These are my live-TeXed notes for the course Math GR8675: Topics in Number Theory taught by Eric Urban at Columbia, Spring 2018.
Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!
01/22/2018
Introduction
The goal of this course is to explain the strategy and introduce the main ingredients in the proof of some results obtained jointly with C. Skinner on the Bloch-Kato conjecture for some polarized motives over an imaginary quadratic field and in particular for elliptic curves.
First let us recall the Bloch-Kato conjecture.
Let
be a number field. Let
be a finite extension. Let
be a finite dimensional
-vector space with a
-action:
. For
a finite place of
. Let
be the inertia subgroup and
be the arithmetic Frobenius. Assume that
is geometric in the sense of Fontaine—Mazur (conjecturally coming from the etale cohomology of an algebraic variety), namely:
is unramified away from finitely many primes. This means that there exists a finite set
of finite places such that for any
,
.
- For any
,
is de Rham (a
-adic Hodge theoretic condition which we do not explain now).
Fixing an embedding
. Associated to
we have its
-function
The local L-factor
where
is a polynomial of degree
, defined as (assume
is crystalline at
) 
Now let us recall the definition of the Bloch-Kato Selmer group
associated to
.
Let
, where
is the 1-dimensional space with the action of
given by the cyclotomic character:
Now we can state the Bloch-Kato conjecture.
(Bloch-Kato)
Two concrete examples are in order.
Let

with trivial action of

. Then

. It follows from Dirichlet's class number formula that

. On the other hand, we have

Notice

where the second equality is by Kummer theory. Therefore

So the Bloch-Kato conjecture is known in this case by Dirichlet's unit theorem that
Let

be an elliptic curve and

. Then

and

(notice the geometric frobenius gives the L-function of

, hence the shift by 1). Recall that we have the Kummer sequence for

:

One can show that

with

(the corank of

) equal to

. The Bloch-Kato conjecture in this case says that

This is a consequence of the rank part of the BSD conjecture plus the finiteness of the Shafarevich-Tate group

.
Now let us consider the following polarized case and state the target theorem of this course.
- Assume
is an imaginary quadratic field.
- Assume
splits in
.
- Assume
is polarized in the sense of
where
(
is the complex conjugation). Notice
and
satisfying the functional equation with center
,
In this case the Bloch-Kato conjecture says that 
- Assume (as the Fontaine—Mazur—Langlands conjecture predicts), there is a cuspidal automorphic representation
on a unitary group
associated to an hermitian space
of dimension
over
, and some Hecke character
of
such that we have isomorphisms of Galois representations 
- Assume for one (hence both, by the polarization)
,
is regular (i.e., all Hodge-Tate weights have multiplicity one) and
. Here we take the geometric convention that
is the Hodge weight of
. (In particular, the last condition excludes the case of elliptic curves.)
- For both
,
is crystalline.
The strategy and the plan
Now let us briefly explain the strategy of the proof.
Step 1
To construct a p-adic deformation of the Galois representation
Namely a (rigid analytic) family of Galois representations (say over a unit disk
),
such that
and there exists an infinite set
such that
, where
is the Galois representation associated to
.
is polarized:
.
is geometrically irreducible.
- For almost all
,
is crystalline at
(with certain particular slopes) and the rank of the monodromy operator is the same
at all
.
Step 2
Use the irreducibility of
to construct a lattice
in
such that
is indecomposable as a Galois representation. It follows that we have two cases: Case A
or Case B: 
Step 3
Use the ramification properties for the family to show that case B is impossible:
as
is finite.
Step 4
We are now in Case A. Use the assumption about particular slopes at
for the family to get the desired extension class 
01/24/2018
Here is a plan of the course:
- A baby case of the strategy.
- Theory of Eisenstein series for unitary groups. Constructing the desired deformation of Galois representations will come from deforming Eisenstein series and the latter is guaranteed by the vanishing of
-values.
-adic deformations of automorphic forms and eigenvarieties. In particular, we will allow ramification at
in order to deal with more general cases beyond crystalline at
(this part is not in the literature).
- Galois representations attached to
-adic automorphic forms for unitary groups (we will only state the results).
- Explain how to deduce the first two steps of the strategy.
- Continuity of crystalline periods (mainly the work of Kisin). This will allow us to exclude Case B and show
is in the Bloch-Kato Selmer group.
- Explain how to remove the conditions on the Hodge-Tate weights so that we can include the case of elliptic curves. Roughly this is achieved by put the Eisenstein series with bad weights in a Coleman family at the cost of only getting nearly holomorphic Eisenstein series. Then one finds overconvergent points with good weights in the Coleman family to apply the argument before.
- Explain how to remove the crystalline assumption at
(this probably cannot be done simultaneously with (g), which needs the finite slope assumption).
- Do the case for CM fields of the form
, where
is imaginary quadratic and
is cyclic. This uses the theory of automorphic induction and the observation that 
- Do the case of higher order vanishing.
Baby case
Let
be a Dirichlet character mod
(not necessarily primitive). Let
be a prime. Let
. Our goal is to show the following implication in the Bloch-Kato conjecture: if
, then 
For

an integer, we define the
real analytic Eisenstein series of weight

, level

and character

(

) to be

Here

,

,

, and

.
We define

where

So the Fourier expansion of

at the cusp

is the Fourier expansion of

at the cusp 0.
Write
, then
has constant term given by
Assume
and
, then the gamma function term has a simple zero and the
-function term does not have a pole except
and
is trivial.
If
, then
has constant term 0 and hence is holomorphic of weight
and level
. Moreover, the Fourier expansion of
(suitably normalized) is given by
When
, we have the
-th Fourier coefficient is
(critical slope).
The Eisenstein series
is eigenform: its
-eigenvalue is
if
and
-eigenvalue is
for
. Therefore the Galois representation associated to
is 
Now Coleman's theory tells us that there exists a family of modular forms
of slope
deforming
. Here
is an eigenform of weight
(varying in an infinite set), level
, and
-eigenvalue of slope
. We have a
-expansion
where
is an analytic function on the
-adic unit disk such that
In particular, when
, we have
. When
is a sufficiently large integer,
is the
-expansion of a modular of weight
and slope
. Its slope is neither zero nor critical (
), so it must be a cusp form. Now the lattice construction gives us an extension
and hence a class in
. By remark 6, the only thing one needs to check is that the class is crystalline when
is trivial, which follows from the continuity of crystalline periods.
01/29/2018
Modular forms on unitary groups
(I was out of town for this lecture and thanks Pak-hin Lee for sending me his notes)
Let
be an imaginary quadratic field with a fixed embedding
, and
be an integer.
Let

be a skew-hermitian matrix (i.e.,

) of signature

(

and

). This means that the hermitian space

can be decomposed as

and

has the form

for some

with

.
Define the associated
unitary group

for any ring

. In particular,

In fact, fix

such that

. Then the conjugation by

,

gives an isomorphism

.
Let

(no confusion with the imaginary quadratic field

) be the maximal compact of

. Then we have

under the previous isomorphism.
One can define an automorphic factor

where

is the complexification of

(

via

). When

,

given by

where

.
Dominant algebraic characters of a Cartan subgroup in

(chosen as the one sent to

by the isomorphism

) are classified by sequences of integers

with

and

. For such a sequence

, let

be the corresponding complex algebraic representation of

.
We define the
automorphic factor 
, which takes values in

. We define
modular forms of weight 
as holomorphic functions

such that

for all

, where

is an arithmetic subgroup (i.e.,

for some

open compact). We also require that

satisfy a moderate growth condition (or cuspidal condition), but we will not make this precise.
Let

be an irreducible cuspidal representation of

, and write

where

(resp.

) is a representation of

(resp.

). Assume that

is a
holomorphic discrete series with its minimal

-type

with highest weight

such that

(this condition is equivalent to that the corresponding Hodge—Tate weights are
regular). In particular,

has dimension

(as the minimal

-type of any discrete series has multiplicity one). Take

and

where

is open compact. Then the function

is

-finite and smooth, so we can consider

where

satisfies

. Since

, we have

for all

. In particular,

is a modular form on

of weight

.
For a cuspidal automorphic representation

on

, and an idele class character

of

. Define the

-function

for

, where

is the base change of

to

. If

is cuspidal (a condition we will assume from now on), this

-function has holomorphic continuation. If

and

are unitary, then there is a functional equation

We will be interested in the central value

.
Eisenstein series on unitary groups
Consider

where

Then

has signature

. In

, consider the isotropic line

and its stabilizer

, with standard Levi decomposition

. Then there is an isomorphism

as before determine a representation of

as follows. Let

be the space of the representation

. Then

acts on

by
Let

be the
modulus character 
We see that

.
For

, define the
induced representation
For a fixed choice of

, define the
Eisenstein series

This is convergent for

if

is tempered (if

is cuspidal, then

is tempered). The general theory of Eisenstein series tells us that

has a meromorphic continuation and a functional equation, which we will not need.
Next we will determine conditions on
,
and
such that the vanishing of
is related to:
is holomorphic at
.
gives a holomorphic modular form on
.
01/31/2018
Let us look more closely at
. Let us choose a weight
Then
as a representation of
. The restriction
contains
with multiplicity one by the classical branching law for
. Moreover, the highest weight of the other components are dominated by
. Thus
is one dimensional as well.
If

, then for all

,

and

, we have

In particular, taking

gives

Taking

and notice

, we obtain
Conversely, if any element in the latter gives rise to such
, then we know that
is one dimensional. In fact, one more condition needs to be satisfied for the converse because
is strictly large than
. Notice
. For
, we have
where
is the image of
in
. One sees that the extra condition is for
, we have
namely
, as desired.
¡õ
Fix
, then Proposition 1 gives an associated
. By Example 3, it is holomorphic when
,
,
. So
. Fix
. Let
. For
and
, we define the function
where
such that
. Let
, then we have an associated modular form
on
(Example 3).
For
, we have
where
is obtained from
by removing the last column (and the
-th row to make the right size). So by Proposition 1 we have
Notice that
is holomorphic, and hence
is a holomorphic function in
. So
is a holomorphic function of
if and only if
In particular, when
, we have
.
Next time we will see that if
, then we will always get a holomorphic Eisenstein series. Otherwise, we get a holomorphic Eisenstein series if either
or if some central
-value vanishes.
02/05/2018
Write

, where

form a fixed basis of

, and

. Let

. We define the Eisenstein series

and
The proof of this theorem is based on Langlands' general theory of Eisenstein series (see for example Moeglin—Waldspurger, Spectral Decomposition and Eisenstein Series for an exposition). Let us give a quick sketch of the theory in the simplest case.
Let

be a reductive group. Let

be a maximal parabolic subgroup. Let

be a cuspidal automorphic representation of

. Let

be the modulus character and

For

, define the Eisenstein series

This series converges when

.
For an automorphic form

on

, its
constant term along a parabolic subgroup

is defined to be
(Langlands)
has a meromorphic continuation to
.
is holomorphic at
if the constant term
is holomorphic at
for all standard parabolic
(equivalently, for just
, see Proposition 2).
Let

. For simplicity further assume that

has order 2, and so

,

. Then

. For

, define the
intertwining operator 
This converges when

.
The first follows from the fact that

is cuspidal. The second uses

and therefore

The integration of the first term is simply

(as

lies in the induced representation), and the integration of the second term is

.
¡õ
Using

, we know

is also a representation of

. One can check that

. The map

is
equivariant for the

-action.
We have a functional equation
After suitably normalize

one can check

. In particular,

,

has the same constant term, and hence they are equal (as their difference has trivial constant term and is perpendicular to all cusp forms).
¡õ
Now let us come back to the unitary case. We have 
We denote
Then the intertwining operator
has an Euler product
where
is a pure tensor, and 
Assume

is unramified, we let

to be a spherical vector. Assume

is further unramified. By Iwasawa decomposition

, where

. We define the spherical vector

to be the unique section such that

. Similarly, we define

.
Let
be a finite set of finite places such that for
,
are unramified. Let
, with
a pure tensor. For
, by the intertwining property, we know that
must be a multiple of the spherical vector
. In fact, Langlands' general theory gives:
For any
finite, we know that
is holomorphic at
(by Harish-Chandra as
is tempered).
02/07/2018
For
, we have
(lies in a 1-dimensional space). Here Harish-Chandra's c-function
is a ratio of
-functions. The key point is that
as the induction
contains the holomorphic discrete series
as a subrepresentation, which always lies in the kernel of the intertwining operator
.
So we conclude that
Here
is holomorphic at
. Since
is tempered, we know all
-values in the denominator does not vanish when
as
is the central value. We also know that
has a simple pole exactly when
. This finishes the proof of Theorem 2 by Theorem 3 (b) and Proposition 2(b).
L-groups, parameters and Galois representations
Let
be a unitary group over a field
associated to a quadratic extension
. Then we have
and
, where
acts on
by the projection onto
, and 
Assume

is a non-archimedean local field, and

is quasi-split and split over an unramified extension of

. Let

be a Borel. Let

be a hyperspecial maximal compact subgroup. Let

. Let

be the space of unramified characters. For

, define

Then

is always 1-dimensional. Let

be the unique irreducible subquotient of

such that

.
For any

in the Weyl group, using intertwining operators one can see

. Thus we have a map

which is an isomorphism (the inverse is given by the Satake isomorphism). We define a map

Identifying

, we obtain an element

such that

for any

. In this way we have an associated parameter

to

. For

an unramified representation of

, we have an associate parameter

, the conjugacy class of

of the associated character

.
02/12/2018
Let
be a parabolic subgroup. The we have an inclusion
. For an unramified representation
on
, we have an unramified representation of
, namely
the unique unramified subquotient of
. Moreover, by the transitivity of the parabolic induction (
, we have 
Take
. Then
where the embedding is given by
. Assume
is a prime where
are unramified. If
splits, then the parameter for
is given by
If
is inert, then the parameter is given by 
For
an automorphic representation on
(holomorphic discrete series at
), the link to Galois representations is given as follows. There exists
such that
Here
.
When
is an irreducible subquotient of the space of Eisenstein series
for
. Then
More generally, for
, we obtain
In this case, we have 
Recall that
, where
. So the two relevant characters are
Let
(motivic weight zero,
), then the two characters are
Notice the difference of the powers of the cyclotomic character
is given by
we know that when
is minimal, the Galois representation is of the form
for some character
. In this case, let
, then
is a twist of
This is exactly the desired shape of Galois representations we are looking for to construct elements in the Bloch-Kato Selmer group.
02/19/2018

-adic deformations of automorphic representations of unitary groups
Recall
is the unitary group for
of signature
. For simplicity (not essential), we will assume that
is split in
. Then
, and the hyperspecial maximal compact subgroup
.
Let

be the
Iwahori subgroup, defined by

Let
For

, we define

. Then one can check that for

, we have

We define

to be the

-algebra generated by

,

, then

is commutative and
![$$U_p\cong \mathbb{Z}_p[T^+/T(\mathbb{Z}_p)].$$](./latex/SkinnerUrban/latex2png-SkinnerUrban_158377288_.gif)
For example, when

, this is generated by the usual

-operator (and the center).
We say a homomorphism

is
finite slope if

for any

. If

is of finite slope, then there exists

such that
Let

be an unramified representation of

. Then we have

for

an unramified character of

. Taking

-invariants, we have an embedding of

-modules

By the Bruhat decomposition

, we know that

has dimension equal to

. More precisely, for any

, there exists an eigenvector

such that

(Notice

, this extra factor comes from the normalization of the Langlands parameter). So any eigenvector of

inside

is attached to a pair

. The values

attached to

correspond to an
ordering of the eigenvalues of the Langlands parameter of

. All such orderings show up if and only if the embedding

is an isomorphism (e.g., the case when

is unitary as

is irreducible).
We normalize

and define

Then

takes values in algebraic integers: in fact there is a Hecke equivariant Eichler-Shimura map,

which embeds the weight

-modular forms into the middle cohomology of the unitary Shimura variety with coefficient in the representation of highest weight

, and the latter has an integral structure preserved under the Hecke action.
We define the
normalized slope of

to be

such that

We say that

is
ordinary if its slope is

.
We say a slope

is
non-critical with respect to the weight

if
Let

. Let

be a finite set of primes containing all primes where

is ramified. We denote

The

gives rise to a character

such that

, and gives the Hecke action on the spherical vector away from

.
We define the
weight space 
to be the rigid analytic variety over

such that for any finite extension

,

Notice

for a finite group

. So

Here

is the open unit disk of radius 1 centered at 1 in

. For a dominant character

, we obtain a corresponding
algebraic weight
![$[\lambda]\in \mathcal{X}(\mathbb{Q}_p)$](./latex/SkinnerUrban/latex2png-SkinnerUrban_87483070_-5.gif)
.
An eigenvariety is going to be a rigid analytic variety
that contains points attached to finite slope automorphic representations of
. If
is of the form
as before, then we say that
is of algebraic weight
. More generally, it is possible to speak about finite slope automorphic representation of a given
-adic weight
. More precisely,
02/21/2018
Construction of the eigenvariety: locally analytic induction
Now let us sketch some ingredients that go into that cohomological approach of the construction of the eigenvariety. Recall we have an injective Eichler-Shimura morphism
So to interpolate the automorphic forms
-adically, we may instead interpolate cohomology with varying coefficient spaces. These coefficient spaces are finite dimensional but with different different dimension. To interpolate them, we will instead embed them into an infinite dimensional space with varying action depending on the weight.
For notice that any
is locally analytic, namely, there exists an integer
such that
is analytic, where
.
Take

. Then

. So

for some

. Find

such that

. Then for

, we have

Notice the last expression is analytic (a convergent power series in

), so

is locally analytic.
For

, we define

where

is

-analytic (i.e. analytic on disks of radius

) such that

here

consists of lower triangular unipotent matrices. By the Iwahori decomposition

we see

is identified with

, the

-analytic functions on

, via the restriction map

. The latter is independent of

.
Define

. It is equal to

, and the decomposition

is unique for

. Notice the natural action

on

extends to a contracting action of

. We will use the right action of

on cohomology to define a
compact operator (as a replacement of finite dimensionality). Explicitly, for

,

, we define the action by

where

In particular,

if

.
We define the action of

on

via

and the action of

via

In particular, for

, the two actions agree:

.
If

is algebraic dominant, we define

to be space of functions

which are also algebraic.
If

is algebraic and

is a simple root, we define

Here for

,

, where

is the half sum of all positive roots,

is a basis vector for the root space

, and

is the differential operator of the left translation. Notice the left translation action of

preserves

(not

), but one can check the image indeed lies in

.
It is clear that
is equivariant for the right
-action. Moreover, for
, we have
The power of
in
will control the integrability of the image.
If

is algebraic dominant, then
See [Urban Annals 2011, Prop. 3.2.12].
¡õ
Let us illustrate Prop. 5 using the simplest example
.
Let

and

be the upper triangular matrices. Let

be an algebraic weight, then

is dominant if and only if

. The two simple roots are

(i.e.,

). The Weyl group is

, where

. So

For

dominant, let

be the algebraic representation of highest weight

. So

can be identified with the space of homogeneous polynomials in

of degree

with action

Define

, if

. Then

Then

identifies

as the algebraic induction

.
We have
, where
is Zariski dense (known as the big cell). In particular,
is determined by its restriction to
:
. This restriction induces an injection
, and the image is the space of polynomials in
of degree
, with the action 
The space
is the space of analytic functions on
such that
Again by restriction
we can identify
as the space of analytic functions on
, with the
-action defined by the same formula
One sees that a function
lies in
if and only if
is a polynomial of degree
, i.e.,
, as in Prop. 5.
One can check by direct computation that
, namely in this case
More conceptually, take a basis of the Lie algebra
,
Then
and
, where
. The function
is invariant under the action of
:
as
,
and
. Moreover, for
: we have
Hence
.
02/26/2018
Construction of the eigenvariety: slope decomposition
Let

be finite slope. We define its
slope 
such that for any

(so

), we have

(Recall for

,

is defined such that

). Notice that if

takes integral values, then

, and hence

. Namely,

lies in the cone generated by the positive roots (which may be
larger than

, the cone generated by the dominant weights). We say the slope

is called
non-critical with respect to

if for any

,

, we have
Suppose

acts on a Banach space

over

. For any

, we define

to be the sum of the generalized eigenspace for

attached to the characters

such that

(i.e.,

.
(Classicality)
Let

. Assume that

is non-critical with respect to

(algebraic dominant). Then the natural map

is an isomorphism.
Notice that

is no longer integral by the definition of non-criticality. Hence the image is killed by

. The result then follows from Prop.
5.
¡õ
Define

to be

. In other words, one replaces the inequalities defining

to strict inequalities.
For any

, the operator

acting on

in completely continuous (i.e. a limit of finite rank operators).
For

, let us show the

-action of

on

is completely continuous. Recall a function

can be identified as a function in

(Definition
33). The norm of

is the sup norm under this identification. For

, we have

. This action is contracting, i.e.,

where

. Write

Then

, and so

, which converges on a
larger disk. Now the claim follows form the following fact: for

, the restriction

is completely continuous. In fact, the truncation

has norm

, hence

is the limit of finite rank operators

.
¡õ
Now let us put things in analytic variation.
Let

be an affinoid subdomain. Then there exists

such that for any

,

is

-analytic. For

, we define

So an element of

is an

-analytic function

where

is the image of

under the map
![$T(\mathbb{Z}_p)\subseteq \mathbb{Z}_p[ [ T(\mathbb{Z}_p)] ]\rightarrow \mathcal{A}(\mathcal{U})$](./latex/SkinnerUrban/latex2png-SkinnerUrban_42252918_-5.gif)
. We can analogously define the

-action on

.
For
we have
i.e.,
gives an analytic variation of
. The cohomology
is not necessarily a Banach space. We have a map
A priori this map is neither injective nor surjective. This is because
may fail to be flat over
, caused by torsion classes that does not vary in family. To resolve this issue, one instead directly works with complexes defining the cohomology and use the slope decomposition of the complexes.
Let

be a congruence subgroup acting freely on

. Then there is a
![$\mathbb{Z}[\Gamma]$](./latex/SkinnerUrban/latex2png-SkinnerUrban_67934979_-5.gif)
-finite free resolution of the trivial

-module

.
This is a consequence of the Borel-Serre compactification

, which has a deformation retract to

such that

is compact. One can then choose a finite triangulation of

, and hence a triangulation of

by pulling back. Let

be the

-chains of the triangulation, a free
![$\mathbb{Z}[\Gamma]$](./latex/SkinnerUrban/latex2png-SkinnerUrban_67934979_-5.gif)
-module. Then the complex

computes the homology of

. But

is contractible, and hence all higher homology groups are trivial, and hence

gives the desired free resolution.
¡õ
Let
be a
-module, then
can be computed using the cohomology of
. So computing the cohomology in
can be computed using complexes whose terms are finitely many copies of
by Prop. 8. The advantage is that now the action of Hecke operators on the cohomology can be lifted to an action on these complexes of Banach spaces (defined uniquely up to homotopy), and the action of each individual
on
is completely continuous and has a slope decomposition. For
, we can decompose
In this way we deduce a slope decomposition on the cohomology.
Since
, we know that
However, we do not have much control over the torsion and the same isomorphism does not hold for the cohomology. Instead of having a control theorem for the cohomology, we simply use that the (alternating) trace of a compact operator on finite slope part of the cohomology
is the same as the trace on
(notice the infinite slope part has trace zero).
For

, where

is the ideal generated by

for

. Then

acts on

and we define

such that for any

,

is equal to the trace of

on

(or finite slope cohomology

). This construction is similar to Wiles' construction of deformation of Galois representations using pseudo-representations.
The finite slope cohomology

has the Hecke action of

, and decomposes into a direction sum

, where

runs over finite slope representation of the Hecke algebra. We define the (alternating) trace

.
In the same way we may also define a Fredholm determinant for each term of the complexes and thus a total determinant by taking alternative product. We will use these analytic families of finite slope distribution to construct the eigenvariety.
03/19/2018
Construction of the eigenvariety: effective finite slope character distribution
Let

. Let

be the two-sided ideal of

generated by

for

. For

an open compact subgroup, we let

(and similarly

).
An irreducible representation

of

is called
finite slope if

, the restriction of

to

, satisifies

for any

. Notice that

is a character since

lies in the center of

and

is irreducible.
For an effective finite slope character distribution

, we define

this infinite sum is completed with respect to an integral structure on

. Then for any

, we have

(by the finiteness assumption b) the operator

is completely continuous). More generally, if

is a

-type, we define

which recovers

when

is trivial. For any

we can consider the
Fredholm determinant 
which is an entire power series with coefficient in

.
For

, one can take

, wehre

runs over the eigenvalues of

on

of slope

. Then we obtain a slope decompostion
Now we construct an eigenvariety attached to
- An analytic family of effective finite character distribution indexed by a weight space
. Namely a map
such that
is an effective finite slope character distribution for any
.
- A
-type
for some
.
Construction of the eigenvariety:
-adic automorphic character distribution
Next step is to contruct an
-family of effective finite slope character distribution which is automorphic. The
(Definition 42, with
replaced by
, see Remark 21) is a finite slope character distribution, but it is not effective in general.
03/26/2018
If

is anisotropic, then

is effective, where

is the dimension of the associated locally symmetric space.
For any

algebraic dominant, and

for

. By Proposition
5 and taking dual, we obtain

where the ideal

. Hence we obtain a congruence of Fredholm determinants

If

is anisotropic and

is regular, then there is only cohomology in the middle degree (Borel-Wallach). In particular,
Assume that
, where
are coprime entire power series with coefficients in
. We need to show that
is actually a constant. If not, then the set of zeros
is non-empty. Pick a point
. Then
is a pole of
. Fix an open neighborhood
of
. Since
is flat (hence open), the image of
in
is also open and thus contains a dense set of algebraic weights. So we may find a point
such that
is algebraic dominant regular, the slope of
is equal to that of
, and
. The congruence mod
implies that
is a pole of a polynomial, a contradiction.
¡õ
More generally, when
is not anisotropic, one needs to replace the cohomology by the cuspidal cohomology, and modify
accordingly.
We say a sequence of dominant regular

is
very regular if for any simple root

,

. In this case, we have

(

-adically), and
(cuspidal character distribution)
- For any converging very regular sequence
, the limit
exists and depends only
.
and is effective.
- For
, we have 

-adic deformation of Eisenstein series
Next we will construct a point on the eigenvariety associated to an Eisenstein series on unitary groups and thus obtain the desired cuspidal
-adic deformations.
03/28/2018
Let
with
regular algebraic dominant. Then by construction
lies on the eigenvariety if and only if
. If
is a cuspidal automorphic representation of
, with
a discrete series of parameter
. For any
occurring in
, the classical multiplicity
is the multiplicity of
in
, which is positive. If
is non-critical, then by Prop. 6 we know that
is equal to the classical multiplicity, hence corresponds to a point on the eigenvariety. But if the slope is critical, then it is not clear that
corresponds to a point in the eigenvariety.
Consider

, and

the character attached to the trivial representation. Then

(appearing only in

). However, for

maximal, we have

. Otherwise, there is a point on the eigenvariety such that

and

(the action on the trivial representation) and one gets a family of cusp forms of slope 1 which specializes in weight 2 to the critical Eisenstein series, which is impossible (see Remark
8).
Now let us come back to the setting of unitary groups. Recall that
splits,
,
is a dominant weight, and
with slope
If
takes integral values, then
is inside the cone generated by the positive simple roots
, namely,
,
, ...,
. By Definition 38,
is non-critical if and only if
does not lie in this cone. Notice that
So non-critical means
,
, ...,
(cf. 30).
Recall that
is a holomorphic Eisenstein series of weight
. Since
splits, we have the Levi
Assume that
is an unramified principal series. Then
is also an unramified principal series.
Choosing
(of dimension
) corresponding to an ordering of the Langlands parameter.
To make such a choice, first we fix a
-stabilization of
(hence an ordering of the Langlands parameter of
) such that the corresponding character of
is non-critical with respect to
. Next we choose
the section corresponding to the ordering
or
Here
are the characters corresponding to
on
. The corresponding slopes are
(called the
-ordinary stabilization) and
(called the critical stabilization) respectively.
We are interested in the case
(to get desired shape of Galois representation) and the case of critical stabilization (to get cuspidal deformation). In this case,
is never non-critical with respect to
, as the extra requirement
is always violated. Nevertheless, the requirement is only violated at the position
, and can salvaged using a Hasse invariant argument as follows.
The critical stabilization of

gives a point on the eigenvariety.
04/02/2018
Galois representations associated to automorphic forms
Today we will review some facts about Galois representations associated to automorphic forms. The Langlands philosophy predicts that to certain cuspidal algebraic automorphic representations
of
, one should attach a compatible system of Galois representations
, where
is the Hecke field of
, characterized by the local Langlands correspondence. This philosophy is now known in many cases.
Let us recall the local Langlands correspondence. Let
be a non-archimedean local field with residue field
and
. We have an exact sequence 
The
Weil group 
is the inverse image of

inside

, where

is the

-Frobenius. A representation

of

is called
smooth if

is trivial on a neighborhood of 1 in

, i.e., there exists

finite index such that

is trivial.
Let
be a prime. Then we have a tame quotient map
sending
to
, where
In particular, we see that
. The following theorem is not hard.
(Monodromy theorem of Grothendieck)
Let

be an

-adic representation. Then there exists a nilpotent endomorphism

(called the
monodromy operator) such that for

(a finite index subgroup of

depending on

), we have
Notice that when
is nontrivial, the representation
is not smooth. To remedy this, one introduces the Weil-Deligne representations instead.
One then associates to
a Weil-Deligne representation
where
for
.
The local Langlands correspondence for
due to Harris-Taylor and Henniart says that there is a bijection
between irreducible smooth representations of
and Frobenius semisimple Weil-Deligne representations of dimension
, characterized by matching
-factors and
-factors on both sides and certain compatibilities. In particular it is compatible with local class field theory:
If
is irreducible, then
with trivial monodromy.
Now let us recall the global results. Let
be a CM extension. Let
be a cuspidal automorphic representation of
. Assume
is cohomological, i.e., there exists
an irreducible algebraic representation of
such that
(equivalently
for
regular algebraic).
is conjugate self-dual:
.
In this case,
descends to a unitary group and one can construct the desired Galois representations using unitary Shimura varieties by comparison of Lefschetz trace formulas and Arthur-Selberg trace formulas, and the stable twisted trace formula (for the purpose of descent). Under the following more special hypothesis the trace formulas simplifies and one can construct the desire Galois representation directly (see [Paris Book Project I]):
is unramified at finite places,
is unramified at places
above those which do not split,
is even.
Finally, one reduces the more general case to this special case using various tricks (quadratic base change and congruences).
We state the most general version of global Galois representations we need as following.
(Chenevier-Harris)
Assume

is cohomological and conjugate self-dual. Let

be a prime of

. Then there exists a Galois representation

such that
- For any
a finite place of
, we have
(see Def. 53 , in particular, the monodromy operator of
has smaller rank). In particular, at unramified places it is given by the local Langlands correspondence.
- If
, then
is de Rham with regular Hodge-Tate weights (i.e., all multiplicities are at most 1). (For example, if
, then the Hodge-Tate weights are given by
, where
,
).
- If
and
is unramified, then
is crystalline and 
We also need quadratic base change for automorphic representations on unitary groups.
04/04/2018

-adic families of Galois representations
Today we will give a sketch of the deformations of Galois representations. Let us come back to the setting of Eisenstein series. Let
be an automorphic representation on the unitary group
of signature
. Let
be a Hecke character of
. Let
be the set of ramification of
and
. Fix
a prime which splits in
.
We assume:
, where
with
.
- If
is ramified then
splits in
.
does not contain primes dividing
and
.
,
.
.
Under these assumptions on
, we have constructed holomorphic Eisenstein series
whose associated Galois representation is
, where
. We further assume that
is irreducible (this is conjectured to be true if
is cuspidal).
- For any
split in
, assume
is given exactly by
(see the thesis of Caraiani).
Now choose
a refinement for
, non-critical with respect to the Hodge-Tate weights of this representation. Here the Hodge-Tate weights are some translation of
which is regular. The crystalline Frobenius at
of the Galois representation associated to the Eisenstein series is given by 
From this choice of a non-critical
-stabilization of
and a suitable choice of
(explained below), one can construct a point on the eigenvariety, and thus there exists a family of Galois representations deforming the Galois representation
of
as in the following theorem.
Let us first explain the proof of Item d) of Theorem 10. To do so, we need more information on the local Langlands correspondence for
.
From the Bernstein decomposition, we know for

an irreducible representation of

, there exists a parabolic

with Levi

and

a cuspidal representation of

such that

is a subquotient of

. We say pairs

of this form up to equivalence to be the
cuspidal support of

. Here

are equivalent if there exists

such that

and

.
We say

are
inertially equivalent if there exists

and

an
unramified character of

such that

. We say

are
inertially equivalent if there exists a cuspidal support

of

and a cuspidal support

such that

. An equivalence class for this equivalence relation is called a
Bernstein component.
The type theory of Bushnell-Kutzko shows that if
is a Bernstein component, then there is a type
, where
open compact, and
a smooth representation of
, such that 
The component of unramified representations is exactly those

such that

(so contains the Steinberg representation as well). In other words, for this component, we have

. Such

has a cuspidal support

, where

is a torus and

is a unramified character.
For representations on the same Bernstein component, we order them using the monodromy operator.
Let

be two nilpotent matrices in

. We say

if

is in the Zariski closure of the set

(so the Jordan normal form of

has more zeros). We say

if

and

.
(Steinberg, Zelevinsky)
There exists

a smooth representation of

such that
Now for
, we choose the local section
as follows. From the Zelevinsky classification, one can see that there exists a subquotient
of
such that
with the monodromy operator only on
. Now choose
a representation of
in Prop. 11 for
. Let
be the corresponding idempotent. Then there exists a local section
such that
.
In this way, for a point in the eigenvariety constructed using the idempotent
, the associated Galois representation satisfies Item d) by Prop. 11.
04/09/2018
Irreducibility
Let us explain the proof of Item c) in Theorem 10. Let
and let
be the points
such that
, and
. For
, a point in
corresponds to a cuspidal automorphic representation which is unramified at
and whose
-stabilization is non-critical (a point on the eigenvariety). For
, the Galois representation
is crystalline at
with the roots of the crystalline Frobenius given by the Langlands parameter of
at
. The
-eigenvalues vary analytically in the family, which gives the analytic functions
in the theorem.
Now let us explain the proof of Item e) in Theorem 10. This part requires our imposed extra assumptions that
is irreducible and satisfies the local-global compatibility at split places
. Assume that
is reducible. Then
, where
irreducible of dimension
(as
is irreducible), and
is a 2-dimensional family such that,
- for any
,
is crystalline at
with Hodge-Tate weights
and crystalline Frobenius eigenvalues
,
,
- at a split place
, the monodromy operator is trivial (as
has correct monodromy and the generic monodromy of
matches with the correct one, the generic monodromy of
must come from
).
with
,
,
, and
.
When specializing to a point in
, we see from a) and c) that the Newton polygon is strictly above the Hodge polygon, hence
is irreducible. Hence
is irreducible.
Now take
an affinoid curve such that
is nonempty for any
. After normalization, we may assume
is smooth. Then
takes value in
which is a Dedekind domain, and we can find (after possibly shrinking
) a free lattice
which is stable under the action of
. When specializing to
we obtain
. Since
is irreducible, we may choose another
-stable lattice
such that
. Now because the monodromy is trivial at
, we know the extension class
is unramified at
. For
, we use the following general lemma about continuity of crystalline periods.
Let

. Assume there exists

Zariski dense subset such that

is Zariski dense for any

. For any

,

is crystalline with eigenvalues

. Then for any

, and

, we have

Here we order

.
It follows from the lemma that
. Hence we obtain (the surjectivity in) the following exact sequence
Hence
is 2-dimensional and thus
is also crystalline at
.
It follows that the extension class
is crystalline at
. So
. The latter group is zero (as
is finite), so
is trivial, a contradiction to the irreducibility of
. Therefore
must be irreducible.
04/11/2018
The Bloch-Kato conjecture
Keep the assumptions before Theorem 10.
There exists a nontrivial extension in

.
Our remaining goal is to explain the strategy of the proof.
Define
(in fact one can fix the first
weights and let the rest move in parallel, in which case
is a curve). Then the same argument as last time shows that
is generically irreducible. Let
be a free lattice such that
has a unique quotient isomorphic to the trivial representation. Then
contains the representation
.

is an extension
If not, then

is the only subrepresentation of

, so we obtain an extension

This extension is not trivial as

is the unique trivial quotient of

. Now for

,

are crystalline Frobenius eigenvalues. By taking

and apply Lemma
1 , we see that

for

,

. In particular, by the same argument we know the extension class of

has class in

, which is a contradiction.
¡õ
It remains to prove the extension class of
is crystalline. To do so, we switch to the dual situation, namely consider free lattice
with unique irreducible quotient
, which gives an extension
whose dual gives the nontrivial class in
.
Let

be a de Rham representation of

and

an extension of the form

Assume that there exists

such that the image

of

in

satisfies

Then

is de Rham.
To prove that

is de Rham, we need to show the surjectivity of

. Tensoring the extension with

and taking

, we obtain an exact sequence

The first and last terms (by Tate-Sen) are both 0. This gives the injectivity of the right vertical map in the commutative diagram with exact rows,
![$$\xymatrix{ D_\mathrm{dR}(E) \ar[r]^-{g} \ar[d] & D_\mathrm{dR}(V) \ar[r]^-{\delta_0} \ar@{->>}[d] & H^1(K, L(1) \otimes B_\mathrm{dR}) \ar@{^(->}[d] \\ (E \otimes B_\mathrm{dR}/B_\mathrm{dR}^+)^{G_K} \ar[r]^-h & (V \otimes B_\mathrm{dR}/B_\mathrm{dR}^+)^{G_K} \ar[r]^-{\delta_1}& H^1(K, L(1) \otimes B_\mathrm{dR}/B_\mathrm{dR}^+).}$$](./latex/SkinnerUrban/latex2png-SkinnerUrban_179885637_.gif)
Here the surjectivity of the middle vertical map follows from that

is de Rham. The assumption on

implies the composition of

and the middle vertical map is surjective. It follows that the map

is surjective, hence

is 0. Thus

is also zero, and thus

is surjective as desired.
¡õ
Now let us finish the proof of Theorem 11. Let
be generated by the eigenvectors of
of eigenvalues
. We claim that
In fact, the first
Hodge-Tate weights
of
are
, and the last
Hodge-Tate weights are
. Since
is weakly admissible, for any
, its Newton polygon is above the Hodge polygon. Hence
Since
and
, we know the claim holds for dimension reason.
Since
for
, we have seen that
. Now take
to be the submodule generated by the eigenvectors with eigenvalues
. Then
. We apply Lemma 3 and obtain that
is de Rham. But an extension of crystalline representations which is de Rham must be semistable. From the assumption on purity, we also know the monodromy is 0, hence the extension
is crystalline, as desired.
04/18/2018
Even order vanishing
Today we will explain the higher rank case.
Assume

vanishes at the center

with
even order, then

.
Recall we have constructed an Eisenstein series and a corresponding point
on the eigenvariety on
of dimension
with a certain
-stabilization and a certain type at ramified places. This provides a family of cuspidal automorphic representations
at for all
in a neighboorhood
of
. Up to a twist the
-function of the Eisenstein series is given by
If the order of vanishing of
is even at the center, then we know the sign of the functional equation
. Hence
. Since
is crystalline at
(hence local sign at
is
) and the local signs at ramified places are determined by the type (and the local sign at
is fixed since the weights are congruent mod
), we know that
is also
for any
. In particular, the
-function of
vanishes at the center.
Thus we can apply Theorem 11 for
on
for which
are not Hodge-Tate weights. Then for
, we obtain an extension
coming from the deformation of an Eisenstein series
on
corresponding to a point
on the eigenvariety
of dimension
. Because
is Zariski dense in
, one can then find a analytic map
(after possibly shrinking
so that the map to the weight space is finite). Let
be an open containing the image of
in
, then we obtain a pseudo-representation
. We have
and in particular,
Therefore we obtain an extension
Here
is the lattice attached to the pseudo-representation such that
has a unique quotient isomorphic to
. The extension we construct is given by the upper right corner
in
Notice that
is trivial again because
is trivial. It remains to show that
is trivial. It suffices to show that the extension
is trivial at
(otherwise there is a
-extension of
unramified at
). For the local triviality it suffices to show that
is Hodge-Tate (equivalently, de Rham in this case). We will show that in fact the entire representation is Hodge-Tate.
We will use the following version of Kisin's lemma.
(Kisin)
Suppose we have a finite slope family of Galois representations of Hodge-Tate type

(

). Suppose for

,

is crystalline at

and the Hodge-Tate weights

moves in parallel for

for any

. Let

. Then
.
- There exists
such that we have an injective map
In other words, these crystalline periods contributes to the Hodge-Tate weights
.
04/23/2018
Now we can finish the proof of Theorem 12. We have constructed a family of Galois representations
where
gives a nontrivial extension
,
at
. Notice that the Hodge-Tate weights of
at
is given by
We apply Lemma 5 to
to obtain
It follows that the extension
is Hodge-Tate and hence is trivial. Hence the extension
is nontrivial. Moreover, the two extensions
and
are linearly independent because
is the unique irreducible quotient of
(otherwise
is also an irreducible quotient).
The rank 3 case: difficulties
Assume now
. Then again we can construct an Eisenstein series
on
which deform into a family of cusp forms
such that
. Then
and hence has the sign of functional equation equal to
. It is expected (but not known) that
generically (for families of modular forms this is a conjecture of Greenberg, see [Howard, Central derivatives of L-functions in Hida families] for the rank
case). Again we can construct a family
on
such that on the vanishing locus
(to obtain a
-adic locus, one also needs to replace complex
-values by
-adic
-values), we have
and
, and is generically irreducible outside
. On
we can even construct a nontrivial extension 
Now the problem is to prove
is trivial using Lemma 5 , we need to further choose the sequence of points
lying in
. However, even the condition that
for
seems difficult to satisfy.
The modular form case
Let
be a cusp eigenform for
of weight
. We assume

- if
, then
comes from a definite quaternion algebra via Jacquet-Langlands
a root of Hecke polynomial at
for
such that
(non-critical condition).
an imaginary quadratic field.
We have the following generalization of Theorem 10.

.
The irreducibility of the family follows from the same argument as before. To exclude Case B, we need to show the desired crystalline property. However, the Hodge-Tate weights are

and Kisin's lemma only gives the information about the first crystalline period, not the first two (as the first two weights need to move in parallel). Instead, we apply Kisin's lemma to the exterior square

, whose Hodge-Tate weights are

. Let

be the extension in Case B

Then

is a subquotient of the exterior square, hence it is semi-stable, with monodromy operator

. It remains to show

.
If not, then there exists
such that
,
. Let
such that
. Then
, and
. But by Kisin's lemma,
, and hence
, and hence
, a contradiction.
¡õ
04/25/2018
Nearly holomorphic modular forms on unitary groups
We first review the
case.
We define the differential operator

Then the third condition in Definition
54 can be replaced by

. Notice it induces an operator which decreases the weight and order
We define the
Maass-Shimura operator 
, which increases the weight and order by
Next let us give equivalent algebraic definitions.
Let

be the universal elliptic curve over the modular curve

. Let

Then

. The Hodge filtration gives an exact sequence

We have a

-Hodge decomposition

.
We define

. Using

, we have an exact sequence

Then

. It follows from the

-Hodge decomposition that this definition agrees with the previous analytic definition. Then induced map from

agrees with the differential operator

. From the Gauss-Manin connection

and the Kodaira-Spencer isomorphism

, we also obtain an algebraic definition of the Maass-Shimura operator induced from

.
One can also define a nearly holomorphic modular form in the same spirit as Katz's modular forms.
A nearly holomorphic modular is a functorial rule

on quadruples

, where

is an elliptic curve,

is a basis of

and

is a basis of

, and

is a

-level structure, such that
![$f(E,\alpha,\omega,\omega')(X)\in R[X]_r$](./latex/SkinnerUrban/latex2png-SkinnerUrban_155114501_-5.gif)
(degree

polynomials; think:

) and

Using Tate's curve we have a

-expansion
![$f(q, X)\in R[X]_r[ [ q ] ]$](./latex/SkinnerUrban/latex2png-SkinnerUrban_197257158_-5.gif)
, and

where

.
Now let us discuss the case of
. Denote
, and
Let
. We introduce the differential operators
defined by the relation 
Let

be an algebraic representation of

. We define

the representation of

defined by

. Let

be the standard representation of the first

and

be the standard representation of the second

, then

.We define

by
We define

. So

, which again decreases the weight and the level. Similarly we can define the Maass-Shimura operator

given by

For example, when

, we have
A
nearly holomorphic modular form of order

, weight

and level

is a

-function

such that
for
.
for any
,
.
Now let us come to the algebraic definition.
Let

be the associated Shimura variety. Let

be the universal (generalized) abelian variety of dimension

, together with a principal polarization, and

-action such that

(both of rank

. Then

.
Let

be the coherent sheaf associated to

on

(so

and

). The Hodge filtration gives an exact sequence

Tensoring with

, we obtain

pulling back along

we obtain an exact sequence

So dualizing we obtain

and hence

We define

Then

agrees with the analytic definition when

, and the map induced by

agrees with

. Finally, Using the Gauss-Manin connection

and the Kodaira-Spencer isomorphism

we also obtain an algebraic definition of

.
Let

. Then using a Mumford object (generalization of the Tate curve), one can define a
-expansion ![$$g(q)\in R[ [ q^{H^+}] ] \otimes V_\rho,$$](./latex/SkinnerUrban/latex2png-SkinnerUrban_148357197_.gif)
where

is the set of hermitian positive definite lattice of

. If

, then we have a
polynomial
-expansion ![$$g(q, X_{i,j})\in R[ [ q^{H^+} ] ] \otimes V_\rho[X_{i,j}]^{\deg\le1}_{1\le i, j\le n}.$$](./latex/SkinnerUrban/latex2png-SkinnerUrban_103858163_.gif)
Then the differential operator

acts on the

-expansion by
Next time we will discuss the nearly holomorphic Eisenstein series and construct points on the eigenvariety when the
-function vanishes. This will complete our proof of Theorem 13.
04/30/2018
Families of nearly holomorphic Eisenstein series
Now we specialize our discussion to
. Let
be a cuspidal automorphic representation of
whose
is a discrete series of weight
. Let
be a parabolic with Levi
. Let
be a primitive eigenform attached to
. We have an associated automorphic form on
given by 
Define
Let
and
. Assume
is in the Langlands quotient of
and
is the spherical vector for
(where
is the set of ramification of
). Then the Eisenstein series
gives a
-form of weight
Notice that this weight is dominant only when
. Computing the constant term as gives the following result.
The Eisenstein series

is a nearly holomorphic of weight

and it is holomorphic if and only if

.
We define families of nearly holomorphic forms by families of polynomial
-expansions.
Let

be an affinoid (not necessarily open) in the weight space of

. Let

be a finite map.
A family of nearly holomorphic forms is a polynomial

-expansion
![$$G\in \mathcal{A}(\mathcal{U})[ [ q^{H^+}] ][X_{i,j}]_1$$](./latex/SkinnerUrban/latex2png-SkinnerUrban_45548101_.gif)
such that there exists

Zariski dense such that for any

,

is an algebraic weight and

is equal to

for

some nearly holomorphic form of weight

. Here for

with

an arithmetic weight, we denote by

the projection of

onto the coordinate along the highest weight vector.
Let
be a Coleman family (for
), where
. Consider the injection
where
. We say that
is classical if
is classification of non-critical weight
.
The strategy is to use the doubling method for

, i.e., we pullback the family of Siegel Eisenstein series on

and pair with the Coleman family on

. The nontrivial computation is to determine the correct sections, especially at

and

.
Another method is to use the ordinary family of Klingen Eisenstein series on
and apply some differential operator to get critical Eisenstein series (in the case of
, one evaluates
at
to get
).
¡õ
Let

be a cuspidal eigenform of weight

. Let

be a root of the Hecke polynomial at

. If

, then there exists a point

on the cuspidal eigenvariety with weight

, crystalline Frobenius eigenvalue

and Hecke eigenvalues away from

given by

.
Like the proof of Proposition
10, we use a lift of the Hasse invariant,

which is a scalar modular form of weight

. Consider

Then by Remark
34, we have

.
Let
be the Coleman family passing through
. By Remark 35, both terms of
can be viewed as sections of the same sheaf, so
makes sense. It is easy to see that
, hence
is holomorphic, and thus
Its projection
to
is a holomorphic form of weight
When
, we have
,
, so the holomorphic family
converges to the nearly holomorphic family of Eisenstein series. If
corresponds to
, then
specializes to
, therefore
.
Now we take a finite slope projector
on an affinoid of the weight space containing the weight
. Let
. Since
, we know that
, and thus
specializes to
. One can check that the slope of
is not of Eisenstein type, hence
is cuspidal. In this way we have constructed a point on the cuspidal eigenvariety.
¡õ