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
![$$\mathcal{R}^\Box_{\bar\rho}(A)\cong\mathfrak{m}_A^{n^2}\cong\Hom_{\mathcal{C}_\mathcal{O}}(\mathcal{O}[ [X_{i,j}] ]_{i,j=1}^n, A).$$](./latex/latex2png-GaloisRepresentations_32344580_.gif)
So
![$\mathcal{O}[ [ X_{i,j}] ]_{i,j=1}^n$](./latex/latex2png-GaloisRepresentations_196908519_-6.gif)
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
![$\End_{k[\Gamma]}(\rho)=\{\phi\in\End_k(V): \phi\rho(\gamma)=\rho(\gamma)\phi,\forall \gamma\in\Gamma\}$](./latex/latex2png-GaloisRepresentations_260279013_-6.gif)
is a division algebra over

.
Suppose
![$\alpha\in\End_{k[\Gamma]}(\rho)$](./latex/latex2png-GaloisRepresentations_73738464_-6.gif)
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
![$k[\Gamma]\rightarrow \End_k(V)$](./latex/latex2png-GaloisRepresentations_251315174_-5.gif)
is surjective.
We say

is
Schur, if
![$\End_{k[\Gamma]}(\rho)=k$](./latex/latex2png-GaloisRepresentations_201823886_-6.gif)
;
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
![$A=\mathbb{F}[\varepsilon]/(\varepsilon^2)$](./latex/latex2png-GaloisRepresentations_57364708_-5.gif)
.
- 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
![$A=\mathbb{F}[\varepsilon]/(\varepsilon^2)$](./latex/latex2png-GaloisRepresentations_57364708_-5.gif)
. 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
![$\mathbb{F}[\Gamma]$](./latex/latex2png-GaloisRepresentations_88906499_-5.gif)
-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
![$\phi: \mathcal{O}[ [X_1,\ldots, X_d] ]\rightarrow R^\Box$](./latex/latex2png-GaloisRepresentations_123969035_-5.gif)
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
![$\mathcal{O}[ [X_1,\ldots, X_d] ]/J_f\twoheadrightarrow \mathcal{O} [[ X_1,\ldots,X_d] ]/J=R^\Box$](./latex/latex2png-GaloisRepresentations_259244193_-5.gif)
. Since

is a lifting of

, we have a map
![$$\eta: \mathcal{O} [[ X_1,\ldots,X_d] ]/J\rightarrow \mathcal{O} [[ X_1,\ldots,X_d] ]/J_f$$](./latex/latex2png-GaloisRepresentations_174759622_.gif)
by the universal property. The composite map
![$$\mathcal{O} [[ X_1,\ldots,X_d] ]/J\xrightarrow{\eta} \mathcal{O} [[ X_1,\ldots,X_d] ]/J_f\twoheadrightarrow \mathcal{O} [[ X_1,\ldots,X_d] ]/J$$](./latex/latex2png-GaloisRepresentations_93773712_.gif)
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
![$R[1/\ell]$](./latex/latex2png-GaloisRepresentations_80258974_-5.gif)
(these are dense in the generic fiber) and pairs

, where

is a finite extension,

is continuous such that
![$L'{}=L(\phi'(R[1/\ell]))$](./latex/latex2png-GaloisRepresentations_156666643_-5.gif)
.
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
![$R^\Box[1/\ell]_x^\wedge:= \lim_j R^\Box[1/\ell]/(\ker\phi_x[1/\ell])^j$](./latex/latex2png-GaloisRepresentations_170406041_-5.gif)
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
![$$\xymatrix@R=0cm@C=0.3cm{\mathcal{C}^0(G,\ad\bar\rho) \ar[r] \ar[rdd] & \mathcal{C}^1(G,\ad^0\bar\rho) \ar[r] \ar[rdd] & \mathcal{C}^2(G,\ad^0\bar\rho) \ar[r] & \mathcal{C}^3(G,\ad^0\bar\rho)\\ & \oplus & \oplus &\oplus \\& \mathcal{C}^0(G,\ad\bar\rho) \ar[r] \ar[rdd] & \displaystyle\bigoplus_{v\in T}\mathcal{C}^1(G_v,\ad^0\bar\rho) \ar[r] & \displaystyle\bigoplus_{v\in S}\mathcal{C}^2(G_v,\ad^0\bar\rho)\\ & & \oplus & &\\ & & \displaystyle\bigoplus_{v\in S\setminus T}\mathcal{C}^1(G_v,\ad^0\bar\rho)/\tilde{\mathcal{L}}(\mathcal{D}_v) \ar[ruu]& }$$](./latex/latex2png-GaloisRepresentations_256398670_.gif)
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
![$R_{\mathcal{A},T}^\mathrm{loc}[ [ X_1,\ldots, X_d] ]\twoheadrightarrow R_\mathcal{A}^{\Box_T}$](./latex/latex2png-GaloisRepresentations_18306381_-7.gif)
. 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 ![$$\chi(G_{F,S},M)=[F: \mathbb{Q}]\dim_\mathbb{F}M-\sum_{v\mid \infty}h^0(G_v,M).$$](./latex/latex2png-GaloisRepresentations_58262687_.gif)
Write

, then
![\begin{align*}
\chi(G,M)-\sum_{v\in S}\chi(G_v,M)&=[F:\mathbb{Q}]\dim_\mathbb{F}M-\sum_{v\mid \infty}h^0(G_v, M)-\sum_{v\mid \ell}\dim_\mathbb{F}M [F_v: \mathbb{Q}_\ell]\\&=-\sum_{v\mid\infty}h^0(G_v,M).
\end{align*}](./latex/latex2png-GaloisRepresentations_247052606_.gif)
(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
![$\mathbb{F}[G_K]$](./latex/latex2png-GaloisRepresentations_135163825_-5.gif)
-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
![$$\xymatrix{0 \ar[r] & H^0(G,M) \ar[r] & \oplus_{v\in S}H^0(G_v,M) \ar[r] & H^2(G,M^D)^\vee &\\
\ar[r] & H^1(G,M) \ar[r] & \oplus_{v\in S}H^1(G_v,M) \ar[r] & H^1(G,M^D)^\vee &\\
\ar[r] & H^2(G,M) \ar[r] & \oplus_{v\in S}H^2(G_v,M) \ar[r] & H^0(G,M^D)^\vee \ar[r] & 0\\}$$](./latex/latex2png-GaloisRepresentations_51080726_.gif)
Back to our situation with
, by definition we have the following exact sequence
![$$\xymatrix{ & H^1(G,M) \ar[r] & \displaystyle\bigoplus_{v\in T}H^1(G_v,M)\bigoplus_{v\in S\setminus T}H^1(G_v,M)/\mathcal{L}(\mathcal{D}_v) \ar[r] & H^2_{\mathcal{A},T} \\ \ar[r] & H^2(G,M) \ar[r] & \displaystyle\bigoplus_{v\in S}H^2(G_v,M) \ar[r] & H^3_{\mathcal{A},T}.}$$](./latex/latex2png-GaloisRepresentations_186561407_.gif)
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 
![$\Spec R^\Box_{\bar\rho,\chi}[1/\ell]$](./latex/latex2png-GaloisRepresentations_195019054_-6.gif)
has finitely many irreducible components and each irreducible component is generically formally smooth (over

) and of dimension

.
We define a closed point

of
![$\Spec R^\Box[1/\ell]$](./latex/latex2png-GaloisRepresentations_148492784_-5.gif)
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
![$R^\Box[1/\ell]^\wedge_x\cong L_x[ [y_1,...,y_{n^2-1}] ]$](./latex/latex2png-GaloisRepresentations_182851690_-5.gif)
. 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
![$$\phi:L_x[ [x_1,...,x_d] ]\twoheadrightarrow R^\Box[1/\ell]_x^\wedge.$$](./latex/latex2png-GaloisRepresentations_259215030_.gif)
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
![$\Spec R^\Box[1/\ell]$](./latex/latex2png-GaloisRepresentations_148492784_-5.gif)
. 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
![$\tau=[(1,0)]$](./latex/latex2png-GaloisRepresentations_47227376_-5.gif)
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 |
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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