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).
Remark 1 The Fontaine—Mazur—Langlands conjecture predicts that, if is irreducible, then it should come from a cuspidal automorphic representation of some reductive group.

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 )

Remark 2 The Fontaine—Mazur—Langlands conjecture predicts that converges for and has a meromorphic continuation to . Moreover, if is semisimple (notice only depends on the semisimplification of ) and does not contain the trivial representation, then should be holomorphic.

Now let us recall the definition of the Bloch-Kato Selmer group associated to .

Remark 3 Recall that if a group (say or ) acts on a vector space , then classifies isomorphism classes of extensions of -representations where is the trivial -representation. Namely, the upper right corner of such extension gives the desired 1-cocycle.
Definition 1 We define where and
Remark 4 Notice that if , then for any finite place , . For , this means we have an exact sequence and if , then we have an exact sequence In other words, is no more ramified than at all places .

Let , where is the 1-dimensional space with the action of given by the cyclotomic character: Now we can state the Bloch-Kato conjecture.

Conjecture 1 (Bloch-Kato)

Two concrete examples are in order.

Example 1 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
Example 2 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.

1. Assume is an imaginary quadratic field.
2. Assume splits in .
3. 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
4. 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
5. 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.)
6. For both , is crystalline.
Theorem 1 (Skinner-Urban) Under the above assumptions (a-f), we have:
1. If , then .
2. If is further even, then .
Remark 5
1. The proof is constructive: we will construct nontrivial elements in the Bloch-Kato Selmer group.
2. There is also a strategy to include the elliptic curve case. But to explain the main ideas, we will first do the above simpler case.
3. We also hope to explain how to relax the crystalline condition and make sure the prime works. If there is time we will also explain what is the obstruction to treat the cases .
4. The assumption that is imaginary quadratic will be used crucially (to ensure is finite). Some cases for general CM field can be treated when is solvable (to use automorphic induction), but one has to be careful about the assumption at .

## 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

1. and there exists an infinite set such that , where is the Galois representation associated to .
2. is polarized: .
3. is geometrically irreducible.
4. 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:

1. A baby case of the strategy.
2. 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.
3. -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).
4. Galois representations attached to -adic automorphic forms for unitary groups (we will only state the results).
5. Explain how to deduce the first two steps of the strategy.
6. 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.
7. 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.
8. Explain how to remove the crystalline assumption at (this probably cannot be done simultaneously with (g), which needs the finite slope assumption).
9. 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
10. 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

Remark 6
1. When , we always have . Moreover, we have if and only if if and only.
2. When , except is trivial. In the latter case , and (as predicted by Bloch-Kato) (i.e., is finite).
Definition 2 For an integer, we define the real analytic Eisenstein series of weight , level and character () to be Here , , , and .
Remark 7
1. converges when and has meromorphic continuation to all .
2. If , then converges and defines a holomorphic form.
3. If , then the meromorphic continuation has no pole at . But it is not clear a priori that is holomorphic. This holomorphy is proved by Shimura by computing the Fourier expansion of and evaluate at . The key here is to verify that the constant term is holomorphic (see below).
Definition 3 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.

Remark 8 If , then the constant term is not holomorphic (as is not holomorphic). Nevertheless, there does exist a holomorphic Eisenstein series of weight 2 with -level and Fourier expansion Notice the -th Fourier coefficient is 1, which is different from . In terms of representation theory, this holomorphic Eisenstein series of weight 2 is explained by a different choice of the local section at . The gamma function term is the intertwining operator at , and the -function term is the product of local intertwining operators over all finite primes . Recall we have an exact sequence For , the local section is chosen to be not lying in the Steinberg subspace, and is nonzero. While for , one can choose to be in the Steinberg subspace and contributes an extra zero, which cancels the pole caused by ! Why does not the same argument in this case construct a nontrivial class in ? Since the slope is 0 rather than the critical slope . The Coleman family (Hida family in this case) passing through this non-critical weight 2 Eisenstein series is a again family of Eisenstein series and the lattice construction breaks down. By the same reasoning we know that there should not exist a holomorphic Eisenstein series of weight 2 of level 1 (though we do have a non-holomorphic Eisenstein series of weight 2 and level 1).

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.

Definition 4 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 .
Definition 5 Define the associated unitary group for any ring . In particular, In fact, fix such that . Then the conjugation by , gives an isomorphism .
Definition 6 Let (no confusion with the imaginary quadratic field ) be the maximal compact of . Then we have under the previous isomorphism.
Remark 9 Another way to construct the maximal compact is as follows. Let and . Here . Then acts on by , where (matrix multiplication). This action is transitive and is the stabilizer of .
Definition 7 One can define an automorphic factor where is the complexification of ( via ). When , given by where .
Definition 8 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 .
Definition 9 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.
Example 3 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 .
Definition 10 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

Definition 11 Consider where Then has signature . In , consider the isotropic line and its stabilizer , with standard Levi decomposition . Then there is an isomorphism
Definition 12 as before determine a representation of as follows. Let be the space of the representation . Then acts on by
Definition 13 Let be the modulus character We see that .
Definition 14 For , define the induced representation
Remark 10
• We may think of as holomorphic functions in the variable . For example, we can take for a fixed . The sections of this form are known as flat sections.
• For and , we see that for all .
• Since , we know that is determined by , where .
• For , we have . Its evaluation at gives a scalar-valued function on given by The data of is the same as the data of .
Definition 15 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.

Proposition 1 Assume . Assume . Then
Proof 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

Definition 16 Write , where form a fixed basis of , and . Let . We define the Eisenstein series and
Theorem 2 Assume is tempered. Then is holomorphic at . Moreover, is a holomorphic as a function of if one of the following conditions is satisfied:
1. if ,
2. if and either
1. , or
2. .

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.

Definition 17 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 .
Definition 18 For an automorphic form on , its constant term along a parabolic subgroup is defined to be
Theorem 3 (Langlands)
1. has a meromorphic continuation to .
2. is holomorphic at if the constant term is holomorphic at for all standard parabolic (equivalently, for just , see Proposition 2).
Definition 19 Let . For simplicity further assume that has order 2, and so , . Then . For , define the intertwining operator This converges when .
Proposition 2 Assume that , then
1. except if .
2. .
Proof 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
Remark 11
1. By the meromorphic continuation of , we know that also has a meromorphic continuation. Moreover, is holomorphic at if and only if is.
2. If is holomorphic at , then is also holomorphic at . Otherwise ( is the order of pole) would be cuspidal, but we know the space of Eisenstein series is perpendicular to cusp forms, a contradiction.
Definition 20 Using , we know is also a representation of . One can check that . The map is equivariant for the -action.
Proposition 3 We have a functional equation
Proof 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

Definition 21 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:

Proposition 4 Here (so ).
Remark 12 The general theory gives the local -factor of the form as the coefficient (the Gindikin-Karpelevich formula).

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).

Remark 13 Now we have constructed as a modular form of weight on . These modular forms can be identified with the global sections the automorphic vector bundle associated to on the unitary Shimura variety . In particular, it has a rational structure and the constructed Eisenstein series is rational (over a number field) because the constant term is rational (this argument dates back to Michael Harris).

## 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

Definition 22 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 .
Definition 23 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.

Remark 14 From the proof of Theorem 2, we see that it is also possible choose a different local section at one place such that to obtain holomorphic Eisenstein series. But this will introduce extra ramification for the Eisenstein series (for the case of , see Remark 8) and we cannot exclude the case (B) in the Step 2 of the strategy (lattice construction).

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 .

Definition 24 Let be the Iwahori subgroup, defined by Let
Definition 25 For , we define . Then one can check that for , we have We define to be the -algebra generated by , , then is commutative and For example, when , this is generated by the usual -operator (and the center).
Definition 26 We say a homomorphism is finite slope if for any . If is of finite slope, then there exists such that
Example 4 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).
Definition 27 A finite slope automorphic representation of is the data of
1. an automorphic representation such that is a discrete series of parameter .
2. a character such that for any and there exists an eigenvector such that . Such a is called a -stabilization of .

In particular, we can view as a representation of occurring in .

Definition 28 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.
Definition 29 We define the normalized slope of to be such that We say that is ordinary if its slope is .
Remark 15 The fact that takes integral values can be viewed as the fact that the Newton polygon attached to the Galois representation is above the Hodge polygon. Then is ordinary means the corresponding Galois representation is ordinary, i.e., the Newton polygon is equal to the Hodge polygon.
Definition 30 We say a slope is non-critical with respect to the weight if
Definition 31 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 .
Definition 32 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 .

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,

Theorem 4 (Urban, Hernandez) Fix an open compact subgroup. Then there exists a rigid analytic variety sitting in the following diagram satisfying the following properties:
1. is flat. In particular, all irreducible components of have dimension equal to .
2. If is a -stabilization of , with automorphic and is holomorphic, then is a point of .
3. If is such that is algebraic and is non-critical with respect to , then there exists automorphic with holomorphic such that is the -stabilization of .
Remark 16 Urban's construction is cohomological and Hernandez's construction is more geometric (using overconvergent forms). The cohomological approach may produce forms which are not holomorphic (only lie in the same discrete -packet), but is more suitable for generalization to parabolic levels.

02/21/2018

Remark 17 The classical cuspidal points are dense in the eigenvariety. These are points , where is an algebraic weight and is a character of that shows up in for cuspidal and .
Remark 18 In fact there is a refinement of the construction: fix an idempotent, then the same statements hold if one replaces by . For example, the original eigenvariety corresponds to taking . More generally, associate to an irreducible smooth representation of (so finite dimensional), we have an idempotent Then if and only if the -type shows up in . This special case will be important for our application to ensure there is no more ramification at ramified places by showing certain -types are preserved in -adic families.

## 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 .

Example 5 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.
Definition 33 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 .
Definition 34 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 .
Definition 35 We define the action of on via and the action of via In particular, for , the two actions agree: .
Definition 36 If is algebraic dominant, we define to be space of functions which are also algebraic.
Remark 19 Since is Zariski dense in , is the same as the space of algebraic functions such that , which is the irreducible algebraic representation of of highest weight . Notice the -action preserves the space of algebraic functions, though the action itself is not algebraic (a -adic twist of the algebraic action).
Definition 37 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.

Proposition 5 If is algebraic dominant, then
Proof See [Urban Annals 2011, Prop. 3.2.12]. ¡õ

Let us illustrate Prop. 5 using the simplest example .

Example 6 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

Remark 20 There is a BGG resolution, It gives a way to compute the cohomology valued in locally algebraic induction in terms of locally analytic inductions. The degree one part of the BGG resolution gives a conceptual proof of Prop. 5.

## Construction of the eigenvariety: slope decomposition

Definition 38 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
Definition 39 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., .
Proposition 6 (Classicality) Let . Assume that is non-critical with respect to (algebraic dominant). Then the natural map is an isomorphism.
Proof 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
Definition 40 Define to be . In other words, one replaces the inequalities defining to strict inequalities.
Proposition 7 For any , the operator acting on in completely continuous (i.e. a limit of finite rank operators).
Proof 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.

Definition 41 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 . 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.

Proposition 8 Let be a congruence subgroup acting freely on . Then there is a -finite free resolution of the trivial -module .
Proof 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 -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).

Definition 42 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.
Definition 43 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 .
Remark 21 Let be the continuous dual of (e.g., when , the differential operator on raises the weight by 2, like Atkin-Serre's theta operator). The above construction also applies to .

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

Definition 44 Let . Let be the two-sided ideal of generated by for . For an open compact subgroup, we let (and similarly ).
Definition 45 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.
Definition 46 A finite slope character distribution is a map for some finite extension such that:
1. there exists a family of irreducible finite slope representations of such that for any , we have where .
2. for any , and , the set is finite.

For such , we simply write where runs over all irreducible finite slope representations. We say is effective if for any .

Definition 47 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 .
Remark 22 By the Fredholm-Riesz-Serre spectral decomposition ([Serre, IHES 1962]), if where is a polynomial with , and an entire power series such that , then there exists a decomposition stable under the action of , where is finite dimensional and and is invertible on (here is the reciprocal polynomial). Moreover, there exists a sequence of polynomials depending algebraically on the coefficients of such that converges to the idempotent projector onto (e.g., in the ordinary case).
Example 7 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

1. 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 .
2. A -type for some .
Theorem 5 Let be the weight space attached to a torus . Let be an -family of effective finite slope character distribution and be a -type. Then there exists a rigid analytic variety sitting in the diagram Here is the finite set of primes such that is not hyperspeicial. It satisfies that a point lies in if and only there exists an irreducible finite slope representation such that
1. ,
2. ,
3. (the restriction of to , see Definition 31).

Moreover, for any , we define to be the hypersurface cut out by the Fredholm determinant (an entire power series with coefficients in ). Then we have a commutative diagram such that the left vertical arrow is finite flat and the bottom horizonal arrow (projection) is finite (hence is equidimensional of dimension ).

Proof (Sketch) Let us explain the local construction (there is no application of the global construction yet).

Let be an affinoid subdomain. Let . Fix , for each decomposition , we define . Varying all such decompositions and gives an admissible covering of . (Notice that Buzzard's eigenvariety machine gives a similar result when comes from a family of so called orthogonalizable Banach modules over ; in contrast, we only have the traces here).

By Remark 22, the decomposition of gives a decomposition of for any . Moreover the projection of onto can be obtained as a squence of polynomials depending algebraically on the coefficients of by taking the limit of , hence is independent of . Now we construct a pseudo-representation For any , this is convergent and defines a pseudo-representation which is actually the trace of the representation of on . (Notice that is the trace of acting on if we had a decomposition in the setting of Buzzard).

Let . We have surjectives onto with unipotent kernel. Hence . The results then follow since is finite over and classifies the Hecke eigensystem of acting on . ¡õ

## 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

Proposition 9 If is anisotropic, then is effective, where is the dimension of the associated locally symmetric space.
Proof 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.

Definition 48 We say a sequence of dominant regular is very regular if for any simple root , . In this case, we have (-adically), and
Theorem 6 (cuspidal character distribution)
1. For any converging very regular sequence , the limit exists and depends only .
2. and is effective.
3. For , we have
Proof (Sketch) The strategy is to use the Hecke equivariant decomposition of the cohomology (due to Franke and Schwermer-J.-S. Li): when is regular, we have One needs to translates the archimedean description of the Eisenstein part into a -adic one.

Let be a standard Levi subgroup. Let be a parabolic with Levi , so . Consider an Weyl group element such that . We define the isomorphism (e.g., for this choice gives the ordinary Eisenstein, rather than the critical Eisenstein series). We define where is the usual constant term (so for any representation of .)

We say that is relevant if

1. has discrete series.
2. The weight spaces are isomorphic (in general the weight space is when has nontrivial center).

It turns out if is relevant, then there is only one parabolic (up to conjugation by ) that is going to contribute. We have

So we define (e.g., when the rank of is zero, we have ). The congruence property (c) for similar to the previous proposition implies (2) that is effective. ¡õ

Remark 23 For a parabolic, and , we define . We say that a finite slope representation is -ordinary if . Let and . Then kills all ordinary contribution, in particular, all the Eisenstein contribution. Hence is effective (in fact a direct factor of .)

## -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.

Example 8 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.

Proposition 10 The critical stabilization of gives a point on the eigenvariety.
Proof There exists a holomorphic form (Hasse invariant) of weight such that the -expansion of is congruent to 1 mod . Write . Then has weight

We can choose a neighborhood of and a factorization of the Fredholm series of some acting on associated to the slope of and get a projector . Now for , we apply the projection and obtain a sum of eigenforms of the same slope as . The projection is nonzero when . These eigenforms are non-critical with respect to and hence corresponds to points on the eigenvariety. Moreover, the system of eigenvalues converging to that of when , and we obtain a point on the eigenvalues associated to . ¡õ

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

Definition 49 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.

Theorem 7 (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.

Definition 50 A Weil-Deligne representation is the data of where,
1. is a smooth representation.
2. a nilpotent endomorphism such that .

It is called irreducible if either or if is regular nilpotent. It is called Frobenius semisimple if is semisimple.

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

1. is cohomological, i.e., there exists an irreducible algebraic representation of such that (equivalently for regular algebraic).
2. 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]):

1. is unramified at finite places,
2. is unramified at places above those which do not split,
3. 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.

Theorem 8 (Chenevier-Harris) Assume is cohomological and conjugate self-dual. Let be a prime of . Then there exists a Galois representation such that
1. 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.
2. 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 , ).
3. If and is unramified, then is crystalline and
Remark 24
1. In a), one expects the equality holds (see the thesis of Caraiani up to Frobenius semi-simplification).
2. In general, for , the Weil-Deligne representation is conjectured to be isomorphic to .The equality in c) means this conjecture holds when is unramified.
Remark 25 For automorphic representations on unitary groups, one can construct Galois representations by first base changing to and then applying the previous Theorem 8. Notice that the latter uses descent to only certain special type of unitary group (e.g., signature at exactly one infinite place).

We also need quadratic base change for automorphic representations on unitary groups.

Theorem 9 (Shin, Morel) Let be a cuspidal cohomological automorphic representations of a unitary group defined by a hermitian space of dimension over . Then there exists an automorphic representation of such that
1. if is unramified and is unramified in ().
2. if splits in .
Remark 26
1. When is not stable, is not necessarily cuspidal.
2. To obtain refined control at places where is ramified, one needs certain simplification in the stable twisted trace formula. This requires in b) that splits in . So in the main theorem of the course one needs to ensure the ramification of is only at split places (which can be arranged due to the freedom of choosing ).

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:

1. , where with .
2. If is ramified then splits in .
3. does not contain primes dividing and .
4. , .
5. .

Under these assumptions on , we have constructed holomorphic Eisenstein series whose associated Galois representation is , where . We further assume that

1. is irreducible (this is conjectured to be true if is cuspidal).
2. 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.

Theorem 10 There exists
• open affinoid,
• a point ,
• Zariski dense,
• a pseudo-representation,
• ,
• , such that,
1. .
2. is unramified outside .
3. For , is crystalline with Hodge-Tate weights with crystalline Frobenius eigenvalues given by
4. For any and , we have
5. is irreducible.
Remark 27 For primes dividing , Item b) is not obvious, and is the content of S. Shah's thesis. Item e) also follows from a)-d).
Remark 28 The monodromy operator may have zeros at certain points in the family, which causes the ramification to drop in Item d). A priori one cannot exclude this possibility (e.g., it shows up in level lowering congruences), but if one imposes the stronger condition that the Galois representation comes directly from Shimura varieties, then one can exactly control the monodromy using geometry. Nevertheless, ensuring the ramification does not increase already suffices for our purpose.

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 .

Definition 51 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 .
Definition 52 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.
Remark 29 Let be the associated Weil-Deligne representation under the local Langlands correspondence. From the very construction of the local Langlands correspondence, we know that Hence the terminology.

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

Example 9 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.

Definition 53 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 .
Proposition 11 (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,

1. for any , is crystalline at with Hodge-Tate weights and crystalline Frobenius eigenvalues , ,
2. 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 ).
3. 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.

Lemma 1 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 .
Remark 30 This generalizes Kisin's lemma: if is of rank 1 for any , then .

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.

Theorem 11 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 .

Lemma 2 is an extension
Proof 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
Remark 31 Notice for the elliptic curve case is empty (due to irregularity), so this argument would not work. Nevertheless, the argument can be salvaged by applying Lemma 1 to the exterior square of .

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 .

Lemma 3 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.
Proof 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, 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.

Theorem 12 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.

Lemma 4 (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
1. .
2. There exists such that we have an injective map In other words, these crystalline periods contributes to the Hodge-Tate weights .
Lemma 5 Assume that is a free lattice which is finite slope of Hodge-Tate type such that fits in an extension Assume that
1. (i.e., 1 does not appear in the first Frobenius eigenvalues).
1. for and .
2. is Hodge-Tate.

Then .

Remark 32 Here . Recall that is Hodge-Tate if and only if . Also recall that Hodge-Tate means the Sen operator is diagonalizable (the valuation of the eigenvalues are the Hodge-Tate weights).
Proof The assertion is clear if 0 is not a Hodge-Tate weight of . Assume otherwise. Suppose we have a sequence of points such that
1. the conclusion holds for all .

Then the conclusion also holds for . In fact, the Sen operator has 0-eigenspace of dimension , so the limit has 0-eigenspace of dimension .

So it remains to prove the result for points such that (as we can then find a sequence of such points converging to ):

1. are integers,
2. for ,
3. for ,
4. .

Here we choose , where is the integer in Kisin's Lemma 4.

For these points, we need to show the surjectivity in the exact sequence By Kisin's Lemma 4 that Since is weakly admissible, we know that . This gives an isomorphism . The desired surjectivity then follows. ¡õ

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).

We have the following generalization of Theorem 10.

Theorem 13 Assume that . Then exists
• a pseudo-representation , where is an open affinoid,
• a set of arithmetic points such that is Zariski dense,
• ,
• ,

such that,

1. , and .
2. is crystalline for with Hodge-Tate weights and Frobenius eigenvalues .
3. For , .
Remark 33 If , then the construction is as before using the eigenvariety for . However, when , the Eisenstein series we are looking for is no longer holomorphic due to irregular Hodge-Tate weights. The idea is then to put into a Coleman (or Hida) family. However, the vanishing of -values in the Coleman family is no longer guaranteed, and the resulting Eisenstein series is only nearly holomorphic (controlled by the order of vanishing).
Corollary 1 .
Proof 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.

Definition 54 Let be a -function. Let be a congruence subgroup. We say that is a nearly holomorphic modular form of order , and of weight for if
1. for any ,
2. is regular (i.e., has a finite limit) at the cusps.
3. , where are holomorphic functions on .

The space of such nearly holomorphic modular forms is denoted by . In particular, .

Definition 55 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
Definition 56 We define the Maass-Shimura operator , which increases the weight and order by

Next let us give equivalent algebraic definitions.

Definition 57 Let be the universal elliptic curve over the modular curve . Let Then . The Hodge filtration gives an exact sequence We have a -Hodge decomposition .
Definition 58 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.

Definition 59 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 (degree polynomials; think: ) and Using Tate's curve we have a -expansion , and where .

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

Definition 60 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
Definition 61 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
Remark 34 It is easy to see (using the canonical line in ) that .
Definition 62 A nearly holomorphic modular form of order , weight and level is a-function such that
1. for .
2. for any ,
3. .

Now let us come to the algebraic definition.

Definition 63 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 .
Definition 64 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 .
Remark 35 We also obtain by and the Poincare duality for .
Definition 65 Let . Then using a Mumford object (generalization of the Tate curve), one can define a -expansion where is the set of hermitian positive definite lattice of . If , then we have a polynomial -expansion 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.

Proposition 12 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.

Definition 66 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 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.
Remark 36 One can similarly define the action of Hecke operators and the finite slope condition.

Let be a Coleman family (for ), where . Consider the injection where . We say that is classical if is classification of non-critical weight .

Theorem 14 There is a family of nearly holomorphic forms such that
1. for any classical, is the polynomial -expansion of the nearly holomorphic Eisenstein series with desired slopes.
2. if , then .
3. if and , then .
Proof 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

Remark 37 The doubling method should generalize well to construct nearly holomorphic families for more general unitary groups.
Corollary 2 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 .
Proof 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