is defined by an infinite sum which converges for Re(s) > 3/2, and it is only a conjecture that this Lfunction has an analytic continuation to the entire complex plane. One very natural way of analytically continuing the L-function is to prove that the curve is modular, because modular forms have nice analytic properties and the analytic continuation of their L-functions is wellknown.

For an elliptic curve over a general number field though, the L-function is currently not known to have an analytic continuation, or even a meromorphic continuation! However, perhaps surprisingly, it turns out that the results above, plus one more good new idea due to Taylor, enabled him to prove meromorphic continuation for a huge class of elliptic curves over totally real fields, and the ideas have now been pushed sufficiently far to show that the L-function of every elliptic curve over every totally real field can be meromorphically continued. We want to say something about how this all happened.

The starting point was Taylor’s paper [Tay02]. The basic idea behind this breakthrough paper is surprisingly easy to explain! Recall first Lemma 1.5, the 3–5 trick. We have an elliptic curve E/Q with E[5] irreducible, and we want to prove that E[5] is modular. We do this by writing down a second elliptic curve A/Q with A[3] irreducible and A[5] ≅ E[5]. Then the trick, broadly, was that A[3] is modular by the Langlands– Tunnell theorem, so A is modular by a modularity lifting theorem, so A[5] is modular, so E[5] is modular. The proof crucially uses the fact that the genus of the modular curve X(5) is zero so clearly does not generalise to much higher numbers.

However, if we are allowed to be more flexible with our base field, then this trick generalises very naturally and easily. Let us say that we have an elliptic curve E over a totally real field F, and we want to prove that E is potentially modular (that is, that E becomes modular over a finite extension field F′ of F, also assumed totally real). Here is a strategy. Say p is a large prime such that E[p] is irreducible. Let us write down a random odd 2-dimensional mod ℓ Galois representation ρ_{ℓ} : Gal(\overline{F} /F) → GL_{2}(\overline{F}_{ℓ}) which is induced from a character; because this representation is induced it is known to be modular. Now let us consider the moduli space parametrising elliptic curves A equipped with

1. an isomorphism A[p] ≅ E[p]

2. an isomorphism A[ℓ] ≅ ρ_{ℓ}.

This moduli problem will be represented by some modular curve, whose connected components will be twists of X(pℓ) and hence, if p and ℓ are large, will typically have large genus. However, such a curve may well still have lots of rational points, as long as I am allowed to look for such things over an arbitrary finite extension F′ of F! So here is the plan: first, consider this moduli problem. Second, find a point on this moduli space defined over F′, for F′ some finite extension of F. Next, ensure that F′ is totally real, and that our modularity lifting theorems are robust enough to apply in the two situations we’ll need them. More precisely, we need one modularity lifting theorem of the form “ρ_{ℓ} is modular over F′ and hence A/F′ is modular”, so A[p] = E[p] is modular, and then another one which says “E[p] is modular over F′, and hence E is modular”.

In 2001, our knowledge of modularity lifting theorems was poorer than it is now (because, for example, this was the era of [BCDT01] and before Kisin’s work on local deformation problems), so this idea would not run as far as it naturally wanted to. Let us sketch some of the issues that arise here. Firstly, if our moduli space has no real points at all for some embedding F → R then we cannot find a point over any totally real extension. So we need to check our moduli problem has real points. Secondly, the modularity lifting theorems available to Taylor in the totally real case were those of Skinner and Wiles so only applied in the ordinary setting (which is not much of a restriction because the curve will be ordinary at infinitely many places) but furthermore only applied in the “distinguished” setting (that is, the characters on the diagonal of the mod p local representation have to be distinct), so the completions of F′ at the primes above p had better not be too big, and similarly the completions of F′ at the primes above ℓ must not be too big either. This results in more local conditions on F′, and so we need to ensure two things: firstly that our moduli problems have points defined over reasonably small extensions of F_{𝔭} and F_{λ} for 𝔭 | p and λ | ℓ, and secondly that there is no local-global obstruction to the existence of a well-behaved F′-point (that is: given that our moduli problem has points over certain “small” local fields, we need to ensure it has a point over a totally real number field whose completions at the primes above p and ℓ are equally “small”). Fortunately, such a local-global theorem was already a result of Moret-Bailly (see [MB89]):

Theorem 1.14 Let K be a number field and let S be a finite set of places of K. Let X be a geometrically irreducible smooth quasi-projective variety over K. Let L run through the finite field extensions of K in which all the primes of S split completely. Then the union of X(L), as L runs through these extensions, is Zariski-dense in X.

Let us see how our plan looks so far. The idea now is that given an elliptic curve E/F, we find a prime p such that the Skinner–Wiles theorem applies to E[p] (so p is an ordinary prime for which E[p] locally at p has two distinct characters on the diagonal) and then write down a random odd prime ℓ and an induced representation ρ_{ℓ} of Gal(\overline{F} /F). Because Moret-Bailly’s theorem needs X geometrically irreducible, let us ensure that ρ_{ℓ} has cyclotomic determinant, and fix an alternating pairing on the underlying vector space (to be thought of as a Weil pairing). Consider now the moduli space of elliptic curves A equipped with isomorphisms A[p] ≅ E[p] and A[l] ≅ ρ_{ℓ} both of which preserve the Weil pairing. If we can find points on this curve defined over the completions of F at all primes above p, ℓ and ∞, then we might hope to conclude. But there are obstacles to finding such points. For example, if λ | ℓ is a prime of F and A[p] is unramified at λ, and b_{λ} is the number of points on A mod λ, then one can read off b_{λ} mod p from A[p], and one also has the Weil bounds on the integer b_{λ}, and these two constraints on b_{λ} might not be simultaneously satisfiable. Because of obstructions of this form, Taylor finds it easier to work not moduli spaces of elliptic curves over F, but with moduli spaces of so-called Hilbert–Blumenthal abelian varieties over F, that is, of higher-dimensional abelian varieties (say g-dimensional) equipped with a certain kind of polarization and endomorphisms by the integers in a second totally real field M of degree g over Q. As is becoming clear, this generalisation of the 3–5 trick is one of these ideas where the great inspiration now has to be offset by the large amount of perspiration that has to get the idea to work. On the other hand, the idea is certainly not restricted to elliptic curves E/F, and applies to a large class of ordinary Galois representations. Taylor managed to put everything together, even in the “pre-Kisin” modularity world, and managed in [Tay02] to prove that a wide class of ordinary 2-dimensional Galois representations of Gal(\overline{Q}/Q) were potentially modular. Let us state his result here.

Theorem 1.15 (Taylor) Let p be an odd prime, let ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{Q}_{p}) be a continuous odd irreducible representation unramified outside a finite set of primes, and assume

ρ|D_{p} = (\begin{matrix}χ^{n}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix})

with χ the cyclotomic character, n ≥ 1, and ψ_{1} and ψ_{2} two finitely ramified characters, such that the mod p reductions of χ^{n}ψ_{1} and ψ_{2} are not equal on the inertia subgroup of D_{p}. Then ρ becomes modular over some totally real number field.

As we said, the reason for the ordinarity assumption is that the result was proved in 2000 before the more recent breakthroughs in modularity lifting theorems. Taylor went on in 2001 in [Tay06] to prove an analogous theorem in the low weight crystalline case.

1.7 Potential Modularity after Kisin.

In this last section we put together Kisin’s modularity lifting theorem methods with Taylor’s potential modularity ideas. Together, the methods can be used to prove much stronger results such as the following:

Theorem 1.16 Let E/F be an elliptic curve over a totally real field. Then there is some totally real extension F′/F such that E/F′ is modular.

In particular, the L-function of E has meromorphic continuation to the whole complex plane. It is difficult to give a precise attribution to this theorem—the history is a little complicated. Perhaps soon after Taylor saw Kisin’s work on local deformation rings he realised that this theorem was accessible, but he could also see that perhaps the Sato–Tate conjecture was accessible too, and turned his attention to this problem instead. Whatever the history, it seems that it was clear to the experts around 2006 and possibly even earlier that the theorem above was accessible. The first published proof that we are aware of is in the appendix by Wintenberger to [Nek09], published in 2009. Much has happened recently in higher-dimensional generalisations of modularity lifting theorems—socalled automorphy lifting theorems, proving that various n-dimensional Galois representations are automorphic or potentially automorphic, and as a result one could also point to, for example, Theorem 8.7 of [BLGHT], where a far stronger (n-dimensional) result is proven from which the theorem follows. Another place to read about the details of the proof of this potential modularity theorem would be the survey article of Snowden [Sno], who sticks to the 2-dimensional situation and does a very good job of explaining what is needed. Snowden fills in various gaps in the literature in order to make his paper relatively self-contained modulo some key ideas of Kisin; if the reader looks at Snowden’s paper then they will see that the crucial ideas are basically due to Taylor and Kisin. One of the problems that needs to be solved is how to do the “component-hopping”: a general modularity lifting theorem might look like “if ρ_{1} is modular and ρ_{2} is congruent to ρ_{1}, and ρ_{1}, ρ_{2} give points on the same components of certain local deformation spaces, then ρ_{2} is modular”. As a result one needs very fine control on the abelian variety that is employed to do the 𝔭–λ trick; however this fine control can be obtained by Moret-Bailly’s theorem. The interested reader should read these references, each of which give the details of the argument.

1.8 Some final remarks.

This survey has to stop somewhere so we just thought we would mention a few things that we have not touched on. In the 2-dimensional case there is of course the work of Khare and Wintenberger, which we have mentioned several times but not really touched on more seriously. As mentioned already, Khare and Wintenberger have done a good job of summarising their strategy in their papers, and the interested reader should start there. Another major area which we have left completely untouched is the higher-dimensional R = T theorems in the literature, concerned with modularity of n-dimensional Galois representations. Here one uses automorphic forms on certain unitary groups to construct Galois representations, and new techniques are needed. The first big result is [CHT08], proving an R = T result at minimal level. After Kisin’s rings are factored into the equation and issues with components are resolved one can prove much stronger theorems; the state of the art at the time of writing seems to be the preprint [BLGGT], which is again very clearly written and might serve as a good introduction to the area. We apologise for not saying more about these recent fabulous works.

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