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!
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.
Two concrete examples are in order.
with trivial action of
. It follows from Dirichlet's class number formula that
. On the other hand, we have
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
be an elliptic curve 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
(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.
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 .
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:
Use the ramification properties for the family to show that case B is impossible: as is finite.
We are now in Case A. Use the assumption about particular slopes at for the family to get the desired extension class
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.
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
an integer, we define the real analytic Eisenstein series
) to be
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.
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.
be a skew-hermitian matrix (i.e.,
) of signature
). This means that the hermitian space
can be decomposed as
has the form
Define the associated unitary group
for any ring
. In particular,
In fact, fix
. Then the conjugation by
gives an isomorphism
(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
is the complexification of
Dominant algebraic characters of a Cartan subgroup in
(chosen as the one sent to
by the isomorphism
) are classified by sequences of integers
. For such a sequence
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
is an arithmetic subgroup (i.e.,
open compact). We also require that
satisfy a moderate growth condition (or cuspidal condition), but we will not make this precise.
be an irreducible cuspidal representation of
, and write
) is a representation of
). Assume that
is a holomorphic
discrete series with its minimal
with highest weight
(this condition is equivalent to that the corresponding Hodge—Tate weights are regular
). In particular,
(as the minimal
-type of any discrete series has multiplicity one). Take
is open compact. Then the function
-finite and smooth, so we can consider
, we have
. In particular,
is a modular form on
For a cuspidal automorphic representation
, and an idele class character
. Define the
is the base change of
is cuspidal (a condition we will assume from now on), this
-function has holomorphic continuation. If
are unitary, then there is a functional equation
We will be interested in the central value
Eisenstein series on unitary groups
, 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
be the modulus character
We see that
, define the induced representation
For a fixed choice of
, define the Eisenstein series
This is convergent for
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 .
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.
, then for all
, we have
In particular, taking
, 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.
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.
form a fixed basis of
. We define the Eisenstein series
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.
be a reductive group. Let
be a maximal parabolic subgroup. Let
be a cuspidal automorphic representation of
be the modulus character and
, define the Eisenstein series
This series converges when
For an automorphic form
, its constant term
along a parabolic subgroup
is defined to be
- 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).
. For simplicity further assume that
has order 2, and so
, define the intertwining operator
This converges when
The first follows from the fact that
is cuspidal. The second uses
The integration of the first term is simply
lies in the induced representation), and the integration of the second term is
, we know
is also a representation of
. One can check that
. The map
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).
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
is unramified, we let
to be a spherical vector. Assume
is further unramified. By Iwasawa decomposition
. 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).
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
is a non-archimedean local field, and
is quasi-split and split over an unramified extension of
be a Borel. Let
be a hyperspecial maximal compact subgroup. Let
be the space of unramified characters. For
is always 1-dimensional. Let
be the unique irreducible subquotient of
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
, we obtain an element
. In this way we have an associated parameter
an unramified representation of
, we have an associate parameter
, the conjugacy class of
of the associated character
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.
-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 .
be the Iwahori subgroup
, defined by
, we define
. Then one can check that for
, we have
to be the
-algebra generated by
is commutative and
For example, when
, this is generated by the usual
-operator (and the center).
We say a homomorphism
is finite slope
is of finite slope, then there exists
be an unramified representation of
. Then we have
an unramified character of
-invariants, we have an embedding of
By the Bruhat decomposition
, we know that
has dimension equal to
. More precisely, for any
, there exists an eigenvector
, this extra factor comes from the normalization of the Langlands parameter). So any eigenvector of
is attached to a pair
. The values
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
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
We say that
if its slope is
We say a slope
with respect to the weight
be a finite set of primes containing all primes where
is ramified. We denote
gives rise to a character
, 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
for a finite group
is the open unit disk of radius 1 centered at 1 in
. For a dominant character
, we obtain a corresponding algebraic
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,
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 .
. Then for
, we have
Notice the last expression is analytic (a convergent power series in
is locally analytic.
, we define
-analytic (i.e. analytic on disks of radius
) such that
consists of lower triangular unipotent matrices. By the Iwahori decomposition
is identified with
-analytic functions on
, via the restriction map
. The latter is independent of
. It is equal to
, and the decomposition
is unique for
. Notice the natural action
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
We define the action of
and the action of
In particular, for
, the two actions agree:
is algebraic dominant, we define
to be space of functions
which are also algebraic.
is algebraic and
is a simple root, we define
is the half sum of all positive roots,
is a basis vector for the root space
is the differential operator of the left translation. Notice the left translation action of
), 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.
is algebraic dominant, then
See [Urban Annals 2011, Prop. 3.2.12].
Let us illustrate Prop. 5 using the simplest example .
be the upper triangular matrices. Let
be an algebraic weight, then
is dominant if and only if
. The two simple roots are
). The Weyl group is
be the algebraic representation of highest weight
can be identified with the space of homogeneous polynomials in
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 .
Construction of the eigenvariety: slope decomposition
be finite slope. We define its slope
such that for any
), we have
is defined such that
). Notice that if
takes integral values, then
, and hence
lies in the cone generated by the positive roots (which may be larger
, the cone generated by the dominant weights). We say the slope
is called non-critical
with respect to
if for any
, we have
acts on a Banach space
. For any
, we define
to be the sum of the generalized eigenspace for
attached to the characters
. Assume that
is non-critical with respect to
(algebraic dominant). Then the natural map
is an isomorphism.
is no longer integral by the definition of non-criticality. Hence the image is killed by
. The result then follows from Prop. 5
. In other words, one replaces the inequalities defining
to strict inequalities.
, the operator
in completely continuous (i.e. a limit of finite rank operators).
, let us show the
is completely continuous. Recall a function
can be identified as a function in
). The norm of
is the sup norm under this identification. For
, we have
. This action is contracting, i.e.,
, 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
is the limit of finite rank operators
Now let us put things in analytic variation.
be an affinoid subdomain. Then there exists
such that for any
, we define
So an element of
is the image of
under the map
. We can analogously define the
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.
be a congruence subgroup acting freely on
. Then there is a
-finite free resolution of the trivial
This is a consequence of the Borel-Serre compactification
, which has a deformation retract to
is compact. One can then choose a finite triangulation of
, and hence a triangulation of
by pulling back. Let
-chains of the triangulation, a free
-module. Then the complex
computes the homology of
is contractible, and hence all higher homology groups are trivial, and hence
gives the desired free resolution.
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).
is the ideal generated by
and we define
such that for any
is equal to the trace of
(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
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.
Construction of the eigenvariety: effective finite slope character distribution
be the two-sided ideal of
an open compact subgroup, we let
An irreducible representation
is called finite slope
, the restriction of
. Notice that
is a character since
lies in the center of
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
-type, we define
is trivial. For any
we can consider the Fredholm determinant
which is an entire power series with coefficient in
, one can take
runs over the eigenvalues of
. 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.
is anisotropic, then
is effective, where
is the dimension of the associated locally symmetric space.
algebraic dominant, and
. By Proposition 5
and taking dual, we obtain
where the ideal
. Hence we obtain a congruence of Fredholm determinants
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.
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
(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.
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.
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
(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.
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
-Frobenius. A representation
is called smooth
is trivial on a neighborhood of 1 in
, i.e., there exists
finite index such that
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)
-adic representation. Then there exists a nilpotent endomorphism
(called the monodromy operator
) such that for
(a finite index subgroup of
), 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.
is cohomological and conjugate self-dual. Let
be a prime of
. Then there exists a Galois representation
- 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.
-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 .
- , 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
a cuspidal representation of
is a subquotient of
. We say pairs
of this form up to equivalence to be the cuspidal support
are equivalent if there exists
are inertially equivalent
if there exists
. We say
are inertially equivalent
if there exists a cuspidal support
and a cuspidal support
. 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
(so contains the Steinberg representation as well). In other words, for this component, we have
has a cuspidal support
is a torus and
is a unramified character.
For representations on the same Bernstein component, we order them using the monodromy operator.
be two nilpotent matrices in
. We say
is in the Zariski closure of the set
(so the Jordan normal form of
has more zeros). We say
a smooth representation of
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.
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.
. Assume there exists
Zariski dense subset such that
is Zariski dense for any
. For any
is crystalline with eigenvalues
. Then for any
, 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.
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
. In particular, by the same argument we know the extension class of
has class in
, which is a contradiction.
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 .
be a de Rham representation of
an extension of the form
Assume that there exists
such that the image
is de Rham.
To prove that
is de Rham, we need to show the surjectivity of
. Tensoring the extension with
, 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,
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.
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.
Even order vanishing
Today we will explain the higher rank case.
vanishes at the center
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.
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
- There exists such that we have an injective map In other words, these crystalline periods contributes to the Hodge-Tate weights .
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
be the extension in Case B
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.
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.
be the universal elliptic curve over the modular curve
. The Hodge filtration gives an exact sequence
We have a
, we have an exact sequence
. 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
is an elliptic curve,
is a basis of
is a basis of
-level structure, such that
Using Tate's curve we have a
Now let us discuss the case of . Denote , and Let . We introduce the differential operators defined by the relation
be an algebraic representation of
. We define
the representation of
be the standard representation of the first
be the standard representation of the second
, which again decreases the weight and the level. Similarly we can define the Maass-Shimura operator
For example, when
, we have
A nearly holomorphic modular form
- for .
- for any ,
Now let us come to the algebraic definition.
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
be the coherent sheaf associated to
). The Hodge filtration gives an exact sequence
, we obtain
pulling back along
we obtain an exact sequence
So dualizing we obtain
agrees with the analytic definition when
, and the map induced by
. Finally, Using the Gauss-Manin connection
and the Kodaira-Spencer isomorphism
we also obtain an algebraic definition of
. Then using a Mumford object (generalization of the Tate curve), one can define a -expansion
is the set of hermitian positive definite lattice of
, then we have a polynomial -expansion
Then the differential operator
acts on the
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.
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.
be an affinoid (not necessarily open) in the weight space of
be a finite map. A family of nearly holomorphic forms
is a polynomial
such that there exists
Zariski dense such that for any
is an algebraic weight and
is equal to
some nearly holomorphic form of weight
. Here for
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
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 ).
be a cuspidal eigenform of weight
be a root of the Hecke polynomial at
, then there exists a point
on the cuspidal eigenvariety with weight
, crystalline Frobenius eigenvalue
and Hecke eigenvalues away from
Like the proof of Proposition 10
, we use a lift of the Hasse invariant,
which is a scalar modular form of weight
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.