These are my live-TeXed notes for the course *18.786: Galois Representations* taught by Sug Woo Shin at MIT, Spring 2014. References.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

02/25/2014

##
Deformations of Galois representations

The global Langlands correspondence is roughly a correspondence between automorphic forms (representations) and -adic Galois representations. Suppose we are given the arrow automorphic representations -adic Galois representations (of course this is highly nontrivial), then it is relatively easy to show this arrow is injective. The strategy to show that this is also surjective (modularity of Galois representations ) is to first show that the reduction is the reduction of the Galois representation coming from a modular form (this is Serre's conjecture in general), then try to study infinitesimal liftings Now both sides then have more algebro-geometric structures (like a map of schemes ). It often turns out to be a closed immersion. If one can then show (e.g., by dimension reason) this is actually an isomorphism (* theorem*), then the surjectivity will follow. Our goal in next few lectures is to study the right hand side, the deformations of Galois representations.

##
Group-theoretic hypothesis

When deforming a representation , we would like to impose a finiteness condition on the profinite , in order to make sense of the the space of deformations and make ring noetherian. Fix a prime and a profinite group. We impose the following -finiteness assumption.

- a) b).
- If is topologically finitely generated, then a) and b) are both satisfied.

- Use Burnside basis theorem (see [Boe] Ex 1.8.1).
- The same generator for will work for the maximal pro- quotient of .
¡õ

- Let be a finite extension ( or ) Then satisfies Hyp: to check a), it is the same thing to check that there exists only finitely many abelian extension of exponent for a given local field. This follows from Kummer theory.
- Let be a finite extension, be a finite set of finite places of , be the maximal extension of unramified outside . Then satisfies Hyp (by global class field theory, see [DDT] 2.41 for details). Notice that itself is not known to be topologically finitely generated.

##
Liftings of mod representations

Let be a finite extension, with uniformizer and residue field .

Let

be the following category. The objects of

are complete noetherian local

-algebras

such that

. Here

is the unique maximal ideal of

and the completeness means that

is an isomorphism. Notice the

-algebra isomorphism

is unique if it exists.

The morphisms in are morphisms of local -algebras (i.e., ). It follows that induces an isomorphism .

Let

be a continuous representation. Define

given by

This is a functor and is called

*the liftings of *.

The functor

is represented by some

.

Consider

. Giving

is the same as giving an element

. Fix a lift

. Then

is in bijection with

, given by

. Hence

So

represents

.

02/27/2014

##
Some background on irreducible representations

We switch notations temperately. Let be a field, be an abstract group, be an finite dimensional -vector spaces. Recall that is *irreducible* if there is no proper -subvector space which is -stable.

(Schur)
If

is irreducible, then

is a division algebra over

.

Suppose

is nonzero, then

is nonzero and

-stable, hence

. Therefore

is invertible.

¡õ
(Schur)
Suppose

is algebraically closed. Let

be irreducible representations. Then

(Burnside's theorem)
If

is irreducible over

, then

is surjective.

We say

is

*Schur*, if

;

*absolutely irreducible* if for any

a field extension,

is irreducible.

is absolutely irreducible if and only if

is irreducible, if and only if

is irreducible and Schur.

See [CR] Section 29.
¡õ

##
Deformations of mod representations

We are back to the usual notation as in the section of lifting of mod representations.

Let

be a continuous representation. We define

the functor of deformations of

by

Here

if there exists

such that

. When

, this happens if and only if

can be chosen in

.

If

is Schur, then

is representable. Say by

, the

*universal deformation ring* of

.

Mazur's original proof uses Schlessinger's criterion of representability. One can also argue as for

(see [DDT] 2.36). Kisin's approach, roughly speaking, is to show that

is the geometric quotient of

by

(see [Boe] 2.1). Also see [Maz97] 10.

¡õ
##
Linear algebraic lemmas

One reduces to the case

is Artinian local

-algebra (since

are of this shape). Then induct on the length of

. The case

is easy and one can further reduce to the case

.

- Assume . Choose a minimal nonzero ideal : one has the filtration where is a -vector space; one can choose to be isomorphic to as an -module. Let such that . Then the induction hypothesis implies that . So we can write , where and . Now the equation tells us that in . Under the identification , we find that in , hence by Schur's lemma, itself must be a scalar too (we only need to be Schur in this part).
- By induction hypothesis, we may assume that lands in . Since as -submodules, we know that either or . In the latter case, it is immediate that lands in . The first case is more difficult. We build the -algebra , where . Then embeds into by . By induction hypothesis, we may assume is indeed an isomorphism. We may replace by and (because quotient by any thing in is fine). This is a much more concrete problem and the rest of the proof can be found in [CHT] Lemma 2.1.10. Here is roughly how it works. Extend -linearly and we obtain that , . Here
- is -linear,
- ,
- .

We want to get rid of and deal with purely matrix algebra. We claim that factors through the surjective map (here we used the *absolutely irreducible* assumption for the surjectivity), i.e., is trivial on . Let , then . By the surjectivity of , we know that because can be anything in . This proves the claim.

Now we are looking for such that for any . This is equivalent to that the coefficient of So we are reduced to the problem of showing that for , there exists such that

- ,
- ,
- .

Conceptually this means that every derivation on is given by the Lie bracket with some element . One can directly show that works.
¡õ

(Brauer-Nesbitt for

-coefficient). Suppose

is

*absolutely irreducible*. If

such that

. Then

.

Use similar reduction and then use Carayol's lemma when

. See [Boe] 2.2.1.

¡õ
([Gee] Ex 3.9) The universal lifting ring

is a power series ring in

variables over the the universal deformation ring

.

03/04/2014

##
Tangent spaces

We are going to work with the universal lifting rings (the same argument works for universal deformation rings). Write for short. Denote its unique maximal ideal by .

The adjoint representation

is given by conjugation

. We denote

(and the corresponding

-module).

One can analyze the tangent space of in terms of group cohomology of .

Write . This is the dimension of cotangent space (ignoring the -direction) of .

.

There is an exact sequence of finite dimensional

-vector spaces

Here any

gives the coboundary

. The result immediately follows.

¡õ
So we can know how big the ring is as long as we know the dimension of and of the adjoint representation.

Choose

such that

generate

as

-vector space, then

is surjective.

This follows from a topological version of Nakayama's lemma. In nice situations, is an isomorphism. In general this is too optimistic but one can further control the the kernel . Notice that . Here is the maximal ideal of . We shall construct an injective map When this vanishes, will be an formal power series ring and hence is formally smooth of dimension : there is no obstruction for liftings (controlled by ) and the space of liftings is -dimensional (controlled by and ).

To construct , we notice that is surjective. For , we can choose a lift of . Notice may not be a homomorphism and this failure is measured by Since , we know that and hence it makes sense to apply in this expression. One checks directly that So

is a continuous 2-cocycle.

- gives a well-defined class .
- if and only if there exists a choice of such that mod is a homomorphism. Here (so ).

The

-linear map

is

*injective*.

It suffices to show that if there exists

as in b) of the previous exercise, then

. Notice

. Since

is a lifting of

, we have a map

by the universal property. The composite map

is the identity map. This shows that

is injective. Let

(in particular

), we want to show that

. Suppose

maps

to

(so

). One checks directly that

. But

by injectivity, hence

. Therefore so

and

.

¡õ
So the number of generators of is and with the number of relations is equal to .

- If , then we have a non-canonical isomorphism .
- In general, .

03/06/2014

##
Generic fiber of universal lifting rings

The following is a basic algebraic fact.

Let

. There is a bijection between closed points of the generic fiber

(these are dense in the generic fiber) and pairs

, where

is a finite extension,

is continuous such that

.

Let be a closed point. Then we know from the above bijection and the universal property that corresponds to a representation . Let to be the category of local Artinian -algebra with residue field .

The ring

pro-represents the functor

given by

##
Deformation problems

In practice, we are more interested in lifting Galois representations with prescribed local behaviors (e.g., requiring good reduction at certain primes for elliptic curves). So one would like to work with certain subspaces of (or in terms of rings, certain quotient rings of ). We would like to make a checklist for technical conditions to define nice subspaces of .

We have a bijection

We explain why

is uniquely determined. Let

be the set of all ideals

such that

. This is nonempty by a). Using b) and c), we know

if and only if

. Moreover

is closed under finite intersection and nested infinite intersection by d) and e). So by Zorn's lemma, there exists a unique minimal ideal

. It is kernel invariant by f).

¡õ
03/11/2014

Typical deformation problems concerns local Galois representations

, where

is a local field. See [CHT] 2.4 for several examples: when

, the Fontaine-Laffaille liftings or ordinary liftings; when

, Taylor-Wiles liftings, are all deformation problems.

For applications, it is important to understand the ring theoretic properties of , e.g., Krull dimension, number of generators, number of relations and so on. We computed these for in terms of Galois cohomology. It is similar for .

Inside

, one can consider the annihilator of the image of

in

,

(think: the subspace cut out by

of the tangent space). We define

be its image in

.

Using the kernel-invariance property of , one can show that

is the full preimage of

.

So one can work directly at the level of cohomology instead of cocycles.

##
Global Galois deformation problems

We begin with a remark on fixing the determinant.

Now let be a number field. Let be a place of and fix . One has the local Galois group (well-defined up to conjugation). Let be a set of finite places of and .

Next we will define a deformation functor and show it is representable and study its ring theoretic properties.

is represented by

. When

, we write it as

.

Let

be all liftings of

and

be the deformation problem given by

. Then

is representable (which is

if

). Then one can construct

as

, where

is the minimal ideal such that

factors through

if and only if

for

. For more details, see [CHT].

¡õ
##
Presenting global deformation rings over local lifting rings

We have seen how to represent over and when happens to be a power series ring over . We are now going to represent over another bigger ring, the *local lifting ring*. This idea is due to Kisin.

Notice has the following universal object:

- , and
- , .

By the kernel-invariance property, . Moreover, it is well-defined element independent of the choice of the representative of the equivalence class. By the universal property of , we know that factors through

Define the

*local lifting ring* to be the completed tensor product

We have a natural map

.

Our next goal is to find the number of the generators and relations for presenting over using Galois cohomology.

As a first thought, suppose consists of all liftings. Write and be the maximal ideals. The the same argument as in Lemma 6 shows that Here sits diagonally. Rewriting this as

Two modifications are needed in general:

- to consider the tangent space over : one should replace by . Concretely, this requires the liftings at to be trivial, i.e., lies in the kernel of Notice the image of is , which we require to be trivial, i.e., . Write and , then if and only if .
- to allow general for : one requires that

The upshot is that is the of the complex

This motivates the definition of the mysterious complex in [Gee].

We define the complex

to be

Here

.

Write

to be the complex in the first row and

to be complex in the second row. So

We define , for or .

Next time we shall study the cohomology Similarly, the number of generators will be given by and the number of generators will be bounded by . Can we compute and ? This needs serious input from Galois cohomology, which we will do next time.

03/13/2014

There exists a surjection

. Here

and the number of the relation is at most

.

The proof goes as in Corollary

4.

¡õ
Our next goal is to compute for in terms of

- the usual local and global Galois cohomology,
- the dimension of the local conditions ,
- the dimension of the "dual Selmer group" (as the error term).

##
Computation of

Assume for simplicity that

- ( causes, e.g., problems at real places; see Kisin's modularity results on 2-adic representations),
- . This implies that there exists a splitting of Galois modules
- all places above of above lies in (this is a harmless assumption).

The fact is that all cohomology groups are finite dimensional over and concentrate in bounded degree. So we can define the *Euler characteristic*

There are four steps to compute .

Step 1
We have
This is clear exact sequence of complexes and the fact that the Euler characteristic is additive in long exact sequences. It follows that The latter is equal to due to the existence of the splitting .

Step 2
We compute in terms of usual Galois cohomology. By definition,
Again, the second term is equal to . Therefore

Step 3
Apply the local and global Euler-Poincare characteristic formula to and to get a formula for .

Step 4
It turns out when . is always easy. By the Euler-Poincare characteristic, to compute , it remains to compute and . The Poitou-Tate duality allows one to understand and in terms of (this is the error term mentioned above) and (easy) of the dual Galois module. When the error term vanishes, is zero so the deformation ring is indeed a power series ring.

To execute the last two steps, We need the following facts.

(Cohomological vanishing)
- Let be a nonarchimedean local field and be a finite -module. Then for . ( has cohomological dimension 2).
- When , for (here we use the assumption that ).
- When is a number field and is a finite -module, for (here we use the assumption that as well: the -cohomological dimension of a number field is 2 when ).

From the long exact sequence in cohomology, it follows that

for

.

Another input is the determination of the Euler-Poincare characteristics.

(Euler-Poincare characteristic)
- When is a nonarchimedean local field of characteristic 0, then This is zero unless , in which case is .
- When is a number field and , then

Write

, then

(here we use the assumption that all primes above

are in

).

The final key inputs are the local and global duality theorems.

Let

be a local or global field. Let

be a finite

-module. Let

be the linear dual and

be the Cartier dual. Here

the twist of

by the cyclotomic character

.

Let

. There is a natural perfect pairing

This gives identification

(Local duality)
Suppose

is a nonarchimedean local field. Then

(Poitou-Tate)
We have a nine term exact sequence

Back to our situation with , by definition we have the following exact sequence

Notice that 1, 4, 5 terms are the same as in Poitou-Tate.

Suppose we have a commutative diagram
where is a subspace. If the top row is exact, then the second row is still exact. Take the first row to be the Poitou-Tate exact sequence and . Then

It follows that Since is absolutely irreducible, we also know that . Combining these with the Euler-Poincare characteristics computation, we obtain desired formulas for and . For explicit expressions, see [Gee] 3.24.

03/18/2014

(I was out of town for AWS 2014, this section is shameless copied from Rong Zhou's typed notes.)

The notation is as above. is a finite extension and is a continuous representations, and fix a character , which reduces to .

We constructed the ring which represents the lifting problem for with the fixed determinant . Its generic fiber has closed points corresponding to -adic liftings of with determinant .

The goal of the next week will be to study the properties (e.g. irreducible components, dimension) of (or ). We split into the two cases and (the second requires some background in -adic hodge theory). This information (irreducible components, dimension) enters into the proof of automorphy lifting theorems. In order to control () or the Krull dimension of , we saw last time that we need to know the , or the Krull dimension of .

##
Local universal lifting rings

has finitely many irreducible components and each irreducible component is generically formally smooth (over

) and of dimension

.

We define a closed point

of

corresponding to the

-adic representation

to be

*smooth* if

. It is shown in [BLGGT], Lemma 1.3.2 that the smooth points are Zariski dense. Thus it suffices to prove that

. Notice this ring is the universal lifting ring for

with coefficients in

(Lemma

9). The idea is to mimic the argument for liftings of

using tangent spaces and Galois cohomology. Define

which, if we fix the determinant, is equal to

Hence there exists a surjection

One shows in the same way as before that

if

.

Thus it suffices to prove and . This follows from the -adic version of local duality and the Euler-Poincare formula. The first gives us (the second equality follows from the smoothness), and the second gives . These two together imply what we wanted.
¡õ

Let

be a nonempty subset of irreducible components of

. We define

to be the largest quotient of

which is

- reduced and -torsion free;
- .

- Let . Then is a deformation problem.
- is equidimensional of dimension . (Note that .

- The non-trivial part is to show that is invariant (this is [BLGGT] Lemma 1.2.2).
- is open and dense in . Let be an irreducible component of and define . One checks that is an irreducible component and is non-empty with In fact, suppose and . Take a sequence of ideals: Then since the quotient is the ring of integers in a finite extension of .
¡õ

Now consider the map which takes finite dimensional Weil-Deligne representations of on -vector space to equivalence classes of triples , where is an representation of the inertia subgroup, is nilpotent, and and commute.

An

*inertial type* is any

in the image. A Weil-Deligne representation is

*of type * if it lies in the preimage of

.

A representation is of

*unramified type* if it is in the preimage of

of any dimension.

03/20/2014

Last time we looked at local universal lifting rings for . We will leave the important Taylor-Wiles deformation (allowing auxiliary primes with ramification) in the homework. Another important deformation problem is the Ihara avoidance defomations due to Taylor (for , Ihara's lemma allows one to raise the level; but Ihara's lemma is not known in higher dimensional. The Ihara avoidance deformation was introduced to bypass Ihara's lemma).

##
Local universal lifting rings

As always, all Galois representations are finite dimensional on -vector spaces. There are more -adic representations of than -adic representations (where the wild inertia is almost killed). The slogan of *-adic Hodge theory* is to try to understand -adic representations of through linear algebraic categories via equivalence (at least fully faithful embeddings) of categories. There exists a hierarchy of -adic Galois representations:

crystalline semistable potentially semistable ( = de Rham) Hodge-Tate all

We will not explain these technical terms but show some analogies with -adic () representations and representations comes from geometry (smooth projective varieties ).

-adic |
crystalline |
semistable |
potentially semistable |
de Rham |

-adic |
unramified |
inertia acting unipotently |
inertia acting potentially unipotent |
all |

smooth projective variety |
good reduction |
semistable reduction |
potentially semistable reduction |
all |

There are two important invariants associated to potentially semistable representations: WD (a Weil-Deligne representation of ) and HT (Hodge-Tate weights, a multiset of integers).

###
The Weil-Deligne functor

When , taking the Weil-Deligne representation gives a functor from *all* -representations to WD-representations of . When , we only define the Weil-Deligne representation functor for potentially semistable representations. Let be a finite Galois extension and be the maximal unramified subextension over . Take to be sufficient large (containing the Galois closure of ), e.g., . Let be the absolute Frobenius on .

We define

to be the category of WD-representations

of

such that

is unramified.

- There exists a dimension preserving functor from the category of potentially semistable representations such that is semistable to the category .
- There exists an equivalence between .

We treat the first functor as a black box (-adic Hodge-Tate theorem). The second functor is purely linear algebraic and can be described more easily. Let be an embedding. Suppose , we define via and , here , is the absolute -adic valuation, and . Then and the isomorphism class does not depend on .

Allowing to be larger and larger, we obtain a functor from potentially semistable -representations to the category of WD-representations of ().

###
Hodge-Tate weights

Given , we will define a collection , where is an unordered multiset of integers. Each in has the following multiplicity It is known that the sum of all these multiplicities is equal to (for any Hodge-Tate representation). These numbers can be read off from some natural filtration (defined after extending coefficients) attached to .

The

-adic cyclotomic character has Hodge-Tate weight

for any

.

When

is an abelian variety and

is given by its

-adic Tate module. Then

with 0 and 1 each occurring

times.

If

is an eigen cuspform of weight

. The associated Galois representation

has

.

If

has finite image, then all Hodge-Tate weights are 0.

###
Potentially semistable local lifting rings