Draft: March 25, 2012
DEFORMATIONS OF GALOIS REPRESENTATIONS ANDAPPLICATIONS — TALK AT THE IHP CONFERENCE ON THE
OCCASION OF THE GALOIS BICENTENNIAL
MATTHEW EMERTON

1. Introduction
The theory of Galois representations with finite field coefficients begins, as far
as I know, with Galois himself. In modern terms, he showed that an irreducible
polynomial f (x) ∈ Q[x] of prime degree p is solvable by radicals if and only if the
Galois group G of f (x) admits an embedding
 ×

Fp Fp
G ,→
⊂ GL2 (Fp ).
0
1
 ×

Fp Fp
(Here the groupis thought of a subgroup of Sym(Fp ), the full permuta0
1


a b
·x=
tion group of Fp , by its affine-linear action on Fp , i.e. via the action
0 1
ax + b. The claimed embedding of G is then obtained by choosing an appropriate
identification between Fp and the p roots of f (x).) Given such a polynomial f (x),thinking of its Galois group G as a quotient of the absolute Galois group GQ of Q,
one obtains a representation ρ : GQ → GL2 (Fp ).1 This is an example of a (twodimensional, mod p) Galois representation.
The basic objective of the theory of deformations of Galois representations is to
study liftings of representations ρ : GQ → GLn (Fp ) to representations ρn : GQ →GLn (Z/pn ), and ultimately to p-adic representations ρ : GQ → GL(Zp ).2
Somewhat more precisely, given ρ, one would like to describe the collection of
all possible liftings, perhaps satisfying some conditions. Typically there are many
liftings (although it is not always obvious a priori that there are any liftings at all),and one thinks of them as lying in a “space”, the Galois deformation space; twoliftings ρ and ρ0 are considered close if they are p-adically close, i.e. if the mod pn
reductions ρn and ρ0n coincide for some power pn (and the larger the value of n, thecloser are the two liftings).
The theory of deformations has another, grander, objective, though, which provided one of the primary motivations for its introduction (by Barry Mazur), and
1The overline in the notation is chosen to indicate that ρ has coefficients in characteristic p,rather than characteristic zero.
2There is a technical distinction between liftings — where one considers the literal homomorphism into a matrix group — and deformations, where one considers such homomorphisms up to
conjugation. However, I will suppress this distinction in this talk. Essentially, if one understands
the liftings of ρ, then one passes to the deformations of ρ by forming an appropriate quotient.
Also, Fp can be replaced by a more general finite field k, and Zp by a the ring of integers in a
finite extension of Zp — and in fact it is technically important to consider these more generalcoefficients, although I will also suppress that point in this talk. Finally, the field Q whose absolute
Galois group we are considering could be replaced by any number field, or also a local field.
1

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MATTHEW EMERTON

which remains the primary motivation for its study. Namely, as Michael Harris explained in his talk, one expects there to be a reciprocity between (certain — namely
algebraic) automorphic forms (for the group GLn , say) and (certain — namely geometric, in the sense of Fontaine and Mazur) n-dimensional p-adic representationsof GQ . A basic strategy for proving such a result is as follows:• Construct Galois representations attached to the automorphic forms of interest, and identify which Galois deformation space(s) they lie in.
• Show that there are many automorphic forms, so that they contribute many
points to the relevant Galois deformation space(s).
• Bound the size of the relevant Galois deformation space(s) from above.
• Combining the previous two points, conclude that every p-adic Galois representation satisfying the appropriate conditions arises from one of the automorphic form under consideration.
This was the strategy introduced by Wiles in his proof (completed by him together with Richard Taylor) of the modularity theorem for semistable elliptic curves
over Q, and hence of Fermat’s Last Theorem, and has remained the basic strategy
in the proof of the many subsequent results in the theory of automorphic forms andGalois representations, such as the proof of the full modularity theorem for elliptic curves over Q (by Breuil, Conrad, Diamond, and Taylor) and the proof of theSato–Tate conjecture for elliptic curves over Q (by Clozel, Harris, Shepherd-Barron,
Shin, and Taylor).
2. An example
I would like to illustrate Galois deformation theory through an example. My goal
is to give a simple, but precise, statement in one particular case, which will give afeel for the kind of statements that are proved in the general theory. Along the way,
I hope to illustrate by example some of the conditions that arise in deformation
theory, as well as the relationship of the theory to Diophantine problems and toautomorphic forms.
Example 1. In fact, I will give two examples. The first goes back in some sense
before Galois, to Gauss. Namely, if pn is a prime, then for any commutative ring A,
we writenµpn (A) := {a ∈ A | ap − 1}.
This is a functor of A; it functorially assigns an abelian group to each ring A. Since
the assignment is made by solving an equation in A, it is a group scheme (in fact afinite flat group scheme) over Spec Z.
In particular, since it is a functor, GQ acts on µpn (Q). This latter group is known
to be cyclic of order pn (it coincides with µpn (C)), and so we obtain a characterχn : GQ → GL1 (Z/pn ) = (Z/pn )× , known as the mod pn cyclotomic character.
These characters are successive liftings of the mod p character χ1 , which we will
also denote by χ, and taken together they give rise to a characterχ : GQ → GL1 (Zp ) = Z×p,
the p-adic cyclotomic character.
Since each χn arises by evaluating the finite flat group scheme µpn on Q, we saythat χn is finite flat over Spec Z. If p is odd, then one can furthermore characterizeeach χn as being the unique lift of χ to a representation with coefficients in Z/pn

DEFORMATIONS OF GALOIS REPRESENTATIONS AND APPLICATIONS

3

that is finite flat over Spec Z. (If p = 2, then χ is the trivial character, which admits
the trivial character as a lift, in addition to the χn .)Example 2a. Let us return to the context of Galois mentioned at the beginning of the introduction, and consider the simplest, and one of the most celebrated,
examples: namely let f (x) = x2 + 1. This is certainly an irreducible quadratic
equation over Q, and Galois’s theorem suggests that we consider the Galois representation

1 F2
ρ : GQ →,0 1
which factors through the quotient Gal(Q(i)/Q) = {1, c} (with cdenoting complex
1 1
conjugation), and is defined by mapping c to the matrix.
0 1
Now this Galois representation has a certain very nice property. Namely, just as
with the cyclotomic characters χn , the representation ρ is finite flat, i.e. arises by
evaluating a finite flat group scheme on Q.
Here is the relevant group scheme: it corresponds to the functor
G1 : A 7→ {a0 + a1 I ∈ A[I] | (a0 + a1 I)2 = 1, a0 a1 = 0}.
Here A is any commutative ring, and I is a formal variable adjoined to A which
satisfies the relation I 2 = −1. The group law is just given by multiplication in thering A[I].
If we compute the values of G1 on Q, we find that G1 (Q) is the two-dimensional
F2 -vector space spanned by −1 + 0 · I and 0 + i · I. (Here i ∈ Q denotes theusual element of square −1.) Since G1 is a functor, GQ acts on G1 (Q), and one
immediately sees that it does so via the representation ρ. Thus ρ does indeed arise
from the finite flat group scheme G1 over Spec Z, and so ρ is finite flat over Spec Z.
We can easily write down some liftings of ρ: For each n ≥ 1, define the finite flatgroup schemenGn : A 7→ a0 + a1 Zn + · · · + a2n −1 Zn2 −1 ∈ A[Zn ] |nn(a0 + a1 Zn + · · · + a2n −1 Zn2 −1 )2 = 1, ai aj = 0 for all i 6= j ,where again A is any commutative ring, and now Zn is a formal variable adjoinednto A which satisfies the relation Zn2 = −1. Squaring, and identifying Zn2 with Zn−1
(and setting Z1 = I), we obtain maps
(1)

Gn → Gn−1 → · · · → G1 .
nFurthermore, if we choose ζn ∈ Q such that ζn2 = −1 (so ζn is a primitive 2n+1 st
root of 1), then one computes that Gn (Q) is the free Z/2n -module of rank twonnspanned by ζn2 + 0 · Zn + · · · + 0 · Zn2 −1 and 0 + ζn · Zn + · · · 0 · Zn2 −1 . Thus theaction of GQ on Gn (Q) gives rise to a representation ρn : GQ → GL2 (Z/2n ), and aconsideration of (1) shows that the ρn are successive lifts of ρ. Again, each of themis (by construction) finite flat over Spec Z.
We may combine all the ρn into a 2-adic lift ρ : GQ → GL2 (Z2 ) of ρ, and usingclass field theory (more or less), one can then prove the following result:ρ is the unique deformation of ρ which has determinant equal to χ (the 2-adic
cyclotomic character), and for which each reduction ρn is finite flat over Spec Z.

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MATTHEW EMERTON

Example 2b. The preceding result is a typical “upper bound” on the size of adeformation space, albeit in a very special (and perhaps not so interesting) context.
However, we can make the context more interesting by introducing an elliptic curve,
as we will now do.
Consider the elliptic curve E cut out by (the projectivization of) the equationy 2 + xy + y = x3 − x2 − x − 14.
This elliptic curve is often denoted X0 (17), because it is isomorphic (over C) to(the completion of) the quotient of the complex upper half-plane by the discrete
group


 a b
| a, b, c, d ∈ Z, ad − 17bc = 1
Γ0 (17) :=17c dof Mobius transformations.
The usual chord–tangent law makes E an algebraic group, and we may consider
its subgroup scheme of 2n -torsion points E[2n ], for any n ≥ 1. If we take the Qpoints, then E[2n ](Q) is free of rank two over Z/2n , and so the GQ -action on it
gives a representation ψn : GQ → GL2 (Z/n). As usual, we write ψ := ψ1 . The ψnare successive lifts one of the other, and we may combine them into a 2-adic lift
ψ : GQ → GL2 (Z2 ) of ψ, the so-called 2-adic Tate module of E.
We can compute the 2-torsion points explicitly: they are the points at which thetangent line to the curve is vertical, i.e. at which
2y + x + 1 = 0.
One finds the three 2-torsion points
P = (11/4, −15/8), Q = (−1 − 2i, i), P + Q = R = (−1 + 2i, −i),as well, of course, as the point O at infinity, which is the identity for the group law.
Clearly the action of GQ on these points factors through Gal(Q(i)/Q), and one sees
immediately that c(P ) = P , while c(Q) = R = P + Q. Thus in fact ψ ∼= ρ.
We can go further and compute E[4](Q), and doing this, one discovers that ψ2
is isomorphic to ρ2 . At this point one might wonder if ψn is isomorphic to ρn for
all n, but general principles show that this is not possible: if E[2n ] were finite flatfor every n, then a theorem of Grothendieck would show that E has good reduction
at every prime, but the curve E becomes a nodal cubic modulo 17 (it acquires anode at the point (7, 13)).
In fact, the group schemes E[2n ] giving rise to the Galois representations ψn are
all finite flat over Spec Z[1/17] (since E has good reduction away from 17), andso we say that the representations ψn are finite flat over Spec Z[1/17]. But theψn have only a weaker property locally at the prime 17: they are semistable at 17.
(This choice of terminology reflects the use of the term semistable to describe nodal
singularities of curves.)
One can then prove the following result:ρ and ψ are the only two 2-adic deformations of ρ whose determinants are equalto χ the 2-adic cyclotomic character, and whose reductions modulo 2n are finite flatover Spec Z[1/17] and semistable at 17, for each n ≥ 1.
The elliptic curve E corresponds (in the sense of the modularity theorem for
elliptic curves over Q) to the unique normalized cuspform of weight 2 on Γ0 (17),fE := q − q 2 − q 4 − 2q 5 + 4q 7 + · · · ,DEFORMATIONS OF GALOIS REPRESENTATIONS AND APPLICATIONS

5

and this statement is a typical result relating a certain deformation space to a spaceof automorphic forms.
More precisely, the condition of finite flatness away from 17, the semistability at
17, and the determinant condition, taken together correspond to the condition ofa modular form being of weight two and level Γ0 (17). The two possible lifts thencorrespond to the two normalized Hecke eigenforms of weight two and level Γ0 (17),namely the Eisenstein series
2E2 := + q + 3q 2 + 4q 3 + 6q 5 + 12q 6 + 8q 7 + · · · ,3and the cuspform fE .3
3. The general theory
The computations of the particular deformation spaces discussed in the preceding
section are relatively straightforward, requiring just class field theory and a little
bit of the theory of congruences of modular forms (together with Wiles’s miraculous
numerical criterion!).4 In general, the problem of computing all the deformationsof a given ρ, and of relating these lifts to automorphic forms, is difficult. Taylor andWiles introduced a method for doing this, which again relies on class field theory,
but in its more subtle manifestations through the theory of Galois cohomology.
It is not possible in the present talk to give many details about the Taylor–Wiles
method, but I will try to give some idea of how it works. For this, let me returnto the strategy described in the introduction, and elaborate a little on each of thesteps.
Constructing Galois representations attached to automorphic forms.
[Much of this may be covered in Michael Harris’s talk.] For automorphic forms
on GL1 , this is more-or-less class field theory. The two-dimensional Galois representations attached to classical modular forms (which are automorphic for GL2 )were constructed by Deligne (and Deligne-Serre in the case of weight one modular
forms), building on earlier ideas of Eichler, Shimura, and Ihara, and following theconjecture of their existence by Serre. The generalization of Deligne’s construction
from classical modular forms to Hilbert modular forms was achieved by Carayol,
Wiles, and Taylor.
The most general currently known results are due to many people, including
Clozel, Kottwitz, Harris and Taylor, Labesse, Shin, and Chenevier. Collectively,
their results attach Galois representations to regular algebraic essentially conjugate
self-dual cuspidal automorphic representations on GLn over a CM field (with n
arbitrary). In order to know precisely which deformation space these Galois representations lie in (i.e. which conditions, such as finite flat or semistable, these Galois
representations satisfy), it is important to know precise local-global compatibility
statements about the Galois representations. In the case of n = 1 this is the compatibility of local and global Artin maps. In the case of n = 2 and Hilbert modular
forms, it is due to Langlands, Deligne, Carayol, T. Saito, Taylor, Skinner, Kisin,
3The fact that our deformation problem captures all the eigenforms of weight 2 and level 17reflects the fact that the unique cuspform fE is in fact congruent to the Eisenstein series mod 2(in fact even mod 4, if we ignore the constant term). If we replaced 17 by another prime, thenthe analogous deformation problem would only capture those modular forms which are congruent
to the Eisenstein series mod 2.
4
See Calegari and Emerton, On the ramification of Hecke algebras at Eisenstein primes.

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MATTHEW EMERTON

and T. Liu. In the case of general n, it is due to Harris–Taylor, Taylor–Yoshida,
Shin, Barnet-Lamb–Gee–Geraghty–Taylor, and Caraiani.
Showing that there are many automorphic forms, and hence many automorphic Galois representations. This is one of the key steps in the Taylor–Wiles method. One quantifies “sufficiently many” in the following way: one augments the level of the automorphic forms under consideration, and shows that thedimension of the space of automorphic forms grows in proportion to the increase inlevel. This is typically achieved by a cohomological argument, and necessitates a
restriction on the nature of the automorphic forms under consideration — e.g. they(or rather the automorphic representations that they generate) should be discrete
series at infinity.
There is another, very fundamental, issue that has to be confronted in this step,which is related to the fact that, in the sketch of the strategy, I have written Galois
deformation space(s), rather than just Galois deformation space. What is intended
by this is that one must identify the candidate ρ whose deformations are to beconsidered.
Essentially, if one wants to employ this method to prove that a particular p-adic
Galois representation, or class of Galois representations, arises from an automorphic
form, then one must first show that its mod p reduction arises from an automorphic
form (so as to know that there any automorphic points in the relevant deformation
space at all). That this should be true goes under the general rubric of Serre’s
conjecture (since Serre proposed the first form of such a conjecture in the case of
two-dimensional mod p representations and classical modular forms), and turns
out to be one of the most difficult parts in implementing the strategy. In Wiles’s
original arguments, he got around the problem of not knowing Serre’s conjecture byappealing to a deep theorem in automorphic forms due to Langlands and Tunnell,and combining it with his celebrated 3 − 5 trick. Since then, the 3 − 5 trick has been
turned (particularly through the work of Taylor) into a basic method in the theory,and in its most refined form was used by Khare and Wintenberger, and Kisin, toprove Serre’s original conjecture for two-dimensional mod p representations.
For higher dimensional Galois representations, one currently has a potential
Serre’s conjecture (developed in the work of Harris, Shepherd-Barron, and Taylor,and further developed since) — that is to say one has Serre’s conjecture, but only
after restricting ρ to a certain open index subgroup of the Galois group on which it
is originally defined. Removing the “potential” from this result is one of the mostimportant, but perhaps one of the most difficult, open problems in the theory.
Bounding the size of a deformation space from above. This is the heart of
the Taylor–Wiles method. The basic problem is that it is hard to measure the sizeof a deformation space a priori. The tangent space to ρ in the deformation spacecan be computed via Galois cohomology, but it is typically large, and a priori one
doesn’t know if this is because the deformation space is actually of high dimension,or just very singular at ρ.
What the Taylor–Wiles method does is very carefully add primes to the levelof the automorphic forms under consideration, and simultaneously consider larger
deformation spaces, in which the conditions at these primes are relaxed. I say “verycarefully” because the heart of the method is that these primes are chosen in sucha way that the tangent space to ρ doesn’t change as the primes are added. On theother hand, by the previous step, one knows that the number of automorphic points

DEFORMATIONS OF GALOIS REPRESENTATIONS AND APPLICATIONS

7

in deformation space is systematically increasing as the primes are added. Passing
to an appropriate limit, one ends up in a situation where all the directions in thetangent space are accounted for by automorphic forms.
The details of how to choose these auxiliary “Taylor–Wiles” primes is one of themost delicate points of the theory, involving subtle points of finite group theory.
At this point, and also in the previous step, one finds that it is in fact technicallymore difficult (even impossible, in general) to work with reducible ρ, and so it iscommon to restrict attention to p-adic Galois representations for which the modp representation ρ is irreducible (or satisfies an even stronger condition, known as“non-degeneracy”, or “having big image”).
There are many other subtleties that arise; for example, adding Taylor–Wiles
primes lets one show that the correct dimension is achieved by automorphic Galois
representations, but doesn’t necessarily show that they fill out all the irreducible
components of deformation space.5 Kisin introduced a technique for dealing withthis problem, and thus pushed the method further; hence people now speak of the“Taylor–Wiles–Kisin method” . . . .
Concluding. Once the Taylor–Wiles method has been successfully carried out,
one knows that all the points in the Galois deformation space of interest, but withthe conditions at the auxiliary, Taylor–Wiles, primes relaxed, are explained by automorphic forms. It is now easy to deduce that all the points in the original Galois
deformation space are also explained by automorphic forms: one just reimposes
the original conditions at the auxiliary primes (whatever they were), and takes into
account local-global compatibility at these primes for the Galois representations
attached to automorphic forms.
Applications. [Return to some recent results such as Sato–Tate, . . . ]Mathematics Department, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637
E-mail address: emerton@math.uchicago.edu

5In fact the issue is more nuanced than this; it is really components in the local deformation
space at p which are at issue, but I will elide this point.