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 $K$ be a number field. Let $E/\mathbb{Q}_p$ be a finite extension. Let $V$ be a finite dimensional $E$-vector space with a $G_K$-action: $\rho_V: G_K\rightarrow \GL_E(V)$. For $v$ a finite place of $K$. Let $I_v\subseteq G_v$ be the inertia subgroup and $\Fr_v$ be the arithmetic Frobenius. Assume that $V$ is geometric in the sense of Fontaine—Mazur (conjecturally coming from the etale cohomology of an algebraic variety), namely:

  • $V$ is unramified away from finitely many primes. This means that there exists a finite set $S$ of finite places such that for any $v\not\in S$, $\rho_V(I_v)=\{1\}$.
  • For any $v|p$, $\rho_V|_{G_v}$ is de Rham (a $p$-adic Hodge theoretic condition which we do not explain now).
Remark 1 The Fontaine—Mazur—Langlands conjecture predicts that, if $V$ is irreducible, then it should come from a cuspidal automorphic representation of some reductive group.

Fixing an embedding $E\hookrightarrow \mathbb{C}$. Associated to $V$ we have its $L$-function $$L(V,s)=\prod_{v<\infty}L_v(V,s).$$ The local L-factor $$L_v(V,s)=P_v(q_v^{-s})^{-1},$$ where $P_v(X)$ is a polynomial of degree $\le d=\dim_E V$, defined as (assume $V$ is crystalline at $v\mid p$) $$P_v(X)=
 \det(1-\rho_V(\Fr_v^{-1})X: V^{I_v}), &v\nmid p\\
 \det(1-\phi_v X: D_\mathrm{crys,v}(V)), & v\mid p.

Remark 2 The Fontaine—Mazur—Langlands conjecture predicts that $L(V,s)$ converges for $\Re(s)\gg0$ and has a meromorphic continuation to $\mathbb{C}$. Moreover, if $V$ is semisimple (notice $L(V,s)$ only depends on the semisimplification of $V$) and does not contain the trivial representation, then $L(V,s)$ should be holomorphic.

Now let us recall the definition of the Bloch-Kato Selmer group $H^1_f(K,V)$ associated to $V$.

Remark 3 Recall that if a group $G$ (say $G_K$ or $G_v$) acts on a vector space $V$, then $H^1(G, V)$ classifies isomorphism classes of extensions of $G$-representations $$0\rightarrow V\rightarrow X\rightarrow E\rightarrow 0,$$ where $E$ is the trivial $G$-representation. Namely, the upper right corner of such extension $$\rho_X=\left(\begin{smallmatrix}\rho_V & * \\ 0 & 1\end{smallmatrix}\right) $$ gives the desired 1-cocycle.
Definition 1 We define $$H^1_f(K,V)=\ker(H^1(G_K, V)\rightarrow \prod_{v<\infty} H^1_{/f}(K_v, V)),$$ where $$H^1_{/f}(K_v, V)=
\ker ( H^1(K_v, V)\rightarrow H^1(I_v,V), & v\nmid p, \\
\ker(H^1(K_v, V)\rightarrow H^1(K_v, V \otimes B_\mathrm{crys}), &v\mid p,
$$ and $$H^1_{/f}(G_v, V)=H^1(K_v, V)/H^1_f(K_v, V).$$
Remark 4 Notice that if $[X]\in H^1_f(K,V)$, then for any finite place $v$, $[X]_v\in H^1_f(K_v, V)$. For $v\nmid p$, this means we have an exact sequence $$0\rightarrow V^{I_v}\rightarrow X^{I_v}\rightarrow E\rightarrow 0, $$ and if $v\mid p$, then we have an exact sequence $$0\rightarrow D_\mathrm{crys}(V)\rightarrow D_\mathrm{crys}(X)\rightarrow E\rightarrow 0.$$ In other words, $X$ is no more ramified than $V$ at all places $v$.

Let $V^\vee(1)=\Hom(V, E(1))$, where $E(1)$ is the 1-dimensional space with the action of $G_K$ given by the cyclotomic character: $$\varepsilon: G_K\rightarrow \mathbb{Z}_p^\times,\quad \sigma(\zeta)=\zeta^{\varepsilon(\sigma)}, \zeta\in \mu_{p^\infty}.$$ Now we can state the Bloch-Kato conjecture.

Conjecture 1 (Bloch-Kato) $$\ord_{s=0} L(V,s)=\rank_E H^1_f(K, V^\vee(1)).$$

Two concrete examples are in order.

Example 1 Let $V=\mathbb{Q}_p$ with trivial action of $G_K$. Then $L(V,s)=\zeta_K(s)$. It follows from Dirichlet's class number formula that $\ord_{s=0} \zeta_K(s)=r_1+r_2-1$. On the other hand, we have $$H^1_f(K,  \mathbb{Q}_p(1))=H^1_f(K, \mathbb{Z}_p(1)) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p.$$ Notice $$H^1(K, \mathbb{Z}_p(1))=\varprojlim_{n}H^1(K, \mu_{p^n})=\varprojlim_n K^\times/(K^\times)^{p^n}=K^\times \otimes_{\mathbb{Z}} \mathbb{Z}_p,$$ where the second equality is by Kummer theory. Therefore $$H^1(K, \mathbb{Q}_p(1))= K^\times \otimes_\mathbb{Z} \mathbb{Q}_p,\quad H^1_f(K, \mathbb{Q}_p(1))=\mathcal{O}_K^\times \otimes_\mathbb{Z} \mathbb{Q}_p.$$ So the Bloch-Kato conjecture is known in this case by Dirichlet's unit theorem that $$r_1+r_2-1=\rank \mathcal{O}_K^\times.$$
Example 2 Let $E$ be an elliptic curve and $V=T_p(E) \otimes_{\mathbb{Z}_p}\mathbb{Q}_p$. Then $V^\vee(1)\cong V$ and $L(V, s)=L(E, s+1)$ (notice the geometric frobenius gives the L-function of $V^\vee=H^1(E, \mathbb{Q}_p)$, hence the shift by 1). Recall that we have the Kummer sequence for $E$: $$0\rightarrow E(K) \otimes \mathbb{Q}_p/\mathbb{Z}_p\rightarrow\Sel_p(K,E)\rightarrow\Sha_p(K,E)\rightarrow 0.$$ One can show that $$\Sel_p(K, E)\cong (\mathbb{Q}_p/\mathbb{Z}_p)^r \times \text{finite group},$$ with $r$ (the corank of $\Sel_p(K,E)$) equal to $\rank H^1_f(K, V)$. The Bloch-Kato conjecture in this case says that $$\ord_{s=1}L(E,s)=\rank_{\mathbb{Q}_p}H^1_f(K,V)$$ This is a consequence of the rank part of the BSD conjecture plus the finiteness of the Shafarevich-Tate group $\Sha_p$.

Now let us consider the following polarized case and state the target theorem of this course.

  1. Assume $K$ is an imaginary quadratic field.
  2. Assume $p$ splits in $K$.
  3. Assume $V$ is polarized in the sense of $$V^\vee(1)\cong V^c,$$ where $\rho_{V^c}(\sigma)=\rho_V(c\sigma c)$ ($c$ is the complex conjugation). Notice $$H^1_f(K, V^\vee(1))=H^1_f(K, V^c)= H^1_f(K, V),$$ and $$L(V^\vee(1), s)=L(V^c,s)=L(V,s)$$ satisfying the functional equation with center $s=0$, $$L(V,-s) ( = L(V^\vee(1), -s)) = \varepsilon(V,s) L(V,s).$$ In this case the Bloch-Kato conjecture says that $$\ord_{s=0}L(V,s)=\rank H^1_f(K,V).$$
  4. Assume (as the Fontaine—Mazur—Langlands conjecture predicts), there is a cuspidal automorphic representation $\pi$ on a unitary group $U(\Phi)$ associated to an hermitian space $(K^d, \Phi)$ of dimension $d$ over $K$, and some Hecke character $\chi$ of $K^\times$ such that we have isomorphisms of Galois representations $$\rho_V\cong R_p(\pi) \otimes \chi_p.$$
  5. Assume for one (hence both, by the polarization) $v\mid p$, $\rho_V|_{G_v}$ is regular (i.e., all Hodge-Tate weights have multiplicity one) and $0,-1\not\in \HT_v(V)$. Here we take the geometric convention that $-1$ is the Hodge weight of $\mathbb{Q}_p(1))$. (In particular, the last condition excludes the case of elliptic curves.)
  6. For both $v\mid p$, $\rho_V|_{G_v}$ is crystalline.
Theorem 1 (Skinner-Urban) Under the above assumptions (a-f), we have:
  1. If $L(V,0)=0$, then $\rank_E H^1_f(K, V)\ge1$.
  2. If $\ord_{s=0}L(V,s)$ is further even, then $\rank_E H^1_f(K, V)\ge2$.
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 $p=2$ works. If there is time we will also explain what is the obstruction to treat the cases $\ord_{s=0}L(V,s)\ge3$.
  4. The assumption that $K$ is imaginary quadratic will be used crucially (to ensure $\mathcal{O}_K^\times$ is finite). Some cases for general CM field $K$ can be treated when $K/\mathbb{Q}$ is solvable (to use automorphic induction), but one has to be careful about the assumption at $p$.

TopThe 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 $$W= E \oplus V \oplus E(1).$$ Namely a (rigid analytic) family of Galois representations (say over a unit disk $\bar D(0,1)$), $$T: G_K\rightarrow A=\mathbb{Q}_p\langle\langle X\rangle\rangle$$ such that

  1. $T_0=\tr(\rho_W)$ and there exists an infinite set $\Sigma\subseteq\bar D(0,1)$ such that $T_x=\tr(\rho_x)$, where $\rho_x$ is the Galois representation associated to $x\in \Sigma$.
  2. $T$ is polarized: $\rho_x^\vee(1)\cong\rho_x^c$.
  3. $T$ is geometrically irreducible.
  4. For almost all $x\in \Sigma$, $\rho_x|_{G_v}$ is crystalline at $v\mid p$ (with certain particular slopes) and the rank of the monodromy operator is the same $\rank N_{\rho_x|_{G_v}}=\rank N_{\rho_W|_{G_v}}$ at all $v\nmid p$.

Step 2 Use the irreducibility of $T$ to construct a lattice $\mathcal{L}$ in $\Frac(A)^d$ such that $\mathcal{L}_0$ is indecomposable as a Galois representation. It follows that we have two cases: Case A $$\rho_{\mathcal{L}_0}\sim
 \varepsilon & {*} & {*}\\ 0 & \rho_V & {*}_A\\ 0 & 0 & 1
$$ or Case B: $$\rho_{\mathcal{L}_0}\sim
\rho_V & {*} & {*}\\ 0 & \varepsilon & {*}_B\\ 0 & 0 & 1

Step 3 Use the ramification properties for the family to show that case B is impossible: $$*_B\in H^1_f(K, \mathbb{Q}_p(1))=0$$ as $\mathcal{O}_K^\times$ is finite.

Step 4 We are now in Case A. Use the assumption about particular slopes at $v|p$ for the family to get the desired extension class $$*_A\in H^1_f(K, V).$$


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 $L$-values.
  3. $p$-adic deformations of automorphic forms and eigenvarieties. In particular, we will allow ramification at $p$ in order to deal with more general cases beyond crystalline at $p$ (this part is not in the literature).
  4. Galois representations attached to $p$-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 ${*}_A$ 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 $p$ (this probably cannot be done simultaneously with (g), which needs the finite slope assumption).
  9. Do the case for CM fields of the form $K=F K_0$, where $K_0$ is imaginary quadratic and $K/\mathbb{Q}$ is cyclic. This uses the theory of automorphic induction and the observation that $$H^1_f(K, V)=H^1_f(K_0, \Ind_{G_{K_0}}^{G_K}V).$$
  10. Do the case of higher order vanishing.

TopBaby case

Let $\chi$ be a Dirichlet character mod $N\ge1$ (not necessarily primitive). Let $p$ be a prime. Let $m\ge1$. Our goal is to show the following implication in the Bloch-Kato conjecture: if $L(1-m,\chi)=0$, then $$H^1_f(\mathbb{Q}_, \mathbb{Q}_p(\chi)(m))\ne0.$$

Remark 6
  1. When $m>1$, we always have $H^1_f=H^1$. Moreover, we have $L(1-m,\chi)=0$ if and only if $\chi(-1)=-(-1)^m$ if and only.
  2. When $m=1$, $L(0,\chi)=0$ except $\chi$ is trivial. In the latter case $L(0,\chi)=\zeta(0)\ne0$, and (as predicted by Bloch-Kato) $H^1_f(\mathbb{Q}, \mathbb{Q}_p(1))=0$ (i.e., $\mathbb{Z}^\times$ is finite).
Definition 2 For $k$ an integer, we define the real analytic Eisenstein series of weight $k$, level $N $ and character $\chi$ ($\chi(-1)=(-1)^k$) to be $$E_{k,N}(z,s,\chi):=\sum_{\gamma\in\Gamma_\infty\backslash \Gamma_0(N)}j(\gamma,z)^{-k}|j(\gamma,z)|^{-2s}\chi(\gamma).$$ Here $\Gamma_\infty=\left(\begin{smallmatrix}1 & {*}\\ 0 & 1\end{smallmatrix}\right) $, $\gamma=\left(\begin{smallmatrix}a & b \\c & d\end{smallmatrix}\right)\in \Gamma_0(N)$, $j(\gamma,z)=cz+d$, and $\chi(\gamma)=\chi(d)$.
Remark 7
  1. $E_{k,N}(z,s,\chi)$ converges when $\Re(s)\gg0$ and has meromorphic continuation to all $s\in \mathbb{C}$.
  2. If $k\ge3$, then $E_{k,N}(z,0,\chi)$ converges and defines a holomorphic form.
  3. If $k=2$, then the meromorphic continuation $E_{k,N}(z,s,\chi)$ has no pole at $s=0$. But it is not clear a priori that $E_{2,N}(z,0,\chi)$ is holomorphic. This holomorphy is proved by Shimura by computing the Fourier expansion of $E_{k,N}(z,s,\chi)$ and evaluate at $s=0$. The key here is to verify that the constant term is holomorphic (see below).
Definition 3 We define $$G_{k,N}=E_{k,N}|_{k,s} \left(\begin{smallmatrix} 0 & 1 \\ N & 0\end{smallmatrix}\right),$$ where $$(f|_{k,s}\gamma)(z)=j(\gamma,z)^{-k}|j(\gamma,z)|^{-2s}f(\gamma,z).$$ So the Fourier expansion of $G_{k,N}$ at the cusp $\infty$ is the Fourier expansion of $E_{k,N}$ at the cusp 0.

Write $z=x+iy$, then $G_{k,N}$ has constant term given by $$\frac{2\pi}{i N}(2y)^{1-k-s}\frac{\Gamma(k+2s-1)}{\Gamma(s)\Gamma(s+k)}\frac{L(k+2s-1,\chi)}{L(k+2s,\chi)}.$$ Assume $k\ge2$ and $s=0$, then the gamma function term has a simple zero and the $L$-function term does not have a pole except $k=2$ and $\chi$ is trivial.

If $(k,\chi)\ne(2,\mathrm{triv})$, then $G_{k,N}(z,0,\chi)$ has constant term 0 and hence is holomorphic of weight $k$ and level $N $. Moreover, the Fourier expansion of $G_{k,N}$ (suitably normalized) is given by $$\sum_{n=1}^\infty\sum_{d|n\atop (d,N)=1}\left(\frac{n}{d}\right)^{d-1}\chi(d) q^n.$$ When $p|N$, we have the $p$-th Fourier coefficient is $p^{k-1}$ (critical slope).

The Eisenstein series $G_{k, Np}(z,0,\chi)$ is eigenform: its $T_\ell$-eigenvalue is $\ell^{k-1}+\chi(\ell)^{-1}$ if $\ell\nmid N$ and $U_\ell$-eigenvalue is $\ell^{k-1}$ for $\ell\mid pN$. Therefore the Galois representation associated to $G_{k,Np}(z,0,\chi)$ is $$\left(\begin{smallmatrix} \varepsilon^{k-1} & \\ & \chi^{-1}\end{smallmatrix}\right).$$

Now Coleman's theory tells us that there exists a family of modular forms $(f_\lambda)$ of slope $k-1$ deforming $G_{Np, k}(z, 0,\chi)$. Here $f_\lambda$ is an eigenform of weight $\lambda$ (varying in an infinite set), level $Np$, and $U_p$-eigenvalue of slope $k-1$. We have a $q$-expansion $$f_\lambda=\sum_{n=1}^\infty a_n(\lambda) q^n,$$ where $a_n(\lambda)$ is an analytic function on the $p$-adic unit disk such that $$a_n(\lambda)=\alpha_n((1+p)^{\lambda-k}-1),\quad  \alpha_n(X)\in \mathbb{Q}_p\langle\langle X\rangle\rangle.$$ In particular, when $\lambda=k$, we have $a_n(k)=\alpha_n(0)$. When $\lambda$ is a sufficiently large integer, $f_\lambda$ is the $q$-expansion of a modular of weight $\lambda$ and slope $k-1$. Its slope is neither zero nor critical ($\lambda-1$), so it must be a cusp form. Now the lattice construction gives us an extension $$\left(\begin{smallmatrix}\varepsilon^{k-1} & {*}\\ 0 & \chi^{-1}\end{smallmatrix}\right),$$ and hence a class in $H^1_f(\mathbb{Q}, \mathbb{Q}_p(\chi)(k-1))$. By remark 6, the only thing one needs to check is that the class is crystalline when $\chi$ is trivial, which follows from the continuity of crystalline periods.

Remark 8 If $(k,\chi)=(2,\mathrm{triv})$, then the constant term is not holomorphic (as $(2y)^{1-k}$ is not holomorphic). Nevertheless, there does exist a holomorphic Eisenstein series of weight 2 with $\Gamma_0(p)$-level and Fourier expansion $$-\frac{\zeta(-1)}{2}+\sum_{n=1}^\infty \sum_{d|n\atop (d,p)=1}d\ q^n.$$ Notice the $p$-th Fourier coefficient is 1, which is different from $p^{k-1}=p$. In terms of representation theory, this holomorphic Eisenstein series of weight 2 is explained by a different choice of the local section $\phi_p$ at $p$. The gamma function term is the intertwining operator $M_\infty(\phi_\infty)$ at $\infty$, and the $L$-function term is the product of local intertwining operators $M_\ell(\phi_\ell)$ over all finite primes $\ell$. Recall we have an exact sequence $$0\rightarrow \mathrm{St}\rightarrow \Ind_{B(\mathbb{Q}_p)}^{\GL_2(\mathbb{Q}_p)}\delta\rightarrow \text{triv}\rightarrow 0.$$ For $k\ge3$, the local section $\phi_p$ is chosen to be not lying in the Steinberg subspace, and $M_p(\phi_p)$ is nonzero. While for $k=2$, one can choose $\phi_p$ to be in the Steinberg subspace and $M_p(\phi_p)=0$ contributes an extra zero, which cancels the pole caused by $\zeta(1)$! Why does not the same argument in this case construct a nontrivial class in $H^1_f(\mathbb{Q}, \mathbb{Q}_p(1))$? Since the slope is 0 rather than the critical slope $k-1$. 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).


TopModular forms on unitary groups

(I was out of town for this lecture and thanks Pak-hin Lee for sending me his notes)

Let $K$ be an imaginary quadratic field with a fixed embedding $K \subset \mathbb{C}$, and $d \geq 1$ be an integer.

Definition 4 Let $\Phi \in M_d(K)$ be a skew-hermitian matrix (i.e., ${}^t \Phi^c = -\Phi$) of signature $(a,b)$ ($b \geq a \geq 0$ and $d = a+b$). This means that the hermitian space $K^d$ can be decomposed as $$K^d \cong (\text{hyperbolic plane})^a \oplus (\text{anisotropic of dim } b-a)$$ and $\Phi$ has the form $$\Phi = \begin{pmatrix} & & 1_a \\ & \theta \\ -1_a \end{pmatrix}$$ for some $\theta \in M_{b-a}(K)$ with ${}^t \theta^c = -\theta$.
Definition 5 Define the associated unitary group $$G(A) = \{ g \in M_d(A \otimes_\mathbb{Z} \O_K) \mid {}^t g^c \Phi g = \Phi\}$$ for any ring $A$. In particular, $$G(\mathbb{R}) \cong U(a,b).$$ In fact, fix $c \in M_d(\mathbb{C})$ such that $c \Phi {}^t\bar{c} = \frac{i}{2} \left(\begin{smallmatrix}1_a & 0 \\ 0 & 1_b \end{smallmatrix}\right)$. Then the conjugation by $c$, $g \mapsto cgc^{-1}$ gives an isomorphism $G(\mathbb{R})\cong U(a,b)$.
Definition 6 Let $K = K_\infty = K_{a,b}$ (no confusion with the imaginary quadratic field $K$) be the maximal compact of $G(\mathbb{R})$. Then we have 
K &\overset{\sim}{\rightarrow} U(a) \times U(b) \\
k &\mapsto ckc^{-1}
under the previous isomorphism.
Remark 9 Another way to construct the maximal compact $K$ is as follows. Let 
\mathcal{D}_{a,b} &= \Bigg\{ x = \begin{pmatrix} z \\ u \\ 1_a \end{pmatrix} \in M_{d \times a}(\mathbb{C}) :z \in M_{a \times a}(\mathbb{C}),\\ &\qquad u \in M_{(b-a) \times a}(\mathbb{C}), i((z-z^*) - u^* \theta u) > 0 \Bigg\}.
and $x_0 = \begin{pmatrix} i 1_a \\ 0 \\ 1_a \end{pmatrix}$. Here $z^*={}^t\bar z$. Then $G(\mathbb{R})$ acts on $\mathcal{D}_{a,b}$ by $g(x) = gx\gamma^{-1}$, where $g \cdot x = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}$ (matrix multiplication). This action is transitive and $K$ is the stabilizer of $x_0$.
Definition 7 One can define an automorphic factor
G(\mathbb{R}) \times \mathcal{D}_{a,b} &\rightarrow K^\mathbb{C} \\
(g,x) &\mapsto J(g,x)
where $K^\mathbb{C}$ is the complexification of $K$ ($K^\mathbb{C} \simeq \GL_a(\mathbb{C}) \times \GL_b(\mathbb{C})$ via $g \mapsto cgc^{-1}$). When $a=b$, $J(g,x)$ given by $$c J(g,x) c^{-1} = (Cz+D, \bar{C}\ {}^tz + \bar{D}),$$ where $g = \left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)$.
Definition 8 Dominant algebraic characters of a Cartan subgroup in $K_{a,b}^\mathbb{C}$ (chosen as the one sent to $\diag_{\GL(a)} \times \diag_{\GL(b)}$ by the isomorphism $K_{a,b}^\mathbb{C} \simeq \GL_a(\mathbb{C}) \times \GL_b(\mathbb{C})$) are classified by sequences of integers $(c_1, c_2, \cdots, c_d)$ with $c_1 \geq c_2 \geq \cdots \geq c_b$ and $c_{b+1} \geq \cdots \geq c_d$. For such a sequence $\tau = (c_1, c_2, \cdots, c_d)$, let $(\rho_\tau, V_\tau)$ be the corresponding complex algebraic representation of $K_{a,b}^\mathbb{C}$.
Definition 9 We define the automorphic factor $J_\tau = \rho_\tau \circ J_{a,b}$, which takes values in $\GL_\mathbb{C}(V_\tau)$. We define modular forms of weight $\tau$ as holomorphic functions $F: \mathcal{D}_{a,b} \rightarrow V_\tau$ such that $$F(\gamma(x)) = J_\tau(\gamma, x) F(x)$$ for all $\gamma \in \Gamma$, where $\Gamma \subset G(\mathbb{Q})$ is an arithmetic subgroup (i.e., $\Gamma = G(\mathbb{Q}) \cap U$ for some $U \subset G(\mathbb{A}_f)$ open compact). We also require that $F$ satisfy a moderate growth condition (or cuspidal condition), but we will not make this precise.
Example 3 Let $\pi \subset L_0^2(G(\mathbb{Q}) \backslash G(\mathbb{A}))$ be an irreducible cuspidal representation of $G(\mathbb{A})$, and write $\pi = \pi_\infty \otimes \pi_f$ where $\pi_\infty$ (resp. $\pi_f$) is a representation of $G(\mathbb{R})$ (resp. $G(\mathbb{A}_f)$). Assume that $\pi_\infty=\pi_\tau^H$ is a holomorphic discrete series with its minimal $K$-type $V_\tau$ with highest weight $\tau = (c_1, \cdots, c_b, c_{b+1}, \cdots, c_d)$ such that $$\begin{cases}
c_1 \geq \cdots \geq c_b, \\
c_{b+1} \geq \cdots \geq c_d, \\
c_b - c_{b+1} \geq d,
\end{cases}$$ (this condition is equivalent to that the corresponding Hodge—Tate weights are regular). In particular, $(\pi_\infty \otimes V_\tau)^K$ has dimension $1$ (as the minimal $K$-type of any discrete series has multiplicity one). Take $\varphi_\infty \in (\pi_\infty \otimes V_\tau)^K$ and $\varphi_f \in \pi_f^U$ where $U \subset G(\mathbb{A}_f)$ is open compact. Then the function $$\varphi = \varphi_\infty \otimes \varphi_f: G(\mathbb{A}) \rightarrow V_\tau$$ is $K$-finite and smooth, so we can consider $$F_\varphi(x) = J_\tau(g, x_0) \varphi(g)$$ where $g \in G(\mathbb{R})$ satisfies $g(x_0) = x$. Since $\varphi_f \in \pi_f^U$, we have $$F_\varphi(\gamma(x)) = J_\tau(\gamma, x) F_\varphi(x)$$ for all $\gamma \in \Gamma = G(\mathbb{Q}) \cap U$. In particular, $\mathcal{F}_\phi$ is a modular form on $G$ of weight $\tau$.
Definition 10 For a cuspidal automorphic representation $\pi$ on $G$, and an idele class character $\chi$ of $K$. Define the $L$-function $$L(\pi, \chi, s):= L(\mathrm{BC}(\pi) \otimes \chi, s)$$ for $\Re(s) \gg 0$, where $\mathrm{BC}(\pi)$ is the base change of $\pi$ to $\GL_d(K)$. If $\mathrm{BC}(\pi)$ is cuspidal (a condition we will assume from now on), this $L$-function has holomorphic continuation. If $\pi$ and $\chi$ are unitary, then there is a functional equation $$L(\pi, \chi, s) = \epsilon(\pi, \chi, s) L(\pi, \chi, 1-s).$$ We will be interested in the central value $L(\pi, \chi, \frac{1}{2})$.

TopEisenstein series on unitary groups

Definition 11 Consider $G' = G(\Phi')$ where $$\Phi' = \begin{pmatrix} & & 1 \\ & \Phi \\ -1 \end{pmatrix} \in \GL_{d+2}(K).$$ Then $G'$ has signature $(a+1, b+1)$. In $K^{d+2}$, consider the isotropic line $(0, \cdots, 0, x)$ and its stabilizer $P$, with standard Levi decomposition $P = MN$. Then there is an isomorphism 
M &\cong G \times \mathbb{G}_{m,K} \\
m(g,t) &\mapsfrom (g,t).
Definition 12 $\pi, \chi$ as before determine a representation of $M(\mathbb{A})$ as follows. Let $V_\pi \subset L_0^2(G(\mathbb{Q}) \backslash G(\mathbb{A}))$ be the space of the representation $\pi$. Then $M(\mathbb{A})$ acts on $V_\pi$ by $$\rho_{\pi,\chi}(m(g,t)) \cdot v = \chi(t) \pi(g) v.$$
Definition 13 Let $\delta$ be the modulus character $$\delta(m) = |\det(m \mid \Lie(N))|.$$ We see that $\delta(m(g,t)) = (t \bar{t})^{-(d+1)}$.
Definition 14 For $s\in \mathbb{C}$, define the induced representation 
I(\rho, s) &= \Ind_{P(\mathbb{A})}^{G'(\mathbb{A})} (\rho_{\pi, \chi} \otimes \delta^s) \\
&= \{ f: G'(\mathbb{A}) \rightarrow V_\pi \mid f(pg) = \delta(p)^{s+1/2} \rho_{\pi, \chi}(p) f(g) \}.
Remark 10
  • We may think of $f_s\in I(\rho,s)$ as holomorphic functions in the variable $s $. For example, we can take $f_s = f_0\cdot \delta^s$ for a fixed $f_0 \in I(\rho, 0)$. The sections $f_s$ of this form are known as flat sections.
  • For $p \in P(\mathbb{Q})$ and $f_s \in I(\rho, s)$, we see that $f_s(pg) = f_s(g)$ for all $g \in G'(\mathbb{A})$.
  • Since $G'(\mathbb{A})=P(\mathbb{A}) K' K_{f,\mathrm{max}}$, we know that $f\in I(\rho)$ is determined by $f|_{K_\mathrm{max}}$, where $K_\mathrm{max}=K'K_{f,\mathrm{max}}$.
  • For $f_s \in I(\rho, s)$, we have $f_s(g) \in V_\pi \subset L_0^2(G(\mathbb{Q}) \backslash G(\mathbb{A}))$. Its evaluation at $1\in G$ gives a scalar-valued function on $G'(\mathbb{A})$ given by $$\tilde{f}_s(g) = f_s(g)(1).$$ The data of $g \mapsto f_s(g)$ is the same as the data of $g \mapsto \tilde{f}_s(g)$.
Definition 15 For a fixed choice of $f_s \in I(\rho, s)$, define the Eisenstein series $$E(f,s,g) = \sum_{\gamma \in P(\mathbb{Q}) \backslash G'(\mathbb{Q})} f_s(\gamma g).$$ This is convergent for $\Re(s) \gg 0$ if $\pi$ is tempered (if $\mathrm{BC}(\pi)$ is cuspidal, then $\pi$ is tempered). The general theory of Eisenstein series tells us that $E(f,s,g)$ has a meromorphic continuation and a functional equation, which we will not need.

Next we will determine conditions on $\pi$, $\chi$ and $s_0 \in \mathbb{C}$ such that the vanishing of $L(\pi, \chi, \frac{1}{2})$ is related to:

  • $E(f,s,g)$ is holomorphic at $s = s_0$.
  • $E(f,s_0,g)$ gives a holomorphic modular form on $\mathcal{D}'$.


Let us look more closely at $I(\rho_\infty)$. Let us choose a weight $$\xi=(c_1,\cdots c_b, p,q, c_{b+1},\cdots ,c_d).$$ Then $$\rho_\xi\cong \rho_{c_1,\cdots,c_b,p} \otimes \rho_{-c_d, \cdots,-c_{b+1},-q}$$ as a representation of $K'{^\mathbb{C}}\cong \GL_{b+1}(\mathbb{C})\times \GL_{a+1}(\mathbb{C})$. The restriction $\rho_\xi|_{K^\mathbb{C}}$ contains $\rho_\tau$ with multiplicity one by the classical branching law for $\GL_n\hookrightarrow \GL_{n+1}$. Moreover, the highest weight of the other components are dominated by $\tau$. Thus $$(V_{\pi_\infty} \otimes V_\xi)^K\cong(V_{\pi_\infty} \otimes V_\tau)^{K}$$ is one dimensional as well.

Proposition 1 Assume $\chi_\infty(z)=z^m \bar z^n$. Assume $m-n=-(p+q)$. Then $$(I(\rho_\infty) \otimes V_\xi)^{K'}\cong (V_{\pi_\infty} \otimes V_\xi)^K.$$
Proof If $\Phi_\infty \in (I(\rho_\infty) \otimes V_\xi)^{K'}$, then for all $k'\in K'$, $p\in P_\infty$ and $g\in G' _ \infty$, we have $$1 \otimes \xi(k') \Phi_\infty(pgk')=\Phi_\infty(pg)=\rho_{\Pi_\infty}(p)\Phi_\infty(g).$$ In particular, taking $g=1$ gives $$\Phi_\infty(pk')=\rho_{\pi_\infty}(p) \otimes \xi(k')^{-1}\Phi_\infty(1).$$ Taking $p=1$ and notice $K'\cap P\supseteq K$, we obtain $$\Phi_\infty(1)\in (V_{\pi_\infty} \otimes V_\xi)^K.$$

Conversely, if any element in the latter gives rise to such $\Phi_\infty$, then we know that $$(I(\rho_\infty) \otimes V_\xi)^{K'}\cong (V_{\pi_\infty} \otimes V_\xi)^K$$ is one dimensional. In fact, one more condition needs to be satisfied for the converse because $K'\cap P$ is strictly large than $K$. Notice $K'\cap P= K'\cap M= K\times U(1)$. For $t\in U(1)$, we have $$m(k,t)\mapsto ( \left(\begin{smallmatrix}g_1 & \\ & t\end{smallmatrix}\right), \left(\begin{smallmatrix}g_2 & \\ & \bar t \end{smallmatrix}\right))\in \GL_{b+1}(\mathbb{C})\times \GL_{a+1}(\mathbb{C}),$$ where $(g_1,g_2)$ is the image of $k$ in $\GL_b(\mathbb{C})\times \GL_a(\mathbb{C})$. One sees that the extra condition is for $t\in U(1)$, we have $$t^p \bar t^{-q}=\chi_\infty^{-1}(t)=t^{-m}\bar t^{-n},$$ namely $m-n=-(p+q)$, as desired.

Fix $\phi_\infty\in (V_{\pi_\infty} \otimes V_\tau)^K$, then Proposition 1 gives an associated $\Phi_\infty\in (I(\rho_\infty) \otimes V_\xi)^{K'}$. By Example 3, it is holomorphic when $p-q\ge d+2$, $c_b\ge p$, $q\ge c_{b+1}$. So $p-q\in\{d+2, \cdots, c_{b+1}-c_b\}$. Fix $\Phi_f\in I(\rho_f)$. Let $\Phi=\Phi_\infty \otimes \Phi_f$. For $h\in G'(\mathbb{A}_f)$ and $z'\in \mathcal{D}'$, we define the function $$\mathcal{F}_h(z')=J_\xi(g', x_0') \Phi(g'h),$$ where $g'\in G_\infty'$ such that $g'(x'_0)=z'$. Let $\phi_h=\Phi(h)$, then we have an associated modular form $f_{\phi_h}(z)$ on $\mathcal{D}$ (Example 3).

For $g'=m(g,t)\in P'$, we have $$\mathcal{F}_h(z')=t^{p+m}\bar t^{-q+n}(t \bar t)^{-(d+1)(s+1/2)}f_{\phi_h}(r(z')),$$ where $r(z')\in \mathcal{D}$ is obtained from $z'\in \mathcal{D}'$ by removing the last column (and the $a+1$-th row to make the right size). So by Proposition 1 we have $$\mathcal{F}_h(z')=|t|^{p-q+m+n-2(d+1)(s+1/2)}f_{\phi_h}(r(z')).$$ Notice that $\phi_h(z)$ is holomorphic, and hence $f_{\phi_h}(r(z'))$ is a holomorphic function in $z'$. So $\mathcal{F}_h(z')$ is a holomorphic function of $z'$ if and only if $$s=s_0:=\frac{k'}{d+1}-\frac{1}{2}+\frac{p-q}{2(d+1)},\quad k'=\frac{m+n}{2}.$$ In particular, when $p-q=d+2$, we have $s_0=s_\mathrm{min}:=\frac{k'+1/2}{d+1}$.

Next time we will see that if $s_0>s_{\mathrm{min}}$, then we will always get a holomorphic Eisenstein series. Otherwise, we get a holomorphic Eisenstein series if either $\chi\ne |\cdot|^{k'}$ or if some central $L$-value vanishes.


Definition 16 Write $\Phi_\infty=\sum_i \Phi_{\infty, i} \otimes v_i$, where $v_i$ form a fixed basis of $V_\xi$, and $\Phi_\infty\in I(\rho_\infty, s)$. Let $\Phi_i=\Phi_{\infty,i} \otimes \Phi_f$. We define the Eisenstein series $$E(\Phi, s, g)=\sum_i E(\Phi_i, s, g) \otimes v_i,$$ and $$E(\mathcal{F}_h, s, z')=\xi(g', x_0') E(\Phi, s, g'h).$$
Theorem 2 Assume $\pi$ is tempered. Then $E(\mathcal{F}_h,s, z')$ is holomorphic at $s=s_0$. Moreover, $E(\mathcal{F}_h, s_0, z')$ is a holomorphic as a function of $z'$ if one of the following conditions is satisfied:
  1. if $s_0>s_\mathrm{min}$,
  2. if $s_0=s_\mathrm{min}$ and either
    1. $\chi|_{\mathbb{A}_\mathbb{Q}^\times}\ne |\cdot |^{2k'}$, or
    2. $L(\pi, \chi^{-1}, k'+1/2)=0$.

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 $G/\mathbb{Q} $ be a reductive group. Let $P=MN\subseteq G$ be a maximal parabolic subgroup. Let $\pi$ be a cuspidal automorphic representation of $M(\mathbb{A})$. Let $\delta_P(p)=|\det \ad(p):\Lie(N)|$ be the modulus character and $$I(\pi,s)=\{\Phi: G(\mathbb{A})\rightarrow V_\pi\text{ smooth, } K\text{-finite}: \Phi(mng)=\delta_P(n)^{s+1/2}\pi(m)\Phi(g)\}.$$ For $\Phi\in I(\pi,s)$, define the Eisenstein series $$E(\Phi, g,s)=\sum_{\gamma\in P(\mathbb{Q})\backslash G(\mathbb{Q})}\Phi(\gamma g).$$ This series converges when $\Re(s)\gg 0$.
Definition 18 For an automorphic form $f$ on $G(\mathbb{A})$, its constant term along a parabolic subgroup $Q\subseteq G$ is defined to be $$f_Q(g):=\int_{N_Q(\mathbb{Q})\backslash N_Q(\mathbb{A})}f(ng)dn.$$
Theorem 3 (Langlands)
  1. $E(\Phi, g,s)$ has a meromorphic continuation to $\mathbb{C}$.
  2. $E(\Phi, g,s)$ is holomorphic at $s=s_0$ if the constant term $E_Q(\Phi, g,s )$ is holomorphic at $s=s_0$ for all standard parabolic $Q\subseteq G$ (equivalently, for just $Q=P$, see Proposition 2).
Definition 19 Let $W_M=N_G(M)/M$. For simplicity further assume that $W_M=\{1,w_0\}$ has order 2, and so $w_0 P w_0=\bar P=M \bar N$, $w_0Mw_0=M$. Then $G=P \coprod P w_0N$. For $\Phi\in I(\pi, s)$, define the intertwining operator $$M(\Phi)(g)=\int_{N(\mathbb{A})}\Phi(w_0ng)dn.$$ This converges when $\Re(s)\gg0$.
Proposition 2 Assume that $\Re(s)\gg0$, then
  1. $E_Q(\Phi, g, s)=0$ except if $Q=P$.
  2. $E_P(\Phi, g, s)=\Phi(g)+M(\Phi)(g)$.
Proof The first follows from the fact that $\pi$ is cuspidal. The second uses $$P(\mathbb{Q})\backslash G(\mathbb{Q})=\{\Id\}\coprod w_0 N(\mathbb{Q}),$$ and therefore $$E_P(\Phi,g,s)=\int_{N(\mathbb{Q})\backslash N(\mathbb{A})}(\Phi(ng)+\sum_{\gamma\in N(\mathbb{Q})}\Phi(w_0\gamma n g))dn.$$ The integration of the first term is simply $\Phi(g)$ (as $\Phi$ lies in the induced representation), and the integration of the second term is $M(\phi)(g)$.
Remark 11
  1. By the meromorphic continuation of $E_P(\Phi, g,s)$, we know that $M(\phi)$ also has a meromorphic continuation. Moreover, $E_P(\Phi, s,g)$ is holomorphic at $s=s_0$ if and only if $M(\phi)(g)$ is.
  2. If $E_P(\Phi, g, s)$ is holomorphic at $s=s_0$, then $E(\Phi, g,s)$ is also holomorphic at $s=s_0$. Otherwise $\lim_{s\rightarrow s_0}(s-s_0)^m E(\Phi, g, s)$ ($m $ 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 $w_0 M w_0=M$, we know $\pi^{w_0}(m):=\pi(w_0 m w_0)$ is also a representation of $M(\mathbb{A})$. One can check that $M(\Phi)\in I(\pi^{w_0}, -s)$. The map $$M_s: I(\pi, s)\rightarrow I(\pi^{w_0},-s)$$ is equivariant for the $G$-action.
Proposition 3 We have a functional equation $$E(\Phi,g,s)=E(M(\Phi), g, -s).$$
Proof After suitably normalize $$M_s: I(\pi, s)\rightarrow I(\pi^{w_0},-s), \quad M_{-s}: I(\pi^{w_0}, -s)\rightarrow I(\pi,s),$$ one can check $M_s\circ M_{-s}=\Id$. In particular, $E(\Phi,g,s)$, $E(M(\Phi), g,-s)$ 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 $$P=\left(\begin{smallmatrix} \bar t^{-1} & \cdot & \cdot\\ & {*}_{a+b} & \cdot \\ & & t  \end{smallmatrix}\right),\quad w_0=\left(\begin{smallmatrix} & & 1 \\& 1_{a+b} & \\ -1 & &  \end{smallmatrix}\right).$$

We denote $$\rho^\vee:=\rho^{w_0}=\rho_{\pi, \chi^\vee},\quad \chi^\vee:=(\chi^c)^{-1}.$$ Then the intertwining operator $M: I(\rho, s)\rightarrow I(\rho^\vee,-s)$ has an Euler product $$M(\Phi,s)(g)= \bigotimes_v M_v(\Phi_v,s)(g_v),$$ where $\Phi$ is a pure tensor, and $$M_v(\Phi_v)(g_v):=\int_{N(\mathbb{Q}_v)}\Phi_v(w_0 n_v g_v)dn_v.$$

Definition 21 Assume $\pi_v$ is unramified, we let $\phi_v\in \pi_v^{K_v}$ to be a spherical vector. Assume $\chi_v$ is further unramified. By Iwasawa decomposition $G'(\mathbb{Q}_v)=P(\mathbb{Q}_v) K_v'$, where $K_v'\cap M(\mathbb{Q}_v)=K_v$. We define the spherical vector $\Phi_v^\mathrm{sph}\in I(\rho_v,s)$ to be the unique section such that $\Phi_v(1)=\phi_v$. Similarly, we define $\Phi_v^{\vee,\mathrm{sph}}\in I(\rho_v^\vee,-s)$.

Let $S$ be a finite set of finite places such that for $v\not\in S\cup\{\infty\}$, $\pi_v,\chi_v$ are unramified. Let $\Phi_f=\Phi_S \otimes \bigotimes_{v\not\in S}\Phi_v^\mathrm{sph}$, with $\Phi_S= \otimes_{v\in S}\Phi_v$ a pure tensor. For $v\not\in S$, by the intertwining property, we know that $M(\Phi_v^\mathrm{sph},s)\in I(\rho_v^\vee,s)$ must be a multiple of the spherical vector $\Phi_v^\mathrm{sph}$. In fact, Langlands' general theory gives:

Proposition 4 $$M(\Phi_v^\mathrm{sph},s)=\frac{L(\pi_v, \chi_v^{-1}, (d+1)s)}{L(\pi_v, \chi_v^{-1},(d+1)s+1)}\frac{L(\chi_v',(2d+2)s)}{L(\chi_v', (2d+2)s+1)}\Phi_v^{\vee,\mathrm{sph}}.$$ Here $\chi_v' = \chi_v|_{\mathbb{Q}_v^\times}$ (so $\chi_\infty' = |\cdot|^{2k'}$).
Remark 12 The general theory gives the local $L$-factor of the form $\frac{L(\ad(M)|\Lie N, qs)}{L(\ad(M)|\Lie N, qs+1)}$ as the coefficient (the Gindikin-Karpelevich formula).

For any $v$ finite, we know that $M_v(\Phi_v,s)$ is holomorphic at $s=s_0$ (by Harish-Chandra as $\pi_v$ is tempered).

02/07/2018 For $v=\infty$, we have $$M_\infty(\Phi_\infty, s)=c(s)\Phi_\infty^\vee \in I(\rho_\infty^\vee, -s) \otimes V_\xi)^{K'}$$ (lies in a 1-dimensional space). Here Harish-Chandra's c-function $c(s)$ is a ratio of $\Gamma$-functions. The key point is that $c(s_0)=0$ as the induction $\Ind_{P(\mathbb{R})}^{G'(\mathbb{R})}(\pi_\tau^H \otimes \delta^{s_0})$ contains the holomorphic discrete series $\pi_\xi^H$ as a subrepresentation, which always lies in the kernel of the intertwining operator $M_\infty$.

So we conclude that $$M(\Phi,s)=c(s)\frac{L^S(\pi, \chi^{-1},(d+1)s)}{L^S(\pi, \chi^{-1},(d+1)s+1)}\frac{L^S(\chi',(2d+2)s)}{L^S(\chi',(2d+2)s+1)}\Phi^\vee.$$ Here $$\Phi^\vee= \bigotimes_{v\not\in S} \Phi_v^{\vee,\mathrm{sph}} \bigotimes_{v\in S} M_v(\Phi_v) \otimes \Phi_\infty^\vee $$ is holomorphic at $s=s_0$. Since $\pi$ is tempered, we know all $L$-values in the denominator does not vanish when $s>s_\mathrm{min}$ as $(d+1)s_\mathrm{min}=k'+1/2$ is the central value. We also know that $L^S(\chi', 2(d+1)s)=L^S(\chi'|\cdot|^{-2k'}, 1)$ has a simple pole exactly when $\chi' = |\cdot|^{2k'}$. This finishes the proof of Theorem 2 by Theorem 3 (b) and Proposition 2(b).

Remark 13 Now we have constructed $E({\mathcal{F}_h}, s_0, z')$ as a modular form of weight $\xi$ on $\mathcal{D}'$. These modular forms can be identified with the global sections the automorphic vector bundle $\omega_\xi$ associated to $\xi$ on the unitary Shimura variety $\Sh'$. 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).

TopL-groups, parameters and Galois representations

Let $H\subseteq \GL_d(E)$ be a unitary group over a field $F$ associated to a quadratic extension $E/F$. Then we have $\hat H=\GL_d(\mathbb{C})$ and $^LH=\hat H \rtimes W_F$, where $W_F$ acts on $\hat H$ by the projection onto $W_{E/F}=\{\Id, j\}$, and $$j(g)=\Phi_d ^{t}g^{-1} \Phi_d^{-1},\quad\Phi_d=\mathrm{antidiag}\{1, -1, 1,-1,\ldots\}.$$

Definition 22 Assume $F$ is a non-archimedean local field, and $H $ is quasi-split and split over an unramified extension of $F$. Let $B=TN\subseteq H$ be a Borel. Let $K\subseteq H=H(F)$ be a hyperspecial maximal compact subgroup. Let $T_K=T\cap K$. Let $X^\mathrm{un}(T)=\Hom(T/T_K, \mathbb{C}^\times)$ be the space of unramified characters. For $\chi\in X^\mathrm{un}(T)$, define $$i_B^H(\chi)=\{f: H\rightarrow \mathbb{C}: f(bg)=\chi\delta^{1/2}(b) f(g)\}.$$ Then $(i_B^H)^K$ is always 1-dimensional. Let $\pi_\chi$ be the unique irreducible subquotient of $i_B^H(\chi)$ such that $\pi_\chi^K\ne0$.
Definition 23 For any $w\in W$ in the Weyl group, using intertwining operators one can see $\pi_{w\chi}\cong\pi_\chi$. Thus we have a map $$X^\mathrm{un}(T)/W\rightarrow\{\text{unramified representations of } H\},$$ which is an isomorphism (the inverse is given by the Satake isomorphism). We define a map $$T/T_K\rightarrow X_*(T)=\Hom(X^*(T), \mathbb{Z}),\quad t\mapsto \lambda_t:\xi\mapsto \val(|\xi(t)|).$$ Identifying $\lambda_t\in X_*(T)\cong X^*(\hat T)$, we obtain an element $g_\chi\in \hat T(\mathbb{C})$ such that $$\chi(t)=\lambda_t(g_\chi),$$ for any $t\in T/T_K$. In this way we have an associated parameter $g_\chi\in \hat T(\mathbb{C})$ to $\chi\in X^\mathrm{ur}(T)$. For $\pi$ an unramified representation of $H $, we have an associate parameter $g_\pi$, the conjugacy class of $(g_\chi, \Frob)\in {}^LH$ of the associated character $\chi$.


Let $P=MN\subseteq H$ be a parabolic subgroup. The we have an inclusion $\iota_M: ^LM\rightarrow{} ^L H$. For an unramified representation $\pi $ on $M$, we have an unramified representation of $H $, namely $i_P^H\pi$ the unique unramified subquotient of $\Ind_P^M \pi$. Moreover, by the transitivity of the parabolic induction ($i_P^H(i_{B\cap M}^M \chi)=i_{B}^H(\chi)$ , we have $$g_{i_P^H(\pi)}=\iota_M(g_\pi).$$

Take $M= G\times \mathbb{G}_{m,K}\subseteq H=G'$. Then $$^LM=(\mathbb{C}^\times\times \GL_d(\mathbb{C})\times \mathbb{C}^\times)\rtimes W_{K/\mathbb{Q}}\hookrightarrow {}^LG'=\GL_{d+2}(\mathbb{C})\rtimes W_{K/\mathbb{Q}}$$ where the embedding is given by $(z,g,z')\mapsto \diag(z,g,z')$. Assume $p$ is a prime where $\pi,\chi$ are unramified. If $p=v\bar v$ splits, then the parameter for $\pi \otimes \chi$ is given by $$(\diag(\chi_v(p), g_{\pi_p}, \chi_{\bar v}(p)^{-1}), \Frob_p).$$ If $p$ is inert, then the parameter is given by $$(\diag(\chi_v(\varpi)^{1/2}, g_{\pi_p}, \chi_{\bar v}(\varpi)^{1/2}),\Frob_p).$$

For $\Pi $ an automorphic representation on $G'$ (holomorphic discrete series at $\infty$), the link to Galois representations is given as follows. There exists $$R_p(\Pi): G_K\rightarrow \GL_{d+2} (\overline{\mathbb{Q}_p})$$ such that $$L^{S\cup \{p\}}(R_p(\Pi),s )=L^{S\cup \{p\}}(\Pi^\vee, s-(1-(d+2))/2).$$ Here $R_p(\Pi)^\vee(1-(d+2))\cong R_p(\Pi)^c$.

When $\Pi $ is an irreducible subquotient of the space of Eisenstein series $E(\Phi, g, 0)$ for $\Phi\in \Ind_P^G(\pi \otimes \chi)$. Then $$L^S(\Pi, s)=L^S(\pi, s) L^S(\chi,s) L^S(\chi^{c,-1},s).$$ More generally, for $E(\Phi, g,s_0)$, we obtain $$L^S(\Pi, s)=L^S(\pi, s) L^S(\chi,s+(d+1)s_0) L^S(\chi^{c,-1},s-(d+1)s_0).$$ In this case, we have $$R_p(\Pi)\cong R_p(\pi)(-1) \oplus \chi_p^{-1}\varepsilon^{(d+1)s_0-(d+1)/2}\oplus \chi_p^c \varepsilon^{-(d+1)s_0-(d+1)/2}.$$

Recall that $(d+1)s_0=k'+(p-q-(d+1))/2$, where $p-q\in\{d+2,\cdots, c_{b+1}-c_b\}$. So the two relevant characters are $$\chi_p^{-1}\varepsilon^{k'}\varepsilon^{(p-q)/2-(d+1)},\quad \chi_p^{c}\varepsilon^{-k'}\varepsilon^{-(p-q)/2}.$$ Let $\chi_1=\chi_p \varepsilon^{-k'}$ (motivic weight zero, $\chi_1^{-1}=\chi_1^c$), then the two characters are $$\chi_1^{-1}\varepsilon^{(p-q)/2-(d+1)}, \quad \chi_1^c\varepsilon^{-(p-q)/2}.$$ Notice the difference of the powers of the cyclotomic character $\varepsilon$ is given by $$p-q-(d+1)\in\{1,2,\cdots, c_{b+1}-c_b-d-1\},$$ we know that when $p-q$ is minimal, the Galois representation is of the form $$R_p(\pi)(-1) \oplus \chi_2 \oplus \chi_2 \varepsilon,$$ for some character $\chi_2$. In this case, let $R=R_p(\pi) \otimes \chi_2^{-1}$, then $R_p(\Pi)$ is a twist of $$R' = R \oplus \varepsilon \oplus \mathbf{1}.$$ 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 $\Phi_v$ at one place such that $M(\Phi_v,s_0)=0$ to obtain holomorphic Eisenstein series. But this will introduce extra ramification for the Eisenstein series (for the case of $\SL_2$, see Remark 8) and we cannot exclude the case (B) in the Step 2 of the strategy (lattice construction).


Top$p$-adic deformations of automorphic representations of unitary groups

Recall $G$ is the unitary group for $\Phi\in M_d(K)$ of signature $(a,b)$. For simplicity (not essential), we will assume that $p$ is split in $K$. Then $G/\mathbb{Q}_p\cong \GL_d/\mathbb{Q}_p$, and the hyperspecial maximal compact subgroup $K_p^\mathrm{max}\cong\GL_d(\mathbb{Z}_p)$.

Definition 24 Let $I\subseteq K_p^\mathrm{max}$ be the Iwahori subgroup, defined by $$I=\{g\in \GL_d(\mathbb{Z}_p): g\bmod{ p}\in B(\mathbb{F}_p)\}.$$ Let $$T^+=\{ t=\diag\{t_1,\ldots, t_d\} \in T(\mathbb{Q}_p): v_p(t_1)\le \cdots\le v_p(t_d)\}\subseteq T(\mathbb{Q}_p).$$
Definition 25 For $t\in T(\mathbb{Q}_p)$, we define $u_t:=\mathbf{1}_{ItI}\in C_c^\infty( G(\mathbb{Q}_p), \mathbb{Z})$. Then one can check that for $t,t'\in T^+$, we have $$u_t {*} u_{t'}=u_{tt'}.$$ We define $U_p$ to be the $\mathbb{Z}_p$-algebra generated by $u_t$, $t\in T^+$, then $U_p$ is commutative and $$U_p\cong \mathbb{Z}_p[T^+/T(\mathbb{Z}_p)].$$ For example, when $d=2$, this is generated by the usual $U_p$-operator (and the center).
Definition 26 We say a homomorphism $\theta :U_p\rightarrow \mathbb{C}$ is finite slope if $\theta(u_t)\ne0$ for any $t\in T^+$. If $\theta$ is of finite slope, then there exists $\alpha_1,\ldots, \alpha_d\in \mathbb{C}^\times$ such that $$\theta(t)=\prod_{i=1}^d \alpha_i^{v_p(t_i)}.$$
Example 4 Let $\pi_p$ be an unramified representation of $G(\mathbb{Q}_p)$. Then we have $$\pi_p\hookrightarrow i(\chi):=\Ind_{B(\mathbb{Q}_p)}^{G(\mathbb{Q}_p)}\chi,$$ for $\chi$ an unramified character of $T(\mathbb{Q}_p)$. Taking $I $-invariants, we have an embedding of $U_p$-modules $$\pi_p^I\hookrightarrow i(\chi)^I.$$ By the Bruhat decomposition $G(\mathbb{Q}_p)=\coprod_{w\in W} BwI$, we know that $i(\chi)^I$ has dimension equal to $\#W=d!$. More precisely, for any $w\in W$, there exists an eigenvector $v_{\chi,w}\in i(\chi)^I$ such that $$u_t v_{\chi,w}=(\chi^w\rho)(t) v_{\chi,w},\quad \rho(t)=\left|\prod_{\alpha\in \Delta^+}\alpha(t)\right|^{1/2}.$$ (Notice $\rho=((d-1)/2, (d-3)/2, \ldots, (1-d)/2)$, this extra factor comes from the normalization of the Langlands parameter). So any eigenvector of $U_p$ inside $i(\chi)^I$ is attached to a pair $(\chi, w)$. The values $(\alpha_1,\ldots, \alpha_d)$ attached to $\theta: U_p\rightarrow \mathbb{C}$ correspond to an ordering of the eigenvalues of the Langlands parameter of $\pi_p$. All such orderings show up if and only if the embedding $\pi_p^I\hookrightarrow i(\chi)^I$ is an isomorphism (e.g., the case when $\chi$ is unitary as $\pi_p$ is irreducible).
Definition 27 A finite slope automorphic representation of $G(\mathbb{A})$ is the data of
  1. an automorphic representation $\pi$ such that $\pi_\infty$ is a discrete series of parameter $\tau=\{c_1,\cdots, c_d\}$.
  2. a character $\theta_p: U_p\rightarrow \mathbb{C}$ such that $\theta_p(u_t)\ne0$ for any $t\in T^+$ and there exists an eigenvector $v\in \pi_p^I$ such that $u_t v=\theta(t) v$. Such a $\theta$ is called a $p$-stabilization of $\pi_p$.

In particular, we can view $\sigma=(\pi, \theta_p)$ as a representation of $G(\mathbb{A}^p) \otimes U_p$ occurring in $\pi^I$.

Definition 28 We normalize $\theta_p$ and define $$\tilde \theta_p(t)=|\lambda_\tau(t)| \theta_p(t), \lambda_\tau=(c_1-a,\ldots, c_b-a, c_{b+1}+b, \cdots, c_d+b).$$ Then $\tilde \theta_p$ takes values in algebraic integers: in fact there is a Hecke equivariant Eichler-Shimura map, $$S_\tau(K, \mathbb{C})\hookrightarrow H^{ab}(S_K, L_{\lambda_\tau}(\mathbb{C})),$$ which embeds the weight $\tau$-modular forms into the middle cohomology of the unitary Shimura variety with coefficient in the representation of highest weight $\lambda_\tau$, and the latter has an integral structure preserved under the Hecke action.
Definition 29 We define the normalized slope of $\tilde \theta_p$ to be $$\mathbf{s}=(s_1,\ldots,s_d)\in \mathbb{Q}^d_{>0}$$ such that $$v(\tilde \theta_p(u_t))=\sum_{i=1}^d s_i v(t_i).$$ We say that $\tilde \theta_p$ is ordinary if its slope is $(0,0,\ldots,0)$.
Remark 15 The fact that $\tilde \theta_p$ takes integral values can be viewed as the fact that the Newton polygon attached to the Galois representation $\rho_{\pi}|_{\mathbb{Q}_p}$ is above the Hodge polygon. Then $\tilde \theta_p$ 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 $\mathbf{s}$ is non-critical with respect to the weight $\lambda=(k_1,\ldots,k_d)$ if $$|s_i-s_{i+1}|<\min_{1\le i\le d}\{k_i-k_{i+1}-1\}.$$
Definition 31 Let $\sigma=(\pi, \theta_p)$. Let $S$ be a finite set of primes containing all primes where $\pi$ is ramified. We denote $$R_{S,p}=C_c^\infty( G(\mathbb{A}_f^{S\cup \{p\}})//K_\mathrm{max}^{S\cup \{p\}}, \mathbb{Z}_p) \otimes U_p.$$ The $\sigma$ gives rise to a character $$\theta_\sigma: R_{S,p}\rightarrow \mathbb{C}^\times$$ such that $\theta_\sigma|_{U_p}=\tilde \theta_p$, and gives the Hecke action on the spherical vector away from $S\cup\{p\}$.
Definition 32 We define the weight space $\mathcal{X}$ to be the rigid analytic variety over $\mathbb{Q}_p$ such that for any finite extension $L/\mathbb{Q}_p$, $$\mathcal{X}(L)=\Hom_\mathrm{cont}(T(\mathbb{Z}_p), L^\times).$$ Notice $T(\mathbb{Z}_p)=\Delta_T \times \mathbb{Z}_p^d$ for a finite group $\Delta_T$. So $$\mathcal{X}(L)\cong\coprod_{\widehat{ \Delta_T}}D_b(1,1)(L).$$ Here $D_b(1,1)$ is the open unit disk of radius 1 centered at 1 in $\bar{\mathbb{Q}}_p^d$. For a dominant character $\lambda \in X^*(T)$, we obtain a corresponding algebraic weight $[\lambda]\in \mathcal{X}(\mathbb{Q}_p)$.

An eigenvariety is going to be a rigid analytic variety $\mathcal{E}$ that contains points attached to finite slope automorphic representations of $G$. If $\sigma$ is of the form $(\pi,\theta_p)$ as before, then we say that $\sigma$ is of algebraic weight $\lambda_\tau$. More generally, it is possible to speak about finite slope automorphic representation of a given $p$-adic weight $\lambda \in \mathcal{X}( \overline{\mathbb{Q}_p})$. More precisely,

Theorem 4 (Urban, Hernandez) Fix $K\subseteq G(\mathbb{A}_\mathbb{F})$ an open compact subgroup. Then there exists a rigid analytic variety $\mathcal{E}_K$ sitting in the following diagram $$\xymatrix{ \mathcal{E}_K \ar@{^(->}[r]^-{x\mapsto (\theta_x, \lambda_x)} \ar[d]^{\pi} & \mathrm{Max}(R_{S,p})\times \mathcal{X} \ar[ld] \\ \mathcal{X}&   }$$ satisfying the following properties:
  1. $\pi$ is flat. In particular, all irreducible components of $\mathcal{E}_K$ have dimension equal to $\dim \mathcal{X}=d$.
  2. If $\theta$ is a $p$-stabilization of $\pi^{IK^p}\ne0$, with $\pi$ automorphic and $\pi_\infty=\pi_\tau^H$ is holomorphic, then $x=(\theta,\lambda_\tau)$ is a point of $\mathcal{E}_K$.
  3. If $x=(\theta_x,\lambda_x)$ is such that $\lambda_x=[\lambda_\tau]$ is algebraic and $\theta_x$ is non-critical with respect to $\lambda_\tau$, then there exists $\pi$ automorphic with $\pi_\infty=\pi_\tau^H$ holomorphic such that $\theta_x$ is the $p$-stabilization of $\pi^{IK^p}$.
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 $L$-packet), but is more suitable for generalization to parabolic levels.


Remark 17 The classical cuspidal points are dense in the eigenvariety. These are points $x=(\theta,\lambda_\tau)$, where $\lambda_\tau$ is an algebraic weight and $\theta$ is a character of $R_{S,p}$ that shows up in $\pi_f^{K^pI}$ for $\pi$ cuspidal and $\pi_\infty=\pi_{\lambda_\tau}^H$.
Remark 18 In fact there is a refinement of the construction: fix $e\in C_c^\infty( G(\mathbb{A})_S, \overline{\mathbb{Q}_p})$ an idempotent, then the same statements hold if one replaces $\pi_f^{K^pI}$ by $e \pi_S \otimes (\pi_f^S)^{K^{S\cup\{p\}}I}$. For example, the original eigenvariety corresponds to taking $e=\frac{1}{\Vol(K_S)}\mathbf{1}_{K_S}$. More generally, associate to an irreducible smooth representation $\rho$ of $K_S$ (so finite dimensional), we have an idempotent $$e_\rho(g)=
  \frac{1}{\dim\rho \Vol(K)} \tr(\rho(g)), & g\in K,\\
  0, & g \not\in K.
\end{cases}.$$ Then $e_\rho \pi_S\ne0$ if and only if the $K_S$-type $\rho$ shows up in $\pi_S$. This special case will be important for our application to ensure there is no more ramification at ramified places by showing certain $K_S$-types are preserved in $p$-adic families.

TopConstruction 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 $$S_\tau(K, \mathbb{C})\hookrightarrow H^\mathrm{ab}(S_K, L_{\lambda_\tau}(\mathbb{C})).$$ So to interpolate the automorphic forms $p$-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 $\lambda\in \mathcal{X}(L)$ is locally analytic, namely, there exists an integer $m=n_\lambda\ge0$ such that $\lambda|_{T_m(\mathbb{Z}_p)}$ is analytic, where $T_m(\mathbb{Z}_p)=\ker (T(\mathbb{Z}_p)\rightarrow T(\mathbb{Z}/p^m))$.

Example 5 Take $T=\mathbb{G}_m$. Then $\lambda: T(\mathbb{Z}_p)=\mathbb{Z}_p^\times\rightarrow L^\times$. So $\lambda(1+p)=x\in L^\times$ for some $|x-1|<1$. Find $n $ such that $x^{p^n}\equiv1\pmod{p}$. Then for $t=(1+p)^{p^ns}\in (1+p)^{p^n \mathbb{Z}_p}$, we have $$\lambda(t)=(x^{p^n})^s=\sum_{i=0}^\infty {s\choose \lambda}(x^{p^n}-1)^i.$$ Notice the last expression is analytic (a convergent power series in $s $), so $\lambda$ is locally analytic.
Definition 33 For $n\ge n_\lambda$, we define $$\mathcal{A}^n_\lambda(L):=\{f: I\rightarrow L\}$$ where $f$ is $n $-analytic (i.e. analytic on disks of radius $p^{-n}$) such that $$f(n^-tg)=\lambda(t) f(g),\quad g\in I, t\in \mathbb{Z}_p, n^-\in N^-_I:=N^-\cap I,$$ here $N^-$ consists of lower triangular unipotent matrices. By the Iwahori decomposition $$I=N_I^-T(\mathbb{Z}_p) N_I^+,$$ we see $\mathcal{A}_\lambda^n(L)$ is identified with $\mathcal{A}^n(N_I^+,L)$, the $n $-analytic functions on $N^+_I$, via the restriction map $f\mapsto f|_{N^+_I}$. The latter is independent of $\lambda$.
Definition 34 Define $\Delta^+:= IT^+I$. It is equal to $N^-_ I T^+ N_I ^+$, and the decomposition $g=n_g^- t_g n_g^+$ is unique for $g\in \Delta^+$. Notice the natural action $I $ on $N^-_I\backslash I=T(\mathbb{Z}_p) N_I^+$ extends to a contracting action of $\Delta^+$. We will use the right action of $\Delta^+$ on cohomology to define a compact operator (as a replacement of finite dimensionality). Explicitly, for $x\in I$, $g\in \Delta^+$, we define the action by $$x*g= \xi(t_g)^{-1} xg,$$ where $$\xi: T(\mathbb{Q}_p)\rightarrow T(\mathbb{Q}_p), \quad t\mapsto \diag\{p^{v_p(t_1)},\ldots, p^{v_p(t_d)}\}.$$ In particular, $\xi(t)=1$ if $t\in T(\mathbb{Z}_p)$.
Definition 35 We define the action of $I $ on $\mathcal{A}_\lambda^n(L)$ via $$(gf)(x)=f(xg),$$ and the action of $\Delta^+$ via $$(g*f)(x)=f(x*g)=f(\xi(t_g)^{-1}xg),$$ In particular, for $g\in I$, the two actions agree: $g*f=gf$.
Definition 36 If $\lambda$ is algebraic dominant, we define $\mathcal{A}_\lambda^\mathrm{alg}(L)$ to be space of functions $f\in \mathcal{A}_\lambda(L)$ which are also algebraic.
Remark 19 Since $I $ is Zariski dense in $G$, $\mathcal{A}_\lambda^\mathrm{alg}(L)$ is the same as the space of algebraic functions $G/L\rightarrow L$ such that $f(n^-tg)=\lambda(t) f(g)$, which is the irreducible algebraic representation of $G$ of highest weight $\lambda$. Notice the $*$-action preserves the space of algebraic functions, though the action itself is not algebraic (a $p$-adic twist of the algebraic action).
Definition 37 If $\lambda$ is algebraic and $\alpha$ is a simple root, we define $$\Theta_\alpha: \mathcal{A}_ \lambda(L)\rightarrow \mathcal{A}_ {s_\alpha*\lambda}(L),\quad f\mapsto \ell(X_\alpha)^{\lambda(H_\alpha)+1} f.$$ Here for $w\in W$, $w*\lambda=w(\lambda+\rho)-\rho$, where $\rho$ is the half sum of all positive roots, $X_\alpha$ is a basis vector for the root space $\mathfrak{g}_\alpha$, and $\ell$ is the differential operator of the left translation. Notice the left translation action of $I $ preserves $\mathcal{A}(I,L)$ (not $\mathcal{A}_\lambda(L)$), but one can check the image indeed lies in $\mathcal{A}_{s_\alpha*\lambda}(L)$.

It is clear that $\Theta_\alpha$ is equivariant for the right $I $-action. Moreover, for $g\in \Delta^+$, we have $$\Theta_\alpha(g*f)=\xi(t_g)^{s_\alpha*\lambda-\lambda}g*\Theta_\alpha(f).$$ The power of $p$ in $\xi(t_g)$ will control the integrability of the image.

Proposition 5 If $\lambda$ is algebraic dominant, then $$\mathcal{A}_\lambda^\mathrm{alg}(L)=\bigcap_\alpha \ker(\Theta_\alpha).$$
Proof See [Urban Annals 2011, Prop. 3.2.12].

Let us illustrate Prop. 5 using the simplest example $G=GL(2)$.

Example 6 Let $G=GL(2)$ and $B\subseteq G$ be the upper triangular matrices. Let $\lambda=(n_1,n_2)$ be an algebraic weight, then $\lambda$ is dominant if and only if $n_1\ge n_2$. The two simple roots are $\alpha=(1,-1),-\alpha=(-1,1)$ (i.e., $\alpha(t_1, t_2)=t_1/t_2$). The Weyl group is $W=\{1,w=s_\alpha\}$, where $w(n_1, n_2)=(n_2,n_1)$. So $$\rho=(1/2, -1/2),\quad w(\lambda+\rho)-\rho=(n_2-1, n_1+1).$$ For $\lambda$ dominant, let $V_\lambda$ be the algebraic representation of highest weight $\lambda$. So $V_\lambda$ can be identified with the space of homogeneous polynomials in $X,Y$ of degree $n_1-n_2$ with action $$(gP)(X,Y)=P((X,Y)g)\det(g)^{n_2}, \quad g\in G, P\in V_\lambda.$$ Define $f_P(g):=P(x,y)$, if $g=\left(\begin{smallmatrix}x & y\\ * & {*} \end{smallmatrix}\right)$. Then $$f_P(\left(\begin{smallmatrix} t_1 & 0 \\ {*} & t_2\end{smallmatrix}\right) ) g )=t_1^{n_1} t_2^{n_2}f_P(g).$$ Then $P\mapsto f_P$ identifies $V_\lambda$ as the algebraic induction $\Ind_{B^-}^G\lambda$.

We have $G=B^-w \coprod B^-N^+$, where $B^-N^+$ is Zariski dense (known as the big cell). In particular, $f_P$ is determined by its restriction to $N^+$: $f_P(\left(\begin{smallmatrix} 1 & z \\ & 1\end{smallmatrix}\right))=P(1,z)=:\tilde P(z)$. This restriction induces an injection $\Ind_{B^-}^G \lambda\rightarrow \Ind_{B^-}^{B^-N^+}\lambda\cong \mathbb{C}[z]$, and the image is the space of polynomials in $z$ of degree $\le n:=n_1-n_2$, with the action $$(g\tilde P)(z)=\det(g) ^{n_2}(a+cz)^n\tilde P(\frac{b+dz}{a+cz}).$$

The space $\mathcal{A}_\lambda(L)$ is the space of analytic functions on $I $ such that $$f((\left(\begin{smallmatrix}t_1 & 0 \\ * & t_2\end{smallmatrix}\right)g)=t_1^{n_1}t_2^{n_2} f(g).$$ Again by restriction $$f\mapsto \tilde f(z)=f(\left(\begin{smallmatrix} 1 & z \\ & 1\end{smallmatrix}\right))=\sum_{k=0}^\infty a_k z^k,$$ we can identify $\mathcal{A}_\lambda(L)$ as the space of analytic functions on $N^+(\mathbb{Z}_p)$, with the $I $-action defined by the same formula $$(g\tilde f)(z)=\det(g) ^{n_2}(a+cz)^n\tilde \tilde f(\frac{b+dz}{a+cz}).$$ One sees that a function $f(z)\in \mathcal{A}_\lambda$ lies in $V_\lambda$ if and only if $f(z)$ is a polynomial of degree $\le n$, i.e., $(d/dz)^{n+1}f=0$, as in Prop. 5.

One can check by direct computation that $\Theta_\alpha(f)\in \mathcal{A}_{s_\alpha*\lambda}$, namely in this case $$(d/dz)^{n+1}f\in \mathcal{A}_{(n_2-1, n_1+1)}.$$ More conceptually, take a basis of the Lie algebra $\mathfrak{sl}(2)$, $$H=\left(\begin{smallmatrix}1 & 0 \\ 0& -1 \end{smallmatrix}\right), X_+=\left(\begin{smallmatrix}0 & 1 \\ 0 & 0 \end{smallmatrix}\right), X_-=\left(\begin{smallmatrix}0 & 0 \\ 1 &0 \end{smallmatrix}\right).$$ Then $\lambda(H)=n$ and $\Theta_\alpha(f)= \ell(X_+)^{n+1}f$, where $(\ell(g)f)(z)=f(g^{-1}z)$. The function $h=\Theta_\alpha(f)$ is invariant under the action of $\ell(X_-)$: $$\ell(X_-) h=\ell(X_1X_+^{n+1})f=\ell([X_-, X_+^{n+1}])f=(n+1)(-\ell(H)+n)f=0,$$ as $\ell(X_-)f=0$, $[X_-, X_+^{n+1}]=(n+1) X_+^n(-H+n)$ and $\ell(H)f=nf$. Moreover, for $t\in N^-\cap I$: we have $$\ell(t) h=\ell(t X_+^{n+1}t^{-1})\ell(t) f=\lambda(t)^{-1}\alpha(t)^{n+1}\ell(X_+^{n+1})f=\lambda^{-1} \alpha^{n+1}(t^{-1}) f.$$ Hence $h\in \mathcal{A}_{\lambda- (n+1)\alpha}=\mathcal{A}_{(n_2-1, n_1+1)}$.


Remark 20 There is a BGG resolution, $$0\rightarrow V_\lambda^\mathrm{loc}\rightarrow \mathcal{A}_\lambda^\mathrm{loc}(L)\rightarrow\cdots\rightarrow \bigoplus_{w\in W, \ell(w)=n}\mathcal{A}_{w*\lambda}(L)\rightarrow\cdots.$$ 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.

TopConstruction of the eigenvariety: slope decomposition

Definition 38 Let $\theta: U_p\rightarrow \mathbb{C}$ be finite slope. We define its slope $\mu_\theta\in X^*(T) \otimes \mathbb{Q}$ such that for any $\mu^\vee\in X_*(T)^+$ (so $\mu^\vee(p)\in T^+$), we have $$(\mu_\theta, \mu^\vee)=v_p(\theta(u_{\mu^\vee(p)}))\in \mathbb{Q}.$$ (Recall for $\mu\in X^*(T), \mu^\vee\in X_*(T)$, $(\mu,\mu^\vee)\in \mathbb{Z}$ is defined such that $\mu\circ \mu^\vee(x)=x^{(\mu,\mu^\vee)}$). Notice that if $\theta$ takes integral values, then $(\mu_\theta, \mu^\vee)\ge0$, and hence $\mu_\theta\in X^*(T)_{\mathbb{Q},+}$. Namely, $\mu_\theta$ lies in the cone generated by the positive roots (which may be larger than $X^*(T)^+_\mathbb{Q}$, the cone generated by the dominant weights). We say the slope $\mu_\theta$ is called non-critical with respect to $\lambda$ if for any $w\in W$, $w\ne1$, we have $$\mu_\theta-\lambda+w*\lambda\not\in X^*(T)_{\mathbb{Q},+}.$$
Definition 39 Suppose $U_p$ acts on a Banach space $M$ over $L$. For any $\mu\in X^*(T)_{\mathbb{Q}}$, we define $M^{\le \mu}$ to be the sum of the generalized eigenspace for $U_p$ attached to the characters $\theta$ such that $\mu_\theta\le \mu$ (i.e., $\mu-\mu_\theta\in X^*(T)_{\mathbb{Q},+}$.
Proposition 6 (Classicality) Let $\Gamma=G(\mathbb{Q})\cap K^p I$. Assume that $\mu$ is non-critical with respect to $\lambda$ (algebraic dominant). Then the natural map $$H^i(\Gamma, V_\lambda)^{\le\mu}\rightarrow H^i(\Gamma, \mathcal{A}_\lambda)^{\le\mu}$$ is an isomorphism.
Proof Notice that $\mu-\lambda +s_\alpha*\lambda$ is no longer integral by the definition of non-criticality. Hence the image is killed by $\Theta_\alpha$. The result then follows from Prop. 5.
Definition 40 Define $T^{++}\subseteq T^+$ to be $\{t\in T^+: \cap_{n\ge1} t^{-n} N\cap I t^m=\{1\}\}$. In other words, one replaces the inequalities defining $T^+$ to strict inequalities.
Proposition 7 For any $t\in T^{++}$, the operator $u_t$ acting on $H^i(\Gamma, \mathcal{A}_\lambda)$ in completely continuous (i.e. a limit of finite rank operators).
Proof For $t\in T^{++}$, let us show the $*$-action of $t $ on $\mathcal{A}_\lambda(L)$ is completely continuous. Recall a function $f\in \mathcal{A}_\lambda(L)$ can be identified as a function in $\mathcal{A}(N^+_I, L)$ (Definition 33). The norm of $f$ is the sup norm under this identification. For $n\in N^+_I$, we have $(t*f)(n)=f(t^{-1}nt)$. This action is contracting, i.e., $$f(x_1,\ldots,x_k)\mapsto f(p^{n_1}x_1, \ldots, p^{n_k} x_k),$$ where $n_i>0$. Write $$f(x_1,\ldots, x_k)=\sum_{\mathbf{n}} a_{\mathbf{n}}x^{\mathbf{n}}\in A(D_k(0,1)).$$ Then $||f||=\sup_{n} |a_\mathbf{n}|$, and so $t*f\in A(D_k(0,p))$, which converges on a larger disk. Now the claim follows form the following fact: for $r>1$, the restriction $$j: A(D_k(0,r))\rightarrow A(D_k(0,1))$$ is completely continuous. In fact, the truncation $$(j-\mathrm{trunc}_N\circ j)(f)=\sum_{\mathbf{n}, \sum n_i>N} a_\mathbf{n} x^\mathbf{n}.$$ has norm $\le 1/r^N$, hence $j$ is the limit of finite rank operators $\mathrm{trunc}_N\circ j$.

Now let us put things in analytic variation.

Definition 41 Let $\mathcal{U}\subseteq \mathcal{X}$ be an affinoid subdomain. Then there exists $n_\mathcal{U}$ such that for any $\lambda \in \mathcal{U}(L)$, $\lambda$ is $n_\mathcal{U}$-analytic. For $n\ge n_\mathcal{U}$, we define $$\mathcal{A}_\mathcal{U}^n:= \mathcal{A}(U) \hat\otimes_{\mathbb{Q}_p}\mathcal{A}^n(N, \mathbb{Q}_p).$$ So an element of $\mathcal{A}_\mathcal{U}^n$ is an $n $-analytic function $$f: I\rightarrow \mathcal{A}(\mathcal{U}), \quad f(n^- t g)=\langle t\rangle_\mathcal{U} f(g),$$ where $\langle t\rangle_\mathcal{U}$ is the image of $t $ under the map $T(\mathbb{Z}_p)\subseteq \mathbb{Z}_p[ [ T(\mathbb{Z}_p)] ]\rightarrow \mathcal{A}(\mathcal{U})$. We can analogously define the $*$-action on $\mathcal{A}_\mathcal{U}^n$.

For $\lambda \in \mathcal{U}(L)$ we have $$\mathcal{A}_\mathcal{U} \otimes_\lambda L\cong \mathcal{A}^n_\lambda(L),$$ i.e., $\mathcal{A}_\mathcal{U}^n$ gives an analytic variation of $\mathcal{A}_\lambda^n(L)$. The cohomology $H^i(\Gamma, \mathcal{A}_\mathcal{U})$ is not necessarily a Banach space. We have a map $$H^i(\Gamma, \mathcal{A}_\mathcal{U}^n) \otimes L \rightarrow H^i(\Gamma, \mathcal{A}^n_\lambda(L)).$$ A priori this map is neither injective nor surjective. This is because $H^*(\Gamma, \mathcal{A}_\mathcal{U})$ may fail to be flat over $\mathcal{A}(\mathcal{U})$, 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 $\Gamma\subseteq G(\mathbb{Q})$ be a congruence subgroup acting freely on $\mathcal{D}$. Then there is a $\mathbb{Z}[\Gamma]$-finite free resolution of the trivial $\Gamma$-module $\mathbb{Z}$.
Proof This is a consequence of the Borel-Serre compactification $\overline{\mathcal{D}}$, which has a deformation retract to $\mathcal{D}$ such that $\Gamma\backslash \overline{\mathcal{D}}$ is compact. One can then choose a finite triangulation of $\Gamma\backslash \overline{\mathcal{D}}$, and hence a triangulation of $\overline{\mathcal{D}}$ by pulling back. Let $C_i(\Gamma)$ be the $i $-chains of the triangulation, a free $\mathbb{Z}[\Gamma]$-module. Then the complex $C_i(\Gamma)$ computes the homology of $\overline{\mathcal{D}}$. But $\overline{\mathcal{D}}$ is contractible, and hence all higher homology groups are trivial, and hence $C_i(\Gamma)$ gives the desired free resolution.

Let $M$ be a $\Gamma$-module, then $H^\cdot(\Gamma, M)$ can be computed using the cohomology of $\Hom_\Gamma(C^\cdot, M)=\Ext_\Gamma^\cdot (\Gamma, M)$. So computing the cohomology in$\mathcal{A}_\lambda$ can be computed using complexes whose terms are finitely many copies of $\mathcal{A}_\lambda$ 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 $u_t$ on $C^\cdot (\Gamma, \mathcal{A}_\lambda)$ is completely continuous and has a slope decomposition. For $h\in \mathbb{Q}_+$, we can decompose $$C^\cdot (\Gamma, \mathcal{A}_\lambda)= C^\cdot(\Gamma, \mathcal{A}_\lambda)^{\le h} \oplus C^\cdot(\Gamma, \mathcal{A}_\lambda)^{>h}.$$ In this way we deduce a slope decomposition on the cohomology.

Since $\mathcal{A}_\mathcal{U} \otimes_\lambda L\cong \mathcal{A}_\lambda$, we know that $$C^\cdot(\Gamma, \mathcal{A}_\mathcal{U}) \otimes_\lambda L\cong C^\cdot(\Gamma, \mathcal{A}_\lambda(L)).$$ 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 $H^\cdot(\Gamma, \mathcal{A}_\lambda)$ is the same as the trace on $C^\cdot(\Gamma, \mathcal{A}_\lambda)$ (notice the infinite slope part has trace zero).

Definition 42 For $f\in C_c(G(\mathbb{A}_f^p), \mathbb{Q}_p) \otimes U_p^{++}$, where $U_p^{++}\subseteq U_p$ is the ideal generated by $u_t$ for $t\in T^{++}$. Then $f$ acts on $C^\cdot (\Gamma, \mathcal{A}_\mathcal{U})$ and we define $I_G^+(f, \lambda)\in \mathcal{A}(\mathcal{U})$ such that for any $\lambda \in\mathcal{U}(L)$, $I_G^+(f,\lambda)$ is equal to the trace of $f$ on $C^\cdot(\Gamma, \mathcal{A}_\lambda)$ (or finite slope cohomology $H^\cdot(\Gamma, \mathcal{A}_\lambda)^\mathrm{fs}$). This construction is similar to Wiles' construction of deformation of Galois representations using pseudo-representations.
Definition 43 The finite slope cohomology $H^q(S_K, \mathcal{A}_\lambda)^\mathrm{fs}$ has the Hecke action of $C_c^\infty(G(\mathbb{A}_f^p)) \otimes U_p$, and decomposes into a direction sum $\sum_{\sigma} m^q(\sigma) \sigma$, where $\sigma$ runs over finite slope representation of the Hecke algebra. We define the (alternating) trace $I_G(f,\lambda)=\sum_{\sigma,q}(-1)^qm^q(\sigma) \tr(f|\sigma)$.
Remark 21 Let $\mathcal{D}_\lambda(L)=\Hom(\mathcal{A}_\lambda(L), L)$ be the continuous dual of $\mathcal{A}_\lambda$ (e.g., when $G=GL(2)$, the differential operator on $\mathcal{D}_\lambda(L)$ raises the weight by 2, like Atkin-Serre's theta operator). The above construction also applies to $\mathcal{D}_\lambda$.

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.


TopConstruction of the eigenvariety: effective finite slope character distribution

Definition 44 Let $\mathcal{H}_p:=C_c^\infty(G(\mathbb{A}_f^p), \mathbb{Q}_p) \otimes U_p$. Let $\mathcal{H}_p'$ be the two-sided ideal of $\mathcal{H}_p$ generated by $u_t$ for $t\in T^{++}$. For $K^p\subseteq G(\mathbb{A}_f^p)$ an open compact subgroup, we let $\mathcal{H}_p(K^p)= \mathbf{1}_{K^p}\mathcal{H}_p \mathbf{1}_{K^p}$ (and similarly $\mathcal{H}_p'(K^p)$).
Definition 45 An irreducible representation $\sigma$ of $\mathcal{H}_p$ is called finite slope if $\omega_\sigma$, the restriction of $\sigma$ to $U_p$, satisifies $\omega_\sigma(u_t)\ne0$ for any $t\in T^{++}$. Notice that $\omega_\sigma$ is a character since $U_p$ lies in the center of $\mathcal{H}_p$ and $\sigma$ is irreducible.
Definition 46 A finite slope character distribution is a map $J: \mathcal{H}_p'\rightarrow L$ for some finite extension $L/\mathbb{Q}_p$ such that:
  1. there exists a family $\{\sigma_i\}_{i\in \mathbb{N}}$ of irreducible finite slope representations of $\mathcal{H}_p$ such that for any $f\in \mathcal{H}_p'$, we have $$J(f)=\sum_{i=1}^\infty m_J(\sigma_i) \tr(f|\sigma_i)$$ where $m_J(\sigma_i)\in \mathbb{Z}$.
  2. for any $h\in \mathbb{Q}$, $K^p\subseteq G(\mathbb{A}_f^p)$ and $t\in T^{++}$, the set $$\{i: m_J(\sigma_i)\ne0, \sigma_i^{K^p}\ne0, v_p(\omega_{\sigma_i}(u_t))\le h\}$$ is finite.

For such $J $, we simply write $$J(f)=\sum_{\sigma} m_J(\sigma) \tr(\sigma(f)),$$ where $\sigma$ runs over all irreducible finite slope representations. We say $J $ is effective if $m_J(\sigma)\ge0$ for any $\sigma$.

Definition 47 For an effective finite slope character distribution $J $, we define $$V_J(K^p)=\sum_{\sigma} m_J(\sigma)\sigma^{K^p},$$ this infinite sum is completed with respect to an integral structure on $\sigma^{K^p}$. Then for any $f\in \mathcal{H}_p'(K^p)$, we have $$J(f)=\tr(f|V_J(K^p))$$ (by the finiteness assumption b) the operator $f$ is completely continuous). More generally, if $\tau$ is a $K^p$-type, we define $$V_J(\tau)= \sum_{\sigma} m_J(\sigma)(e_\tau \sigma),$$ which recovers $V_J(K^p)$ when $\tau$ is trivial. For any $f\in \mathcal{H}_p'$ we can consider the Fredholm determinant $$P_J(f)(X)=\det (1- Xf: V_J(K^p)),$$ which is an entire power series with coefficient in $L$.
Remark 22 By the Fredholm-Riesz-Serre spectral decomposition ([Serre, IHES 1962]), if $$P_J(f)(X)= Q(X)S(X),$$ where $Q(X)$ is a polynomial with $Q(0)=1$, and $S(X) $ an entire power series such that $(Q,S)=1$, then there exists a decomposition stable under the action of $f$, $$V_J(K^p)=N \oplus F,$$ where $N $ is finite dimensional and $$\det(1-Xf: N)=Q(X),$$ and $Q^*(f)$ is invertible on $F$ (here $Q^*(X)=X^{\deg Q} Q(1/X)$ is the reciprocal polynomial). Moreover, there exists a sequence $P_n(X)$ of polynomials depending algebraically on the coefficients of $Q, S$ such that $P_n(f)$ converges to the idempotent projector onto $N $ (e.g., $P_n=u_t^{n!}$ in the ordinary case).
Example 7 For $t\in T^{++}$, one can take $Q(X)=\prod_\alpha (1-\alpha X)$, wehre $\alpha$ runs over the eigenvalues of $u_t$ on $V_J(K^p)$ of slope $\le h$. Then we obtain a slope decompostion $$V_J(K^p)=V_J(K^p)^{\le h} \oplus V_J(K^p)^{>h}.$$

Now we construct an eigenvariety attached to

  1. An analytic family of effective finite character distribution indexed by a weight space $\mathcal{X}$. Namely a map $$J:\mathcal{H}_p\rightarrow \mathcal{O}(\mathcal{X})$$ such that $J_\lambda:=\omega_\lambda\circ J$ is an effective finite slope character distribution for any $\lambda\in\mathcal{X}(L)$.
  2. A $K^p$-type $\tau$ for some $K^p\subseteq G(\mathbb{A}_f^p)$.
Theorem 5 Let $\mathcal{X}/\mathbb{Q}_p$ be the weight space attached to a torus $T$. Let $J $ be an $\mathcal{X}$-family of effective finite slope character distribution and $\tau$ be a $K^p$-type. Then there exists a rigid analytic variety $\mathcal{E}_{J,\tau}$ sitting in the diagram $$\xymatrix{\mathcal{E}_{J,\tau} \ar[d] \ar@{^(->}[r] & \mathrm{Max}( R_{S,p}) \times \mathcal{X} \ar[ld]  \\ \mathcal{X} }$$ Here $S$ is the finite set of primes $\ell$ such that $K^p_\ell$ is not hyperspeicial. It satisfies that a point $x=(\theta_x, \lambda_x)$ lies in $\mathcal{E}_{J,\tau}(L)$ if and only there exists an irreducible finite slope representation $\sigma$ such that
  1. $m_{J_\lambda}(\sigma)\ne0$,
  2. $e_\tau \sigma\ne0$,
  3. $\theta_x=\theta_\sigma$ (the restriction of $\sigma$ to $R_{S,p}\subseteq \mathcal{H}_p(K^p)$, see Definition 31).

Moreover, for any $f\in \mathcal{H}_p'(K^p)^\times$, we define $\mathcal{Z}_J(f)\subseteq \mathbb{A}^{1, \mathrm{rig}}\times \mathcal{X}$ to be the hypersurface cut out by the Fredholm determinant $P_J(f)(X)$ (an entire power series with coefficients in $\mathcal{O}(\mathcal{X})$). Then we have a commutative diagram $$\xymatrix{\mathcal{E}_{J,\tau} \ar[d] \ar[r] &\mathrm{Max}(R_{S,p})\times \mathcal{X} \ar[d] \\ \mathcal{Z}_J(f) \ar[r] & \mathcal{X} }$$ such that the left vertical arrow $(\theta, \lambda)\mapsto (\theta(f)^{-1},\lambda)$ is finite flat and the bottom horizonal arrow (projection) is finite (hence $\mathcal{E}_{J,\tau}$ is equidimensional of dimension $\dim \mathcal{X}$).

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

Let $U\subseteq \mathcal{X}$ be an affinoid subdomain. Let $J_U: \mathcal{H}_p'\xrightarrow{J} \mathcal{O}(\mathcal{X})\rightarrow \mathcal{O}(U)$. Fix $t\in T^{++}$, for each decomposition $P_{J_U}(X)=Q(X)S(X)$, we define $\mathcal{Z}_Q:=\mathrm{Max}(\mathcal{O}(U)[X]/Q^*(X))$. Varying all such decompositions and $U$ gives an admissible covering of $\mathcal{Z}_J$. (Notice that Buzzard's eigenvariety machine gives a similar result when $J $ comes from a family of so called orthogonalizable Banach modules $M_U$ over $\mathcal{O}(U)$; in contrast, we only have the traces here).

By Remark 22, the decomposition of $P_{J_U}$ gives a decomposition of $P_{J_\lambda}$ for any $\lambda \in U(L)$. Moreover the projection of $V_{J_\lambda}(\tau)$ onto $N_{Q_\lambda}$ can be obtained as a squence of polynomials $P_n(X)$ depending algebraically on the coefficients of $Q, S$ by taking the limit of $P_n(u_t)$, hence is independent of $\lambda$. Now we construct a pseudo-representation $$T_Q: \mathcal{H}_p(K^p)\rightarrow \mathcal{O}(U), \quad f\mapsto \lim_{n\rightarrow\infty} J(P_n(u_t)f).$$ For any $\lambda\in U(L)$, this is convergent and defines a pseudo-representation which is actually the trace of the representation of $\mathcal{H}_p(K^p)$ on $V_{J_\lambda}(\tau)$. (Notice that $T_Q$ is the trace of $\mathcal{H}_p(K^p)$ acting on $N_Q$ if we had a decomposition $M_U= N_Q \oplus F_Q$ in the setting of Buzzard).

Let $h_Q= \mathcal{O}(U) \hat\otimes R_{S,p}/\ker T_Q$. We have $h_Q \otimes_\lambda L$ surjectives onto $h_{Q_\lambda}$ with unipotent kernel. Hence $\mathrm{Max}(h_Q)(L)=\mathrm{Max}(h_{Q_\lambda})$. The results then follow since $h_{Q_\lambda}$ is finite over $L$ and $\mathrm{Max}(h_{Q_\lambda})$ classifies the Hecke eigensystem of $R_{S,p}$ acting on $N_{Q_\lambda}$.

TopConstruction of the eigenvariety: $p$-adic automorphic character distribution

Next step is to contruct an $\mathcal{X}$-family of effective finite slope character distribution which is automorphic. The $I_G^+(f,\lambda)$ (Definition 42, with $\mathcal{A}_\lambda$ replaced by $\mathcal{D}_\lambda$, see Remark 21) is a finite slope character distribution, but it is not effective in general.


Proposition 9 If $G$ is anisotropic, then $(-1)^{d} I_G^+(f,\lambda)$ is effective, where $d=\dim \mathcal{S}_G$ is the dimension of the associated locally symmetric space.
Proof For any $\lambda$ algebraic dominant, and $f=f^p \otimes u_t$ for $t\in T^{++}$. By Proposition 5 and taking dual, we obtain $$I_G^+(f,\lambda)\equiv\tr(f, H^\cdot(\tilde{\mathcal{S}}_G, V_\lambda^\vee))\pmod{N(\lambda,t)},$$ where the ideal $N(\lambda, t):=\inf_{w\ne\Id} |t^{w*\lambda-\lambda}|_p$. Hence we obtain a congruence of Fredholm determinants $$P_{I_{G,\lambda}^+}(f,X)\equiv \prod_q \det(1-fX| H^q(\tilde{\mathcal{S}}_G, V_\lambda^\vee))^{(-1)^q}\pmod{N(\lambda,t)}.$$ If $G$ is anisotropic and $\lambda$ is regular, then there is only cohomology in the middle degree (Borel-Wallach). In particular, $$P_{(-1)^dI_{G,\lambda}^+}(f,X)\equiv \det(1-fX|H^d(\mathcal{S}_G, V_\lambda^\vee))\pmod{N(\lambda,t)}.$$

Assume that $P_{(-1)^dI_G^+}(f, X)=N(X)/D(X)$, where $N, D$ are coprime entire power series with coefficients in $\mathcal{O}(\mathcal{X})$. We need to show that $D $ is actually a constant. If not, then the set of zeros $Z'{}=Z(D)\backslash Z(D)\cap Z(N)$ is non-empty. Pick a point $x=(\theta,\lambda)\in Z'$. Then $x $ is a pole of $P_{I_G^+}(f,X)$. Fix an open neighborhood $W\subseteq Z'$ of $x $. Since $Z(D)\rightarrow\mathcal{X}$ is flat (hence open), the image of $W$ in $\mathcal{X}$ is also open and thus contains a dense set of algebraic weights. So we may find a point $x'{}=(\theta',\lambda')\in W$ such that $\lambda'$ is algebraic dominant regular, the slope of $\theta'$ is equal to that of $\theta$, and $v_p(N(\lambda', t))> v_p(\theta)$. The congruence mod $N(\lambda',t)$ implies that $x'$ is a pole of a polynomial, a contradiction.

More generally, when $G$ is not anisotropic, one needs to replace the cohomology by the cuspidal cohomology, and modify $I_G^+(f,\lambda)$ accordingly.

Definition 48 We say a sequence of dominant regular $\lambda_n\rightarrow \lambda$ is very regular if for any simple root $\alpha$, $\lambda_n(H_\alpha)\rightarrow+\infty$. In this case, we have $N(\lambda_n,t)\rightarrow0$ ($p$-adically), and $$I_G^+(f,\lambda)=\lim_{n\rightarrow\infty} \tr(f|H^\cdot(\tilde{\mathcal{S}}_G, V_{\lambda_n}^\vee)).$$
Theorem 6 (cuspidal character distribution)
  1. For any converging very regular sequence $\lambda_n\rightarrow \lambda$, the limit $$I_{G,\mathrm{cusp}}^+(f,\lambda)=(-1)^d\lim_{n\rightarrow\infty}\tr(f, H^d_\mathrm{cusp}(\tilde{\mathcal{S}}_G, V_{\lambda_n}^\vee))$$ exists and depends only $\lambda$.
  2. $I_{G,\mathrm{cusp}}^+(f,\lambda)\in \mathcal{O}(\mathcal{X})$ and is effective.
  3. For $f=f^p \otimes u_t$, we have $$I_{G,\mathrm{cusp}}^+(f,\lambda)\equiv(-1)^d \tr(f, H^d_\mathrm{cusp}(\tilde{\mathcal{S}}_G, V_\lambda^\vee))\pmod{N(\lambda,t)}.$$
Proof (Sketch) The strategy is to use the Hecke equivariant decomposition of the cohomology (due to Franke and Schwermer-J.-S. Li): when $\lambda$ is regular, we have $$H^\cdot(\tilde{\mathcal{S}}_G, V_\lambda^\vee)=H^\cdot_\mathrm{cusp}(\tilde{\mathcal{S}}_G, V_\lambda^\vee) \oplus H^\cdot_\mathrm{Eis}(\tilde{\mathcal{S}}_G, V_\lambda^\vee).$$ One needs to translates the archimedean description of the Eisenstein part into a $p$-adic one.

Let $M\supseteq T$ be a standard Levi subgroup. Let $P$ be a parabolic with Levi $M=P/N$, so $T_M\cong T$. Consider an Weyl group element $w\in W(G)$ such that $w B w^{-1}\in P$. We define the isomorphism $$T\rightarrow T_M, t\mapsto w t w^{-1}$$ (e.g., for $\GL(2)$ this choice gives the ordinary Eisenstein, rather than the critical Eisenstein series). We define $$\mathcal{H}_p(G)\rightarrow \mathcal{H}_p(M), \quad f^p \otimes u_t\mapsto f_M^\mathrm{reg}=f_M^p \otimes |t^{(1-w)\rho}| u_t^M,$$ where $$f_M^p(m)=\int_{ K_\mathrm{max}^pN_P(\mathbb{A}_f^p)} f(k^{-1}mnk)dkdn$$ is the usual constant term (so $\tr (f_M,\sigma)=\tr (f, \Ind_{P(\mathbb{A}_f)}^{G(\mathbb{A}_f)})$ for any representation $\sigma$ of $M(\mathbb{A}_f)$.)

We say that $M$ is relevant if

  1. $M(\mathbb{R})$ has discrete series.
  2. The weight spaces $\mathcal{X}_G \cong \mathcal{X}_M$ are isomorphic (in general the weight space is $\Hom(T(\mathbb{Z}_p)/\overline{Z_G(\mathbb{Q})\cap K})$ when $G$ has nontrivial center).

It turns out if $M$ is relevant, then there is only one parabolic (up to conjugation by $G(\mathbb{A}_f^p)$) that is going to contribute. We have

$$\tr(f, H^\cdot_\mathrm{Eis}(, V_\lambda^\vee))=\sum_{M \mathrm{rel.}}\sum_{w\in W^M_\mathrm{Eis}} (-1)^{\ell(w)-\dim N} \tr(f_M^\mathrm{reg}, H_\mathrm{cusp}^\cdot(\tilde{\mathcal{S}}_M, V_{\lambda+(1-w^{-1})\rho}^M).$$

So we define $$I_{G,\mathrm{cusp}}^+(f,\lambda)=I_G^+(f,\lambda)-\sum_{M,w}(-1)^{\ell(w)-\dim N} I_{M,\mathrm{cusp}}^+(f_M^\mathrm{reg}, \lambda+(1-w^{-1})\rho).$$ (e.g., when the rank of $M$ is zero, we have $I_{M,\mathrm{cusp}}^+=I_M^+$). The congruence property (c) for $I_{G,\mathrm{cusp}}^+(f,\lambda)$ similar to the previous proposition implies (2) that $I_{G, \mathrm{cusp}}^+(f,\lambda)$ is effective.

Remark 23 For $P$ a parabolic, and $t\in Z_M^{++}$, we define $e_P=\lim_{n\rightarrow\infty}(u_t)^{n!}$. We say that a finite slope representation $\sigma$ is $P$-ordinary if $e_P\sigma=\sigma$. Let $e_P':=1-e_P$ and $e':=\prod_P e_P'$. Then $e'$ kills all ordinary contribution, in particular, all the Eisenstein contribution. Hence $I_G^+(f\circ e', \lambda)$ is effective (in fact a direct factor of $I_{G,\mathrm{cusp}}^+$.)

Top$p$-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 $p$-adic deformations.


Let $x=(\theta,\lambda)$ with $\lambda$ regular algebraic dominant. Then by construction $x $ lies on the eigenvariety if and only if $m(\theta, I_{G,\mathrm{cusp},\lambda}^+)>0$. If $\pi$ is a cuspidal automorphic representation of $G(\mathbb{A})$, with $\pi_\infty$ a discrete series of parameter $\lambda$. For any $\theta$ occurring in $\pi_f^{K^pI}$, the classical multiplicity $m_\mathrm{cl}(\theta,\lambda)$ is the multiplicity of $\theta$ in $H^d(\mathcal{S}_G(K^p), V_\lambda^\vee)$, which is positive. If $\theta$ is non-critical, then by Prop. 6 we know that $m(\theta, I_{G,\mathrm{cusp},\lambda}^+)$ 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 $x $ corresponds to a point in the eigenvariety.

Example 8 Consider $G=\GL(2)$, and $\theta$ the character attached to the trivial representation. Then $m_\mathrm{cl}(\theta, 0)=1$ (appearing only in $H^0$). However, for $K^p$ maximal, we have $m(\theta,I_{G,\mathrm{cusp},0}^+)=0$. Otherwise, there is a point on the eigenvariety such that $\theta(T_\ell)=\ell+1$ and $\theta(U_p)=p$ (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 $p$ splits, $G/\mathbb{Q}_p=\GL_n/\mathbb{Q}_p$, $$\lambda_\tau=(c_1-a,\ldots, c_b-a, c_{b+1}+b,\ldots,c_d+b)$$ is a dominant weight, and $\theta: U_p\rightarrow \mathbb{C}$ with slope $$\mu_\theta:=(\mu_1,\ldots,\mu_d)=(v_p(\alpha_1), \ldots, v_p(\alpha_n))\in X^*(T) \otimes \mathbb{Q}.$$ If $\theta$ takes integral values, then $\mu_\theta$ is inside the cone generated by the positive simple roots $e_i=(0, \ldots,0, 1, -1, \ldots, 0)$, namely, $\mu_1\ge0$, $\mu_1+\mu_2\ge0$, ..., $\mu_1+\cdots+\mu_{n-1}\ge0$. By Definition 38, $\mu_\theta$ is non-critical if and only if $\mu_\theta-(s_{e_i}*\lambda-\lambda)$ does not lie in this cone. Notice that $$s_{e_i}*\lambda-\lambda=(0,\ldots, 0,\lambda_{i+1}-\lambda_i-1, \lambda_i-\lambda_{i+1}+1,\ldots, 0).$$ So non-critical means $\mu_1< \lambda_1-\lambda_2+1$, $\mu_1+\mu_2<\lambda_2-\lambda_3+1$, ..., $\mu_1+\cdots+\mu_{n-1}<\lambda_{n-1}-\lambda_n+1$ (cf. 30).

Recall that $E(\Phi, s_0)$ is a holomorphic Eisenstein series of weight $\xi=(c_1,\ldots, c_b, p,q, c_{b+1},\ldots, c_d)$. Since $p$ splits, we have the Levi $$M(\mathbb{Q}_p)=\GL_d(\mathbb{Q}_p)\times \mathbb{Q}_p^\times\times \mathbb{Q}_p^\times.$$ Assume that $\pi_p$ is an unramified principal series. Then $\Ind_{P(\mathbb{Q}_p)}^{\GL_{d+2}(\mathbb{Q}_p)} \pi_p \otimes \chi_p$ is also an unramified principal series. Choosing $$\Phi_p\in (\Ind_{P(\mathbb{Q}_p)}^{\GL_{d+2}(\mathbb{Q}_p)} \pi_p \otimes \chi_p)^I$$ (of dimension $(d+2)!$) corresponding to an ordering of the Langlands parameter.

To make such a choice, first we fix a $p$-stabilization of $\pi_p$ (hence an ordering of the Langlands parameter of $\pi_p$) such that the corresponding character of $U_p(G)$ is non-critical with respect to $\lambda_\tau$. Next we choose $\Phi_p$ the section corresponding to the ordering $$\theta'{}=(\alpha_1, \alpha_2, \ldots, \alpha_b, \chi_1(p)p^{1/2}, \chi_2(p) p^{-1/2}, \alpha_{b+1},\ldots, \alpha_d)$$ or $$\theta'{}=(\alpha_1, \alpha_2, \ldots, \alpha_b, \chi_2(p)p^{1/2}, \chi_1(p) p^{-1/2}, \alpha_{b+1},\ldots, \alpha_d).$$ Here $\chi_1,\chi_2$ are the characters corresponding to $\chi_p$ on $\mathbb{Q}_p^\times\times \mathbb{Q}_p^\times$. The corresponding slopes are $$\mu_{\theta'}=(\mu_1,\ldots, \mu_b,0,0,\mu_{b+1},\ldots, \mu_d)$$ (called the $P$-ordinary stabilization) and $$\mu_{\theta'}=(\mu_1,\ldots, \mu_b, p-q-1-d, -(p-q-1-d), \mu_{b+1},\ldots, \mu_d)$$ (called the critical stabilization) respectively.

We are interested in the case $p-q=d+2$ (to get desired shape of Galois representation) and the case of critical stabilization (to get cuspidal deformation). In this case, $\mu_{\theta'}$ is never non-critical with respect to $\lambda_\xi$, as the extra requirement

$$\mu_1+\cdots+\mu_b+1<p-q-(d+2)+1=1,$$ is always violated. Nevertheless, the requirement is only violated at the position $b+1$, and can salvaged using a Hasse invariant argument as follows.

Proposition 10 The critical stabilization of $E(\Phi, s_0)$ gives a point on the eigenvariety.
Proof There exists a holomorphic form $A$ (Hasse invariant) of weight $$(\underbrace{p-1,\ldots, p-1}_{b+1},\underbrace{0,\ldots,0}_{a+1})$$ such that the $q$-expansion of $A$ is congruent to 1 mod $p$. Write $g=E(\Phi, s_0)$. Then $gA^s$ has weight $$\xi_s=(c_1+(p-1)s,\ldots, c_b+(p-1)s, q+d+2+(p-1)s, q, c_{b+1},\ldots, c_d).$$

We can choose $\mathcal{U}\subseteq \mathcal{X}$ a neighborhood of $\xi$ and a factorization of the Fredholm series of some $u_t$ acting on $R\Gamma(S_{G'}(K^pI), \mathcal{D}_\mathcal{U})$ associated to the slope of $g$ and get a projector $e_\mathcal{U}$. Now for $\xi_s\in\mathcal{U}$, we apply the projection $e_{\mathcal{U}}(gA^s)$ and obtain a sum of eigenforms of the same slope as $g$. The projection is nonzero when $s\gg 0$. These eigenforms are non-critical with respect to $\xi_s$ and hence corresponds to points on the eigenvariety. Moreover, the system of eigenvalues converging to that of $g$ when $s\rightarrow \infty$, and we obtain a point on the eigenvalues associated to $g$.


TopGalois 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 $\pi$ of $\GL_n/K$, one should attach a compatible system of Galois representations $\rho_\lambda: G_K\rightarrow \GL_n(E_{\pi,\lambda})$, where $E_\pi$ is the Hecke field of $\pi$, characterized by the local Langlands correspondence. This philosophy is now known in many cases.

Let us recall the local Langlands correspondence. Let $K$ be a non-archimedean local field with residue field $\kappa$ and $q=\# \kappa$. We have an exact sequence $$1\rightarrow I_K\rightarrow G_K\rightarrow G_{\kappa}\cong \hat{\mathbb{Z}} \rightarrow 1.$$

Definition 49 The Weil group $W_K$ is the inverse image of $\Phi^\mathbb{Z}\subseteq G_\kappa$ inside $G_K$, where $\Phi$ is the $q$-Frobenius. A representation $(\rho, V)$ of $W_K$ is called smooth if $\rho$ is trivial on a neighborhood of 1 in $W_K$, i.e., there exists $I'\subseteq I_K$ finite index such that $\rho|_{I'}$ is trivial.

Let $\ell\nmid q$ be a prime. Then we have a tame quotient map $G_K\rightarrow \mathbb{Z}_\ell(1)$ sending $g$ to $t_\ell(g)$, where $$g(\sqrt[\ell^n]{\pi_K})=\zeta_{\ell^n}^{t_\ell(g)}\cdot\sqrt[\ell^n]{\pi_K}.$$ In particular, we see that $t_\ell(\Phi g\Phi^{-1})=q t_\ell(g)$. The following theorem is not hard.

Theorem 7 (Monodromy theorem of Grothendieck) Let $\rho: G_K\rightarrow \GL(V)$ be an $\ell$-adic representation. Then there exists a nilpotent endomorphism $N_\rho\in\End(V)$ (called the monodromy operator) such that for $g\in I'$ (a finite index subgroup of $I_K$ depending on $\rho$), we have $$\rho(g)=\exp(t_\ell(g) N_\rho).$$

Notice that when $N_\rho$ is nontrivial, the representation $\rho$ is not smooth. To remedy this, one introduces the Weil-Deligne representations instead.

Definition 50 A Weil-Deligne representation is the data of $(\rho, V,N)$ where,
  1. $\rho: W_K\rightarrow \GL(V)$ is a smooth representation.
  2. $N\in \End(V)$ a nilpotent endomorphism such that $\rho(\Phi) N \rho(\Phi)^{-1}=q N$.

It is called irreducible if either $\dim V=1$ or if $N $ is regular nilpotent. It is called Frobenius semisimple if $\rho$ is semisimple.

One then associates to $\rho: G_K\rightarrow \GL(V)$ a Weil-Deligne representation $\WD(\rho)=(\rho_\Phi, V, N_\rho)$ where $\rho_\Phi(\Phi^m g)=\rho(\Phi^m g)\exp (-t_\ell(g) N_\rho)$ for $g\in I_K$.

The local Langlands correspondence for $\GL_n$ due to Harris-Taylor and Henniart says that there is a bijection $\pi\mapsto \rec_K(\pi)$ between irreducible smooth representations of $\GL_n(K)$ and Frobenius semisimple Weil-Deligne representations of dimension $n $, characterized by matching $L$-factors and $\varepsilon$-factors on both sides and certain compatibilities. In particular it is compatible with local class field theory: $$\rec_K(\chi)=\chi\circ\Art_K^{-1}.$$ If $\pi=i_B^{\GL_n}(\chi)$ is irreducible, then $$\rec_K(\pi)= \bigoplus_{i=1}^n \chi_i\circ\Art_K^{-1},$$ with trivial monodromy.

Now let us recall the global results. Let $K/F$ be a CM extension. Let $\pi$ be a cuspidal automorphic representation of $\GL_n(\mathbb{A}_K)$. Assume

  1. $\pi$ is cohomological, i.e., there exists $W_\lambda$ an irreducible algebraic representation of $G_\infty=G(K \otimes \mathbb{R})$ such that $$H^\cdot(\mathfrak{g},K; \pi_\infty \otimes W_\lambda)\ne0.$$ (equivalently $\chi_{\pi_\infty}=\chi_\lambda$ for $\lambda$ regular algebraic).
  2. $\pi$ is conjugate self-dual: $\pi^\vee\cong \pi\circ c$.

In this case, $\pi$ 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. $K/F$ is unramified at finite places,
  2. $\pi_v$ is unramified at places $v$ above those which do not split,
  3. $[F: \mathbb{Q}]$ 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 $\pi$ is cohomological and conjugate self-dual. Let $\mathfrak{p}|p$ be a prime of $E_\pi$. Then there exists a Galois representation $$\rho_{\pi, \mathfrak{p}}\rightarrow \GL_n(E_{\pi,\mathfrak{p}})$$ such that
  1. For any $v\nmid p$ a finite place of $K$, we have $$\WD(\rho_{\pi, \mathfrak{p}}|_{G_{K,v}})\prec\rec_{K_v}( \pi_v \otimes |\det|^{(1-n)/2}),$$ (see Def. 53 , in particular, the monodromy operator of $\WD$ has smaller rank). In particular, at unramified places it is given by the local Langlands correspondence.
  2. If $v|p$, then $\rho_{\pi, \mathfrak{p}}|_{G_{K_v}}$ is de Rham with regular Hodge-Tate weights (i.e., all multiplicities are at most 1). (For example, if $F=\mathbb{Q}$, then the Hodge-Tate weights are given by $\lambda_i+n-i$, where $\lambda=(\lambda_1,\ldots,\lambda_n)$, $\lambda_1\ge\cdots\ge\lambda_n$).
  3. If $v|p$ and $\pi_v$ is unramified, then $\rho_{\pi,\mathfrak{p}}|_{G_{K_v}}$ is crystalline and $$\det(1- T\phi| D_\mathrm{crys}(\rho_{\pi,\mathfrak{p}}|_{G_{K_v}}))=\det(1-T\Phi|\rec_{K_v}(\pi_v \otimes |\det|^{(1-n)/2})).$$
Remark 24
  1. In a), one expects the equality holds (see the thesis of Caraiani up to Frobenius semi-simplification).
  2. In general, for $v|p$, the Weil-Deligne representation $D_\mathrm{pst}(\rho_{\pi,\mathfrak{p}}|_{G_{K_v}})$ is conjectured to be isomorphic to $\rec_{K_v}(\pi_v \otimes|\det|^{(1-n)/2})$.The equality in c) means this conjecture holds when $\pi_v$ is unramified.
Remark 25 For automorphic representations on unitary groups, one can construct Galois representations by first base changing to $\GL_{n,K}$ and then applying the previous Theorem 8. Notice that the latter uses descent to only certain special type of unitary group (e.g., signature $(n-1,1)$ at exactly one infinite place).

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

Theorem 9 (Shin, Morel) Let $\pi$ be a cuspidal cohomological automorphic representations of a unitary group $G$ defined by a hermitian space of dimension $n $ over $K$. Then there exists $\Pi_K=\mathrm{BC}(\pi)$ an automorphic representation of $\GL_n(\mathbb{A}_K)$ such that
  1. $\Pi_{K,w}=\mathrm{BC}(\pi_v)$ if $\pi_v$ is unramified and $v$ is unramified in $K/F$ ($w|v$).
  2. $\Pi_{K,w}=\mathrm{BC}(\pi_v)$ if $v$ splits in $K/F$.
Remark 26
  1. When $\pi$ is not stable, $\mathrm{BC}(\pi)$ is not necessarily cuspidal.
  2. To obtain refined control at places where $\pi_v$ is ramified, one needs certain simplification in the stable twisted trace formula. This requires in b) that $v$ splits in $K/F$. So in the main theorem of the course one needs to ensure the ramification of $\pi$ is only at split places (which can be arranged due to the freedom of choosing $K $).


Top$p$-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 $\pi$ be an automorphic representation on the unitary group $G$ of signature $(a,b)$. Let $\chi$ be a Hecke character of $K^\times$. Let $S$ be the set of ramification of $\pi$ and $\chi$. Fix $p$ a prime which splits in $K$.

We assume:

  1. $\pi_\infty=\pi_\tau^H$, where $\tau=(c_1,\dots, c_b; c_{b+1}, c_d)$ with $c_b-c_{b+1}\ge d+2$.
  2. If $\pi_v$ is ramified then $v$ splits in $K$.
  3. $S$ does not contain primes dividing $\disc(K)$ and $p$.
  4. $\chi|_{\mathbb{A}_\mathbb{Q}^\times}=|\cdot|^{2k'}$, $k'=(m+n)/2$.
  5. $L(\mathrm{BC}(\pi)\times \chi^{-1},k'+1/2)=0$.

Under these assumptions on $\pi,\chi$, we have constructed holomorphic Eisenstein series $E(\Phi, s_0)$ whose associated Galois representation is $R \oplus \varepsilon \oplus \mathbf{1}$, where $R=\rho_{\pi,p} \otimes \chi_{p}^{-1}$. We further assume that

  1. $R: G_K\rightarrow \GL_d(\overline{\mathbb{Q}_p})$ is irreducible (this is conjectured to be true if $\mathrm{BC}(\pi)$ is cuspidal).
  2. For any $v\nmid p$ split in $K$, assume $\mathrm{WD}(R|_{G_v})$ is given exactly by $\rec_{K_v}(\pi_v)$ (see the thesis of Caraiani).

Now choose $\alpha=(\alpha_1,\ldots,\alpha_d)$ a refinement for $R|_{G_{\mathbb{Q}_p}}$, non-critical with respect to the Hodge-Tate weights of this representation. Here the Hodge-Tate weights are some translation of $$c_1-a+n-1,\ldots c_b-a+n-b, c_b+b+n-b-1,\ldots, c_d+b),$$ which is regular. The crystalline Frobenius at $p$ of the Galois representation associated to the Eisenstein series is given by $$(\alpha_1,\ldots,\alpha_b,1,p^{-1}, \alpha_{b+1},\ldots \alpha_d).$$

From this choice of a non-critical $p$-stabilization of $\pi$ and a suitable choice of $\Phi_f^p=\Phi_S \otimes \Phi^{S \cup \{p\}}_\mathrm{sph}$ (explained below), one can construct a point on the eigenvariety, and thus there exists a family of Galois representations deforming the Galois representation $R$ of $\pi \otimes \chi^\vee$ as in the following theorem.

Theorem 10 There exists
  • $\mathcal{U}\subseteq \mathcal{X}$ open affinoid,
  • a point $x_0\in \mathcal{U}$,
  • $\Sigma\subseteq \mathcal{U}(\overline{\mathbb{Q}_p})$ Zariski dense,
  • $T: G_K\rightarrow \mathcal{A}(\mathcal{U})$ a pseudo-representation,
  • $\phi_1,\ldots, \phi_d\in\mathcal{A}(\mathcal{U})$,
  • $\kappa:=(\kappa_1,\ldots, \kappa_d)\in \mathcal{A}(\mathcal{U})^d$, such that,
  1. $T_{x_0}=\tr(R)+\varepsilon_{\mathrm{cyc}}+1$.
  2. $T$ is unramified outside $S\cup \{p\}$.
  3. For $x\in \Sigma$, $\rho_x |_{G_{\mathbb{Q}_p}}$ is crystalline with Hodge-Tate weights $(k_1,\ldots, k_{d+2})=\kappa(x)$ with crystalline Frobenius eigenvalues given by $$(\phi_1(x)p^{k_1}, \phi_2(x)p^{k_2},\ldots,\phi_{d+2}(x)p^{k_{d+2}}).$$
  4. For any $v\in S$ and $x\in \Sigma$, we have $$\dim(\rho_x)^{I_v}\ge 2+\dim(R)^{I_v}.$$
  5. $T$ is irreducible.
Remark 27 For primes dividing $\disc K$, 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 $\rho_x$ 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 $\mathcal{G}=\GL(n, \mathbb{Q}_\ell)$.

Definition 51 From the Bernstein decomposition, we know for $\pi$ an irreducible representation of $\mathcal{G}$, there exists a parabolic $P\subseteq \mathcal{G}$ with Levi $M$ and $\sigma$ a cuspidal representation of $M$ such that $\pi$ is a subquotient of $i_P^\mathcal{G}\sigma$. We say pairs $(M, \sigma)$ of this form up to equivalence to be the cuspidal support of $\pi$. Here $(M,\sigma)\sim (M',\sigma')$ are equivalent if there exists $g\in \mathcal{G}$ such that $gMg^{-1}=M'$ and $\sigma\circ \Int(g^{-1})\cong\sigma'$.
Definition 52 We say $(M,\sigma)\sim_I(M',\sigma')$ are inertially equivalent if there exists $g\in\mathcal{G}$ and $\chi$ an unramified character of $M'$ such that $\sigma\circ\Int(g) \otimes \chi\cong \sigma'$. We say $\pi\sim_I\pi'$ are inertially equivalent if there exists a cuspidal support $(M,\sigma)$ of $\pi$ and a cuspidal support $(M',\sigma')$ such that $(M,\sigma)\sim_I(M',\sigma')$. An equivalence class for this equivalence relation is called a Bernstein component.
Remark 29 Let $\rec_K(\pi)=(r_\pi, N_\pi)$ be the associated Weil-Deligne representation under the local Langlands correspondence. From the very construction of the local Langlands correspondence, we know that $$\pi\sim_I\pi'\Leftrightarrow r_\pi|_{I_K}\cong r_{\pi'}|_{I_K}.$$ Hence the terminology.

The type theory of Bushnell-Kutzko shows that if $\Omega$ is a Bernstein component, then there is a type $(U,\lambda)$, where $U\subseteq \mathcal{G}$ open compact, and $\lambda$ a smooth representation of $U$, such that $$\pi \in \Omega\Leftrightarrow \Hom_U(\lambda, \pi)\ne0.$$

Example 9 The component of unramified representations is exactly those $\pi$ such that $\pi^{\mathrm{Iw}}\ne0$ (so contains the Steinberg representation as well). In other words, for this component, we have $(U,\lambda)=(\mathrm{Iw},\mathbf{1})$. Such $\pi$ has a cuspidal support $(M,\sigma)$, where $M$ is a torus and $\sigma$ is a unramified character.

For representations on the same Bernstein component, we order them using the monodromy operator.

Definition 53 Let $N, N'$ be two nilpotent matrices in $M_n(\mathbb{C})$. We say $N\prec N'$ if $N $ is in the Zariski closure of the set $g N' g^{-1}, g\in \GL_n(\mathbb{C})$ (so the Jordan normal form of $N $ has more zeros). We say $\pi\prec_I \pi'$ if $\pi\sim_I\pi'$ and $N_\pi\prec N_{\pi'}$.
Proposition 11 (Steinberg, Zelevinsky) There exists $\lambda$ a smooth representation of $\GL_n(\mathcal{O}_K)$ such that $$\pi'\prec_I \pi\Leftrightarrow\Hom_{\GL_n(\mathcal{O}_K)}(\lambda, \pi')\ne0.$$

Now for $\ell\in S$, we choose the local section $$\Phi_\ell\in I_\ell(\pi, \chi,s_0)=\Ind_{P(\mathbb{Q}_\ell)}^{\GL_{d+2}(\mathbb{Q}_\ell)} \pi_\ell \otimes \chi_\ell|\cdot|^{s_0(d+1)}$$ as follows. From the Zelevinsky classification, one can see that there exists a subquotient $\sigma$ of $I_\ell(\pi,\chi,s_0)$ such that $$\rec_K(\sigma)=\rec_K(\pi) \otimes \chi_\ell|\cdot|^{s_0(d+1)+1/2} \otimes \chi_\ell|\cdot|^{s_0(d+1)-1/2},$$ with the monodromy operator only on $\rec_K(\pi)$. Now choose $\lambda$ a representation of $\GL_{d+2}(\mathbb{Z}_\ell)$ in Prop. 11 for $\sigma$. Let $e_\lambda$ be the corresponding idempotent. Then there exists a local section $\Phi_\ell\in I_\ell(\pi,\chi,s_0)$ such that $e_\lambda \Phi_\ell\ne0$.

In this way, for a point in the eigenvariety constructed using the idempotent $e_S=\prod_{\ell\in S}e_\ell$, the associated Galois representation satisfies Item d) by Prop. 11.



Let us explain the proof of Item c) in Theorem 10. Let $c>0$ and let $\Sigma_c\subseteq \mathcal{U}(\overline{\mathbb{Q}_p})$ be the points $x$ such that $\kappa_i(x)\in \mathbb{Z}$, and $\kappa_i(x)-\kappa_{i+1}(x)>c$. For $c\gg0$, a point in $\Sigma_c$ corresponds to a cuspidal automorphic representation which is unramified at $p$ and whose $p$-stabilization is non-critical (a point on the eigenvariety). For $x\in \Sigma_c$, the Galois representation $\rho_x$ is crystalline at $p$ with the roots of the crystalline Frobenius given by the Langlands parameter of $\pi_x$ at $p$. The $u_t$-eigenvalues vary analytically in the family, which gives the analytic functions $\phi_i$ in the theorem.

Now let us explain the proof of Item e) in Theorem 10. This part requires our imposed extra assumptions that $R$ is irreducible and satisfies the local-global compatibility at split places $v\nmid p$. Assume that $T$ is reducible. Then $T=T_1+T_2$, where $T_1$ irreducible of dimension $d$ (as $\rho_{x_0}$ is irreducible), and $T_2$ is a 2-dimensional family such that,

  1. for any $x\in \Sigma_c$, $\rho_{2,x}$ is crystalline at $p$ with Hodge-Tate weights $\kappa_1(x)<\kappa_2(x)$ and crystalline Frobenius eigenvalues $p^{\kappa_1(x)}\phi_1(x)$, $p^{\kappa_2(x)}\phi_2(x)$,
  2. at a split place $v\nmid p$, the monodromy operator is trivial (as $R$ has correct monodromy and the generic monodromy of $T$ matches with the correct one, the generic monodromy of $T$ must come from $T_1$).
  3. $T_{2, x_0}=1+\varepsilon$ with $\kappa_1(x_0)=-1$, $\kappa_2(x_0)=0$, $\phi_1(x_0)=p$, and $\phi_2(x_0)=p^{-1}$.

When specializing to a point in $\Sigma_c$, we see from a) and c) that the Newton polygon is strictly above the Hodge polygon, hence $\rho_{2, x}|_{G_v}$ is irreducible. Hence $T_2$ is irreducible.

Now take $\mathcal{V}\subseteq \mathcal{U}$ an affinoid curve such that $\mathcal{V}\cap \Sigma_c$ is nonempty for any $c\gg0$. After normalization, we may assume $\mathcal{V}$ is smooth. Then $T_2|_{\mathcal{V}}$ takes value in $\mathcal{A}(\mathcal{V})$ which is a Dedekind domain, and we can find (after possibly shrinking $\mathcal{V}$) a free lattice $\mathcal{L}\subseteq F(\mathcal{V})^2$ which is stable under the action of $G_K$. When specializing to $x_0$ we obtain $\rho_{2, x_0}\cong 1+ \varepsilon$. Since $\rho_2$ is irreducible, we may choose another $G_K$-stable lattice $\mathcal{L}'$ such that $\rho_{\mathcal{L}',x_0}\cong \left(\begin{smallmatrix} \varepsilon & {*}\\ 0 &1\end{smallmatrix}\right) $. Now because the monodromy is trivial at $v\nmid p$, we know the extension class $c\in H^1(G_K, L(1))$ is unramified at $v\nmid p$. For $v|p$, we use the following general lemma about continuity of crystalline periods.

Lemma 1 Let $\rho: G_{\mathbb{Q}_p}\rightarrow \GL_n(\mathcal{A}(\mathcal{V}))$. Assume there exists $\Sigma\subseteq \mathcal{V}(\overline{\mathbb{Q}_p})$ Zariski dense subset such that $\Sigma_c$ is Zariski dense for any $c$. For any $x\in \Sigma_c$, $\rho_x$ is crystalline with eigenvalues $p^{\kappa_i(x)}\phi_i(x)$. Then for any $x\in \Sigma$, and $k=1,\ldots,n$, we have $$D_\mathrm{crys}(\wedge^k \rho_x)^{\phi=\prod_{i=1}^k p^{\kappa_i(x)}\phi_i(x)}\ne0.$$ Here we order $\kappa_1(x)\le \kappa_2(x)\le\cdots\le\kappa_n(x)$.
Remark 30 This generalizes Kisin's lemma: if $D_\mathrm{crys}(\rho_x)^{\phi=p^{\kappa_1(x)}\phi_1(x)}$ is of rank 1 for any $x\in \Sigma_c$, then $D_\mathrm{crys}(\rho_{x_0})^{\phi=p^{\kappa_1(x_0)}\phi_1(x_0)}\ne0$.

It follows from the lemma that $D_\mathrm{crys}(\rho_{\mathcal{L}',x_0})^{\phi=1}\ne0$. Hence we obtain (the surjectivity in) the following exact sequence $$0\rightarrow D_\mathrm{crys}(L(1))\rightarrow D_\mathrm{crys}(\rho_{\mathcal{L}',x_0})\rightarrow D_\mathrm{crys}(L)\rightarrow 0.$$ Hence $D_\mathrm{crys}(\rho_{\mathcal{L}', x_0})$ is 2-dimensional and thus $\rho_{\mathcal{L}',x_0}$ is also crystalline at $v$.

It follows that the extension class $c$ is crystalline at $v\mid p$. So $c\in H^1_f(K, L(1))$. The latter group is zero (as $\mathcal{O}_K^\times$ is finite), so $c$ is trivial, a contradiction to the irreducibility of $\rho_2$. Therefore $T$ must be irreducible.


TopThe Bloch-Kato conjecture

Keep the assumptions before Theorem 10.

Theorem 11 There exists a nontrivial extension in $H^1_f(K, R^\vee(1))=H^1_f(K, R)$.

Our remaining goal is to explain the strategy of the proof.

Define $$\mathcal{U}'{}=\{x\in \mathcal{U}: \kappa_i(x)-\kappa_{i+1}(x)=\kappa_i(x_0)-\kappa_{i+1}(x_0), \forall i\ne b+1\},$$ (in fact one can fix the first $b$ weights and let the rest move in parallel, in which case $\mathcal{U}'$ is a curve). Then the same argument as last time shows that $T|_{\mathcal{U}'}$ is generically irreducible. Let $\mathcal{L}\subseteq F(\mathcal{U}')^{d+2}$ be a free lattice such that $\mathcal{L}_{x_0}$ has a unique quotient isomorphic to the trivial representation. Then $\mathcal{L}_{x_0}$ contains the representation $L(1)$.

Lemma 2 $E=\mathcal{L}_{x_0}/L(1)$ is an extension $$0\rightarrow R\rightarrow E\rightarrow L\rightarrow 0.$$
Proof If not, then $L(1)$ is the only subrepresentation of $E$, so we obtain an extension $$0\rightarrow L(1)\rightarrow \mathcal{L}_{x_0}/R\rightarrow L\rightarrow 0.$$ This extension is not trivial as $L$ is the unique trivial quotient of $\mathcal{L}_{x_0}$. Now for $x\in \mathcal{U}'\cap \Sigma$, $\phi_i(x)p^{\kappa_i(x)}$ are crystalline Frobenius eigenvalues. By taking $\phi_1'(x)=\phi_i(x)p^{\kappa_i(x)-\kappa_{i+1}(x)}$ and apply Lemma 1 , we see that $D_\mathrm{crys}(\mathcal{L}_{x_0})^{\phi=\alpha_i}\ne0$ for $\alpha_i=\phi_i(x_0)p^{\kappa_i(x_0)}$, $i=1,\ldots,b+1$. In particular, by the same argument we know the extension class of $\mathcal{L}_x/R$ has class in $H^1_f(K, L(1))=0$, which is a contradiction.
Remark 31 Notice for the elliptic curve case $\mathcal{U}'\cap \Sigma$ 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 $\mathcal{L}_x$.

It remains to prove the extension class of $E$ is crystalline. To do so, we switch to the dual situation, namely consider free lattice $\mathcal{L}$ with unique irreducible quotient $R $, which gives an extension $$0\rightarrow L(1)\rightarrow \mathcal{L}_{x_0}/L\rightarrow R\rightarrow0,$$ whose dual gives the nontrivial class in $H^1(K, R^\vee(1))$.

Lemma 3 Let $V$ be a de Rham representation of $G_K$ and $E$ an extension of the form $$0\rightarrow L(1)\rightarrow E\rightarrow V\rightarrow 0.$$ Assume that there exists $D\subseteq D_\mathrm{crys}(E)$ such that the image $g(D)$ of $D$ in $D_\mathrm{crys}(V)\subseteq D_\mathrm{dR}(V)$ satisfies $$g(D) \oplus \Fil^0D_\mathrm{dR}(V)=D_\mathrm{dR}(V).$$ Then $E$ is de Rham.
Proof To prove that $E$ is de Rham, we need to show the surjectivity of $g: D_\mathrm{dR}(E)\rightarrow D_\mathrm{dR}(V)$. Tensoring the extension with $B_\mathrm{dR}^+$ and taking $H^i(K, -)$, we obtain an exact sequence $$H^0(K, t B_\mathrm{dR}^+)\rightarrow \Fil^0D_\mathrm{dR}(E)\rightarrow \Fil^0D_\mathrm{dR}(V)\rightarrow H^1(K, tB_\mathrm{dR}^+).$$ The first and last terms (by Tate-Sen) are both 0. This gives the injectivity of the right vertical map in the commutative diagram with exact rows, $$\xymatrix{ D_\mathrm{dR}(E) \ar[r]^-{g} \ar[d] &  D_\mathrm{dR}(V) \ar[r]^-{\delta_0} \ar@{->>}[d]  &  H^1(K, L(1) \otimes B_\mathrm{dR}) \ar@{^(->}[d] \\  (E \otimes B_\mathrm{dR}/B_\mathrm{dR}^+)^{G_K} \ar[r]^-h &  (V \otimes B_\mathrm{dR}/B_\mathrm{dR}^+)^{G_K} \ar[r]^-{\delta_1}&  H^1(K, L(1) \otimes B_\mathrm{dR}/B_\mathrm{dR}^+).}$$ Here the surjectivity of the middle vertical map follows from that $V$ is de Rham. The assumption on $D$ implies the composition of $g$ and the middle vertical map is surjective. It follows that the map $h$ is surjective, hence $\delta_1$ is 0. Thus $\delta_0$ is also zero, and thus $g$ is surjective as desired.

Now let us finish the proof of Theorem 11. Let $D_1\subseteq D_\mathrm{crys}(R)$ be generated by the eigenvectors of $\phi$ of eigenvalues $\alpha_1,\ldots,\alpha_b$. We claim that $$D_1 \oplus \Fil^0(D_\mathrm{dR}(R))=D_\mathrm{dR}(R).$$ In fact, the first $b$ Hodge-Tate weights $\kappa_1(x_0),\ldots, \kappa_b(x_0)$ of $R $ are $<-1$, and the last $a$ Hodge-Tate weights are $>0$. Since $D_\mathrm{crys}(R)$ is weakly admissible, for any $D\subseteq D_\mathrm{crys}(R)$, its Newton polygon is above the Hodge polygon. Hence $$D_1\cap \Fil^0 D_\mathrm{dR}(R)=0.$$ Since $\dim D_1=b$ and $\dim \Fil^0 D_\mathrm{dR}(R)=a$, we know the claim holds for dimension reason.

Since $D_\mathrm{crys}(\mathcal{L}_{x_0})^{\phi=\alpha_i}$ for $i=1,\ldots, b+1$, we have seen that $D_\mathrm{crys}(\mathcal{L}_{x_0}/L)^{\phi=\alpha_i}\ne0$. Now take $D$ to be the submodule generated by the eigenvectors with eigenvalues $\alpha_1,\ldots, \alpha_b$. Then $g(D)=D_1$. We apply Lemma 3 and obtain that $E$ 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 $E$ is crystalline, as desired.


TopEven order vanishing

Today we will explain the higher rank case.

Theorem 12 Assume $L(R, s)$ vanishes at the center $s=0$ with even order, then $\dim H^1_f(K, R)\ge2$.

Recall we have constructed an Eisenstein series and a corresponding point $x_0$ on the eigenvariety on $G'=G_{a+1,b+1}$ of dimension $d+2$ with a certain $p$-stabilization and a certain type at ramified places. This provides a family of cuspidal automorphic representations $\Pi_x'$ at for all $x $ in a neighboorhood $\mathcal{U}$ of $x_0$. Up to a twist the $L$-function of the Eisenstein series is given by $$L(R \oplus \varepsilon \oplus 1,s)=L(R, s)\zeta(s+1)\zeta(s).$$ If the order of vanishing of $L(R,s) $ is even at the center, then we know the sign of the functional equation $\varepsilon(R)=+1$. Hence $\varepsilon(R \oplus \varepsilon \oplus 1)=-1$. Since $R_x'$ is crystalline at $p$ (hence local sign at $p$ is $+1$) and the local signs at ramified places are determined by the type (and the local sign at $\infty$ is fixed since the weights are congruent mod $p-1$), we know that $\varepsilon(R_x')$ is also $-1$ for any $x\in \mathcal{U}$. In particular, the $L$-function of $R_x'$ vanishes at the center.

Thus we can apply Theorem 11 for $R_x'$ on $G'$ for which $0,-1$ are not Hodge-Tate weights. Then for $x\in \Sigma$, we obtain an extension $$0\rightarrow L(1)\rightarrow E_x\rightarrow R_x'\rightarrow 0,$$ coming from the deformation of an Eisenstein series $E(\Pi_x' \otimes \chi)$ on $G''=G_{a+2,b+2}$ corresponding to a point $\mathrm{Eis}(x)$ on the eigenvariety $\mathcal{E}_{G''}$ of dimension $d+4$. Because $\Sigma$ is Zariski dense in $\mathcal{U}$, one can then find a analytic map $\mathcal{U}\rightarrow\mathcal{E}_{G''}$ (after possibly shrinking $U$ so that the map to the weight space is finite). Let $\mathcal{U}''$ be an open containing the image of $\mathcal{U}$ in $\mathcal{E}_{G''}$, then we obtain a pseudo-representation $T''_ {\mathcal{U}''}$. We have $T''_{\mathrm{Eis}(x)}=1+ \varepsilon +\tr (R_x)$ and in particular, $$T''_{\mathrm{Eis}(x_0)}=1+\varepsilon +1+\varepsilon+\tr R.$$ Therefore we obtain an extension $$0\rightarrow \mathcal{A}(\mathcal{U})(1)\rightarrow \mathcal{E}_\mathcal{U}\rightarrow R_\mathcal{U}\rightarrow 0.$$ Here $R_\mathcal{U}$ is the lattice attached to the pseudo-representation such that $R_{x_0}$ has a unique quotient isomorphic to $R $. The extension we construct is given by the upper right corner ${*}_3$ in $$
\varepsilon & {*}_1 & {*}_2 & {*}_3\\ & \mathbf{1} & {*} & {*} \\ & & \varepsilon & {*}\\ & & & R
 $$ Notice that $*_1$ is trivial again because $\mathcal{O}_K^\times$ is trivial. It remains to show that $*_2$ is trivial. It suffices to show that the extension $$0\rightarrow \varepsilon\rightarrow {*}_2\rightarrow \varepsilon\rightarrow 0$$ is trivial at $p$ (otherwise there is a $\mathbb{Z}_p$-extension of $K$ unramified at $p$). For the local triviality it suffices to show that $*_2$ 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 $(n_1,\ldots,n_r)$ ($\sum n_k=d$). Suppose for $x\in\Sigma$, $T_x$ is crystalline at $p$ and the Hodge-Tate weights $\kappa_i$ moves in parallel for $n_k\le i\le n_{k+1}$ for any $k$. Let $Q_x(X)=\prod_{i=1}^{n_1}(X-\phi_i(x)p^{\kappa_i(x)})$. Then
  1. $\dim D_\mathrm{crys}(R_{x_0})^{Q_{x_0}(\phi)=0}=n_1$.
  2. There exists $N\gg 0$ such that we have an injective map $$D_\mathrm{crys}(R_{x_0})^{Q_{x_0}(\phi)=0}\rightarrow (R_{x_0} \otimes B_\mathrm{dR}/t^{\kappa_{n_1}(x_0)+N}B^+_\mathrm{dR})^{G_{\mathbb{Q}_p}}.$$ In other words, these crystalline periods contributes to the Hodge-Tate weights $\le \kappa_{n_1}(x_0)+N$.
Lemma 5 Assume that $E_\mathcal{U}$ is a free lattice which is finite slope of Hodge-Tate type $(n_1,1,n_2,\ldots)$ such that $E_\mathcal{U}$ fits in an extension $$0\rightarrow \mathcal{A}(U)\rightarrow E_\mathcal{U}\rightarrow R_\mathcal{U}\rightarrow0.$$ Assume that
  1. $Q_x(1)\ne0$ (i.e., 1 does not appear in the first $n_1$ Frobenius eigenvalues).
  1. $\kappa_i(x)>0$ for $i>n_1+1$ and $x\in \Sigma$.
  2. $R_{x_0}$ is Hodge-Tate.

Then $\dim E_{x_0}(0)=\dim R_{x_0}(0)+1$.

Remark 32 Here $V(0):= (V \otimes \mathbb{C}_p)^{G_{\mathbb{Q}_p}}\subseteq \mathrm{gr}^0 D_\mathrm{dR}(V)$. Recall that $V$ is Hodge-Tate if and only if $\dim V=\sum\dim V(i)$. Also recall that Hodge-Tate means the Sen operator $\Phi_\mathrm{sen}$ 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 $R_{x_0}$. Assume otherwise. Suppose we have a sequence of points $x_n\in \Sigma$ such that
  1. the conclusion holds for all $x_n$.
  2. $x_n\rightarrow x_0$ ($p$-adically).

Then the conclusion also holds for $x_0$. In fact, the Sen operator $\Phi_\mathrm{sen}(x_n)$ has 0-eigenspace of dimension $1+\dim R_{x_0}$, so the limit $\Phi_\mathrm{sen}(x_0)$ has 0-eigenspace of dimension $\ge 1+\dim R_{x_0}$.

So it remains to prove the result for points $z\in \mathcal{U}$ such that (as we can then find a sequence of such points converging to $x_0$):

  1. $\kappa_i(z)$ are integers,
  2. $\kappa_i(z)=\kappa_i(x_0)$ for $i=1,\ldots, n_1$,
  3. $\kappa_i(z)>c$ for $i>n_1+1$,
  4. $Q_z(1)\ne0$.

Here we choose $c\ge \kappa_{n_1}(x_0)+N$, where $N $ is the integer in Kisin's Lemma 4.

For these points, we need to show the surjectivity in the exact sequence $$0\rightarrow L \rightarrow (E_{z} \otimes B_\mathrm{dR}/t^c B^+_\mathrm{dR})^{G_{{\mathbb{Q}}_p}}\rightarrow (R_{z} \otimes B_\mathrm{dR}/t^c B^+_\mathrm{dR})^{G_{{\mathbb{Q}}_p}}\rightarrow 0. $$ By Kisin's Lemma 4 that $$D_\mathrm{crys}(E_z)^{Q_z(\phi)=0}\hookrightarrow (E_z \otimes B_\mathrm{dR}/t^{c}B^+_\mathrm{dR})^{G_{\mathbb{Q}_p}}.$$ Since $D_\mathrm{crys}$ is weakly admissible, we know that $D_\mathrm{crys}(R_{z})^{Q_z(1)=0}\oplus \Fil^c D_\mathrm{dR}(R_{z})=D_\mathrm{dR}(R_{z})$. This gives an isomorphism $D_\mathrm{crys}(R_z)^{Q_z(\phi)=0}\cong (R_{z} \otimes B_\mathrm{dR}/t^c B^+_\mathrm{dR})^{G_{{\mathbb{Q}}_p}}$. The desired surjectivity then follows.


Now we can finish the proof of Theorem 12. We have constructed a family of Galois representations $$0\rightarrow \mathcal{A}(\mathcal{U})(1)\rightarrow E_\mathcal{U}\rightarrow R_\mathcal{U}\rightarrow 0,$$ where $R_\mathcal{U}$ gives a nontrivial extension ${*}$, $$0\rightarrow L(1)\rightarrow R_{x_0}/L\rightarrow R\rightarrow 0$$ at $x_0$. Notice that the Hodge-Tate weights of $E_\mathcal{U}$ at $x_0$ is given by $$(\kappa_1(x_0), \ldots, \kappa_b(x_0), -1, -1, 0, \ldots).$$ We apply Lemma 5 to $E_\mathcal{U}(-1)$ to obtain $$\dim\mathrm{gr}^1 (D_\mathrm{dR}(E_{x_0}))=\dim \mathrm{gr}^1 (D_\mathrm{dR}(R_{x_0}))+1.$$ It follows that the extension ${*}_2$ is Hodge-Tate and hence is trivial. Hence the extension ${*}_3$ is nontrivial. Moreover, the two extensions ${*}$ and ${*}_3$ are linearly independent because $R $ is the unique irreducible quotient of $E_{x_0}$ (otherwise $\varepsilon$ is also an irreducible quotient).

TopThe rank 3 case: difficulties

Assume now $\ord_{s=0}L(R, s)=3$. Then again we can construct an Eisenstein series $E(R)$ on $G'$ which deform into a family of cusp forms $\mathcal{F}'$ such that $\mathcal{F}' _ {x_0}=E(R)$. Then $\ord_{s=0}L(E(R),s)=2$ and hence has the sign of functional equation equal to $+1$. It is expected (but not known) that $L(\mathcal{F}' _ x, 0)\ne0$ 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 $\le1$ case). Again we can construct a family $\mathcal{F}''$ on $G''$ such that on the vanishing locus $\mathcal{Z}=\{x: L(\mathcal{F}' _ x,0)=0\}$ (to obtain a $p$-adic locus, one also needs to replace complex $L$-values by $p$-adic $L$-values), we have $\mathcal{F}'' _ x = E(R_{\mathcal{F}' _ x})$ and $R_{\mathcal{F}'' _ x}\cong R_{\mathcal{F}' _ x}\oplus \varepsilon \oplus 1$, and is generically irreducible outside $\mathcal{Z}$. On $\mathcal{Z}$ we can even construct a nontrivial extension $$0\rightarrow \mathcal{A}(\mathcal{Z})(1)\rightarrow E_\mathcal{Z}\rightarrow R_{\mathcal{F}'_Z}\rightarrow 0.$$

Now the problem is to prove $*_2$ is trivial using Lemma 5 , we need to further choose the sequence of points $z$ lying in $\mathcal{Z}$. However, even the condition that $\kappa_i(z)=\kappa_i(x_0)$ for $i\le n_1$ seems difficult to satisfy.

TopThe modular form case

Let $f$ be a cusp eigenform for $\Gamma_0(N)$ of weight $k=2m\ge2$. We assume

  • $(p, N)=1$
  • if $k=2$, then $f$ comes from a definite quaternion algebra via Jacquet-Langlands
  • $\alpha$ a root of Hecke polynomial at $p$ for $f$ such that $v_p(\alpha)<k-1$ (non-critical condition).
  • $K/\mathbb{Q}$ an imaginary quadratic field.

We have the following generalization of Theorem 10.

Theorem 13 Assume that $L(f,m)=0$. Then exists
  • a pseudo-representation $T: G_K\rightarrow \mathcal{A}(\mathcal{U})$, where $\mathcal{U}\subseteq \mathcal{X}=\Hom_\mathrm{cont}((\mathbb{Z}_p^\times)^4, \overline{\mathbb{Q}_p}^\times)$ is an open affinoid,
  • $\Sigma\subseteq U$ a set of arithmetic points such that $\Sigma_c$ is Zariski dense,
  • $\phi_1,\ldots\phi_4,\kappa_1,\ldots\kappa_4\in \mathcal{A}(\mathcal{U})$,
  • $x_0\in \mathcal{U}$,

such that,

  1. $T_{x_0}=\tr V_f(m)+ \varepsilon+1$, $\kappa(x_0)=(-m,-1,0,m-1)$ and $\phi(x_0)=(\alpha p^{-m},1, p^{-1}, p^{m-1}\alpha^{-1})$.
  2. $T_x$ is crystalline for $x\in \Sigma$ with Hodge-Tate weights $\kappa_i(x)$ and Frobenius eigenvalues $\phi_i(x)p^{\kappa_i(x)}$.
  3. For $\ell\ne p$, $\dim R_x^{I_\ell}=\dim V_f(m)^{I_\ell}+2$.
Remark 33 If $m>1$, then the construction is as before using the eigenvariety for $U(2,2)$. However, when $m=1$, the Eisenstein series we are looking for is no longer holomorphic due to irregular Hodge-Tate weights. The idea is then to put $f$ into a Coleman (or Hida) family. However, the vanishing of $L$-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 $\dim H^1_f(K, V_f(m))\ge1$.
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 $(-1,-1,0,0)$ 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 $\wedge^2 \mathcal{L}_{x_0}$, whose Hodge-Tate weights are $(-2,-1,-1,-1,-1,0)$. Let $E$ be the extension in Case B $$0\rightarrow \varepsilon\rightarrow E\rightarrow 1\rightarrow 0.$$ Then $W=E \otimes R$ is a subquotient of the exterior square, hence it is semi-stable, with monodromy operator $N_W=N_E \otimes \Id$. It remains to show $N_E=0$.

If not, then there exists $v\in D_\mathrm{st}(E)$ such that $\phi v=v$, $N v\ne0$. Let $v'\in D_\mathrm{crys}(R)$ such that $\phi(v')=\alpha p^{-m} v'$. Then $v \otimes v'\in D_\mathrm{st}(E) \otimes D_\mathrm{crys}(R)=D_\mathrm{st}(E \otimes R)$, and $N(v \otimes v')=Nv \otimes v'\ne0$. But by Kisin's lemma, $D_\mathrm{crys}(\wedge^2 \mathcal{L}_{x_0})^{\phi = \alpha p^{-m}}\ne0$, and hence $D_\mathrm{crys}(E \otimes R)^{\phi = \alpha p^{-m}}\ne0$, and hence $v \otimes v'\in \ker N$, a contradiction.


TopNearly holomorphic modular forms on unitary groups

We first review the $\SL_2$ case.

Definition 54 Let $f: \mathcal{H}\rightarrow \mathbb{C}$ be a $C^\infty$-function. Let $\Gamma\subseteq \SL_2(\mathbb{Z})$ be a congruence subgroup. We say that $f$ is a nearly holomorphic modular form of order $\le r$, and of weight $k\ge0$ for $\Gamma$ if
  1. $f|_{k}\gamma=f$ for any $\gamma\in\Gamma$,
  2. $f$ is regular (i.e., has a finite limit) at the cusps.
  3. $f(z)=f_0(z)+\frac{1}{y} f_z(z)+\cdots \frac{1}{y^r}f_r(z)$, where $f_0,\ldots,f_r$ are holomorphic functions on $\mathcal{H}$.

The space of such nearly holomorphic modular forms is denoted by $\mathcal{N}^r_k(\Gamma)$. In particular, $\mathcal{N}_k^0(\Gamma)=\mathcal{M}_k(\Gamma)$.

Definition 55 We define the differential operator $$\varepsilon: C^\infty(\mathcal{H})\rightarrow C^\infty(\mathcal{H}),\quad (\varepsilon f)(z)=8i\pi y^2\partial f/\partial \bar z.$$ Then the third condition in Definition 54 can be replaced by $\varepsilon^{r+1}f=0$. Notice it induces an operator which decreases the weight and order $$\varepsilon: \mathcal{N}_k^r(\Gamma)\rightarrow \mathcal{N}_{k-2}^{r-1}(\Gamma).$$
Definition 56 We define the Maass-Shimura operator $\delta_k: \mathcal{N}_k^r(\Gamma)\rightarrow \mathcal{N}_{k+2}^{r+1}(\Gamma)$, which increases the weight and order by $$\delta_k f(z)=\frac{1}{2i\pi}\left(\frac{\partial f}{\partial z}+ \frac{k}{2i y} f\right)=\frac{1}{2\pi} y^{-k}\frac{\partial( y^k f)}{\partial z}.$$

Next let us give equivalent algebraic definitions.

Definition 57 Let $p: \mathcal{E}\rightarrow X$ be the universal elliptic curve over the modular curve $X=X(\Gamma)/\mathbb{Q}$. Let $$\omega=p_*\Omega_{\mathcal{E}/X}(\log \partial \mathcal{E}),\quad \mathcal{H}^1_\mathrm{dR}=R^1 p_*(\Omega^\cdot_{\mathcal{E}/X}(\log \partial \mathcal{E})).$$ Then $\mathcal{M}_k(\Gamma)/\mathbb{Q}= H^0(X/\mathbb{Q}, \omega^{\otimes k})$. The Hodge filtration gives an exact sequence $$0\rightarrow \omega \rightarrow\mathcal{H}^1_\mathrm{dR}\rightarrow \omega^\vee\rightarrow 0.$$ We have a $C^\infty$-Hodge decomposition $\mathcal{H}^1_\mathrm{dR}=\omega \oplus \omega^\vee$.
Definition 58 We define $\mathcal{H}^r_k=\omega^{\otimes k-r} \otimes \Sym^r(\mathcal{H}^1_\mathrm{dR})$. Using $\omega^{\otimes r}\hookrightarrow \Sym^r(\mathcal{H}^1_\mathrm{dR})$, we have an exact sequence $$0\rightarrow \omega^r \rightarrow \mathcal{H}^r_k\rightarrow \mathcal{H}^{r-1}_{k-2}\rightarrow 0.$$ Then $\mathcal{N}_k^r(\Gamma)/\mathbb{Q}= H^0(X/\mathbb{Q}, \mathcal{H}^r_k)$. It follows from the $C^\infty$-Hodge decomposition that this definition agrees with the previous analytic definition. Then induced map from $\mathcal{H}^r_k\rightarrow \mathcal{H}^{r-1}_{k-2}$ agrees with the differential operator $\varepsilon$. From the Gauss-Manin connection $$\mathcal{H}^1_\mathrm{dR}\rightarrow \mathcal{H}^1_\mathrm{dR} \otimes \Omega_{X/\mathbb{Q}}(\log \partial X)$$ and the Kodaira-Spencer isomorphism $\Omega_{X/\mathbb{Q}}(\log\partial X)\cong \omega^{\otimes 2}$, we also obtain an algebraic definition of the Maass-Shimura operator induced from $\mathcal{H}^r_k\rightarrow \mathcal{H}^{r+1}_{k+2}$.

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 $f$ on quadruples $(E/R, \alpha, \omega,\omega')$, where $E/R$ is an elliptic curve, $\omega$ is a basis of $H^0(E, \omega)$ and $(\omega,\omega')$ is a basis of $H^1_\mathrm{dR}(E)$, and $\alpha$ is a $\Gamma$-level structure, such that $f(E,\alpha,\omega,\omega')(X)\in R[X]_r$ (degree $\le r$ polynomials; think: $X=\frac{1}{8\pi y}$) and $$f(E, \alpha, a\omega, a^{-1}\omega'+b \omega)(X)=a^k f(E,\alpha,\omega,\omega')(a^{-2} X-a^{-1}b).$$ Using Tate's curve we have a $q$-expansion $f(q, X)\in R[X]_r[ [ q ] ]$, and $$(\varepsilon f)(q, X)=d/dX f(q, X),\quad (\delta_k f)(q, X)=X^k D(X^{-k} f(q, X)),$$ where $D:=q \partial /\partial q- X^2 \partial/ \partial X$.

Now let us discuss the case of $G=U(n,n)$. Denote $\mathcal{D}=\{z \in M_n(\mathbb{C}): i(\bar z ^t- z)>0\}$, and $$\Xi(z):=(i (\bar z - z^t), i(\bar z ^t- z))\in \GL_n(\mathbb{C})\times \GL_n(\mathbb{C}).$$ Let $r(z)=i (\bar z- z^t)=(r_{i,j})$. We introduce the differential operators $\partial r_{i,j}$ defined by the relation $$\partial/\partial \bar z_{kl}=\sum_{i,j} \partial r_{i,j}/\partial \bar z_{k,\ell}\cdot \partial/ \partial r_{i,j}.$$

Definition 60 Let $(\rho, V_\rho)$ be an algebraic representation of $\GL_n(\mathbb{C})\times \GL_n(\mathbb{C})$. We define $(S, M_n)$ the representation of $\GL_n\times \GL_n$ defined by $(g_1, g_2)A=g_1A g_2^t$. Let $\St^+$ be the standard representation of the first $\GL_n$ and $\St^-$ be the standard representation of the second $\GL_n$, then $S=\St^+ \otimes \St^-$.We define $$\varepsilon_\rho(f): G(\mathbb{A}_f)\times \mathcal{D}\rightarrow \Hom(M_n(\mathbb{C}), V_\rho)=V_{\rho \otimes S^\vee}$$ by $$\varepsilon_\rho(f) (g_f, z)(A)=\sum_{i,j} A_{i,j} \partial/\partial r_{i,j}f(g_f,z)\in V_\rho.$$
Definition 61 We define $\varepsilon_\rho^{r+1}=\varepsilon_{\rho \otimes (S^\vee)^{\otimes r}}\circ \cdots \varepsilon \varepsilon_{\rho \otimes S^\vee} \circ \varepsilon_\rho$. So $\varepsilon_\rho^{r}: \mathcal{N}_\rho^r(K)\rightarrow \mathcal{N}_{\rho \otimes S^\vee}^{r-1}(K)$, which again decreases the weight and the level. Similarly we can define the Maass-Shimura operator $$\delta_\rho: \mathcal{N}_\rho^r\rightarrow \mathcal{N}_{\rho \otimes S}^{r+1}$$ given by $$\delta_\rho(f)(g_f,z)=\sum_{i,j} \rho(\Xi(z))^{-1} \partial/ \partial z_{i,j}(\rho(\Xi(z)) f(g_f, z)) \otimes E_{i,j}\in V_{\rho \otimes S}.$$ For example, when $\rho=\det^k$, we have $$\delta_k(f)=\sum_{i,j}\partial f/\partial z_{i,j} E_{i,j}+ k\tr (r(z)^t (E_{i,j}))f.$$
Remark 34 It is easy to see (using the canonical line in $S \otimes S^\vee$) that $\varepsilon_{\det^k \otimes S}\circ \delta_k(f)=k f$.
Definition 62 A nearly holomorphic modular form of order $\le r$, weight $\rho$ and level $K$ is a$C^\infty$-function $f: G(\mathbb{A}_f)\times \mathcal{D}\rightarrow V_\rho$ such that
  1. $f(gk, z)=f(g,z)$ for $k\in K$.
  2. $f(\gamma g)=f(g)$ for any $\gamma\in G(\mathbb{Q})$,
  3. $\varepsilon_\rho^{r+1}(f)=0$.

Now let us come to the algebraic definition.

Definition 63 Let $X=\Sh_{G,K}$ be the associated Shimura variety. Let $\mathcal{G}\rightarrow \bar X$ be the universal (generalized) abelian variety of dimension $2n$, together with a principal polarization, and $\mathcal{O}_K$-action such that $\omega_{A/R}=\omega^+ \oplus \omega^-$ (both of rank $n)$. Then $\mathcal{H}^1_\mathrm{dR}(A/R)=\mathcal{H}^+ \oplus \mathcal{H}^-$.
Definition 64 Let $\omega_\rho$ be the coherent sheaf associated to $\rho$ on $X$ (so $\omega_{\St^+}=\omega^+$ and $\omega_{\St^-}=\omega^-$). The Hodge filtration gives an exact sequence $$0\rightarrow \omega^+\rightarrow \mathcal{H}^+\rightarrow (\omega^-)^\vee\rightarrow0.$$ Tensoring with $\omega^-$, we obtain $$0\rightarrow \omega^+\otimes \omega^-\rightarrow \mathcal{H}^+ \otimes \omega^{-}\rightarrow (\omega^-)^\vee \otimes  \omega^-\rightarrow 0,$$ pulling back along $\mathcal{O}_X\hookrightarrow  (\omega^-)^\vee \otimes  \omega^-$ we obtain an exact sequence $$0\rightarrow \omega^+ \otimes \omega^-\rightarrow \mathcal{J}\rightarrow \mathcal{O}_X\rightarrow 0.$$ So dualizing we obtain $$0\rightarrow \mathcal{O}_X\rightarrow \mathcal{J}^\vee\rightarrow \omega_{S^\vee}\rightarrow 0, $$ and hence $$0\rightarrow \omega_\rho\rightarrow \omega_{\rho} \otimes \mathcal{J}^\vee\rightarrow\omega_{\rho \otimes S^\vee}\rightarrow 0.$$ We define $$\omega_\rho^r=\Hom(\Sym^r(\mathcal{J}), \omega_\rho).$$ Then $$\mathcal{N}_\rho^r(K, R)=H^0(X/R, \omega_\rho^r)$$ agrees with the analytic definition when $R=\mathbb{C}$, and the map induced by $$\omega_\rho^r\rightarrow \omega_{\rho \otimes S^\vee}^{r-1} $$ agrees with $\varepsilon_\rho$. Finally, Using the Gauss-Manin connection $\mathcal{H}^+\rightarrow \mathcal{H}^+ \otimes \Omega_{X/K}(\log \partial X)$ and the Kodaira-Spencer isomorphism $$\Omega_{X/K}(\log \partial X) \cong \omega_S=\omega^+ \otimes \omega^-,$$ we also obtain an algebraic definition of $\delta_\rho$.
Remark 35 We also obtain $\mathcal{J}^\vee\hookrightarrow \omega_{S^\vee} \otimes \mathcal{J}$ by $\mathcal{O}_X\hookrightarrow \omega_S \otimes \omega_{S^\vee}$ and the Poincare duality for $\mathcal{H}^1_\mathrm{dR}$.
Definition 65 Let $g\in H^0(X/R, \omega_\rho)$. Then using a Mumford object (generalization of the Tate curve), one can define a $q$-expansion $$g(q)\in R[ [ q^{H^+}] ] \otimes V_\rho,$$ where $H^+$ is the set of hermitian positive definite lattice of $\mathrm{M}_n(K)$. If $g\in H^0(X/R, \omega^1_\rho)$, then we have a polynomial $q$-expansion $$g(q, X_{i,j})\in R[ [ q^{H^+} ] ] \otimes V_\rho[X_{i,j}]^{\deg\le1}_{1\le i, j\le n}.$$ Then the differential operator $\varepsilon$ acts on the $q$-expansion by $$(\varepsilon g)(q)=\sum_{i,j} \partial/\partial X_{i,j} g(q, X_{i,j}) \otimes E_{i,j}^\vee \in R[ [ q^{H^+} ]] \otimes V_{\rho \otimes S^\vee}.$$

Next time we will discuss the nearly holomorphic Eisenstein series and construct points on the eigenvariety when the $L$-function vanishes. This will complete our proof of Theorem 13.


TopFamilies of nearly holomorphic Eisenstein series

Now we specialize our discussion to $G=U(2,2)$. Let $\pi$ be a cuspidal automorphic representation of $\GL(2)$ whose $\pi_\infty$ is a discrete series of weight $k=2m\ge 2$. Let $P=MN\subseteq G$ be a parabolic with Levi $M=U(1,1) \times \mathbb{G}_m$. Let $f$ be a primitive eigenform attached to $\pi$. We have an associated automorphic form on $U(1,1)$ given by $$\phi_f(\gamma g_\infty k, z)=(c_\infty i+d_\infty)^{-m}(\bar c_\infty i+\bar d_\infty) f(g_\infty i, z).$$

Define $$I(s)=\Ind_{P(\mathbb{A})}^{G(\mathbb{A})}(\pi \otimes \delta^{s/3}).$$ Let $\Phi \in I(s_0)$ and $\Phi_s= \Phi\cdot \delta^{(s-s_0)/2}$. Assume $\Phi_v$ is in the Langlands quotient of $I(s_0)$ and $\Phi_v$ is the spherical vector for $v\not\in S$ (where $S$ is the set of ramification of $\pi$). Then the Eisenstein series $E_{\kappa_m}(f):=E(\Phi, s_0)$ gives a $C^\infty$-form of weight $$\kappa_m:=(m, 2, -2, -m).$$ Notice that this weight is dominant only when $m\ge 2$. Computing the constant term as gives the following result.

Proposition 12 The Eisenstein series $E_{\kappa_m}(f)$ is a nearly holomorphic of weight $\kappa_m$ and it is holomorphic if and only if $L(f,m)=0$.

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

Definition 66 Let $\mathcal{V}\subseteq \mathcal{X}_4$ be an affinoid (not necessarily open) in the weight space of $U(2,2)$. Let $w: \mathcal{U}\rightarrow \mathcal{V}$ be a finite map. A family of nearly holomorphic forms is a polynomial $q$-expansion $$G\in \mathcal{A}(\mathcal{U})[ [ q^{H^+}] ][X_{i,j}]_1$$ such that there exists $\Sigma\subseteq \mathcal{U}$ Zariski dense such that for any $x\in \Sigma$, $w(x)$ is an algebraic weight and $G_x:=w_x(G)$ is equal to $\pi_{w(x)}(g_x(q, X_{i,j}))$ for $g_x$ some nearly holomorphic form of weight $w(x)$. Here for $\kappa=(k_1,\ldots, k_4)$ with $k_2-k_3\ge4$ an arithmetic weight, we denote by $\pi_\kappa$ the projection of $V_\kappa$ 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 $\mathcal{F}\in \mathcal{A}(W)[ [q] ]$ be a Coleman family (for $\GL(2)$), where $W\subseteq \mathcal{X}_1=\Hom_\mathrm{cont}( \mathbb{Z}_p^\times, \overline{\mathbb{Q}_p}^\times)$. Consider the injection $$\iota: \mathcal{X} _ 1 '\hookrightarrow \mathcal{X} _ 4,\quad \xi\mapsto (\xi^{1/2}, [ 2 ], [ -2 ], \xi^{-1/2}),$$ where $[k]=(x\mapsto x^k)$. We say that $x\in W(\overline{\mathbb{Q}_p})$ is classical if $\mathcal{F}_x $ is classification of non-critical weight $k_x=2m_x$.

Theorem 14 There is a family of nearly holomorphic forms $E(\mathcal{F})\in\mathcal{A}(W)[ [ q^{H^+}] ][X_{i,j}]_1$ such that
  1. for any $x\in W(\overline{\mathbb{Q}_p})$ classical, $E(\mathcal{F})_{\iota (x)}$ is the polynomial $q$-expansion of the nearly holomorphic Eisenstein series $E_{\kappa_{m_x}}(\mathcal{F}_x)$ with desired slopes.
  2. if $\kappa_x=2m_x\ge 2$, then $E_{\kappa_{m_x}}(\mathcal{F}_x)\ne0$.
  3. if $\kappa_x=2m_x\ge 2$ and $L(\mathcal{F}_x, m_x)=0$, then $\varepsilon(E_{\kappa_{m_x}}(\mathcal{F}_x))=0$.
Proof The strategy is to use the doubling method for $U(1,1) \times U(2,2)\hookrightarrow U(3,3)$, i.e., we pullback the family of Siegel Eisenstein series on $U(3,3)$ and pair with the Coleman family on $U(1,1)$. The nontrivial computation is to determine the correct sections, especially at $p$ and $\infty$.

Another method is to use the ordinary family of Klingen Eisenstein series on $U(2,2)$ and apply some differential operator to get critical Eisenstein series (in the case of $\GL(2)$, one evaluates $\delta(E^\mathrm{ord}_{k-2})$ at $k=2$ to get $E_2^\mathrm{crit}$).

Remark 37 The doubling method should generalize well to construct nearly holomorphic families for more general unitary groups.
Corollary 2 Let $f$ be a cuspidal eigenform of weight $k=2m \ge2$. Let $\alpha$ be a root of the Hecke polynomial at $p$. If $L(f,m)=0$, then there exists a point $x_\mathrm{Eis}(f)$ on the cuspidal eigenvariety with weight $(m,2,-2,-m)$, crystalline Frobenius eigenvalue $(\alpha, p, p^{-1},\alpha^{-1})$ and Hecke eigenvalues away from $p$ given by $E_{\kappa_m}(f)$.
Proof Like the proof of Proposition 10, we use a lift of the Hasse invariant, $A\in H^0(X, \omega_{\det(\St^+)^{k_0}})$ which is a scalar modular form of weight $k_0\in (p-1)\mathbb{Z}$. Consider $$B_s=\frac{1}{k_0 s}\delta_{k_0s}(A^s)\in H^0(X, \omega_{\det^{k_0s} \otimes S}^1)= H^0(X, \omega_{\det^{k_0s}} \otimes \mathcal{J}^\vee).$$ Then by Remark 34, we have $\varepsilon(B_s)=A^s$.

Let $\mathcal{F}$ be the Coleman family passing through $f$. By Remark 35, both terms of $$G_{x,s}^1=A^s E(\mathcal{F}_x)-\varepsilon(E(\mathcal{F}_x)) B_s$$ can be viewed as sections of the same sheaf, so $G_{x,s}^1$ makes sense. It is easy to see that $\varepsilon(G_{x,s}^1)=0$, hence $G^1_{x,s}$ is holomorphic, and thus $$G_{x,s}^1\in H^0(X, \omega_{\det^{k_0s}} \otimes \omega_S \otimes \omega_{S^\vee} \otimes \omega_{\kappa_m}).$$ Its projection $G_{x,s}$ to $\omega_{\det^{k_0s}} \otimes \omega_{\kappa_m}$ is a holomorphic form of weight $$\kappa_{x,s}=(m+k_0s, 2+k_0s, -2,-m).$$ When $s\rightarrow 0$, we have $A_s\rightarrow 1$, $B_s\rightarrow 0$, so the holomorphic family $G_{x,s}\rightarrow E(\mathcal{F}_x)$ converges to the nearly holomorphic family of Eisenstein series. If $x_0$ corresponds to $f$, then $G_{x,s}$ specializes to $E(\mathcal{F}_{x_0})\ne0$, therefore $G_{x,s}\ne0$.

Now we take a finite slope projector $e$ on an affinoid of the weight space containing the weight $\kappa_{x,s}$. Let $K_{x,s}=e G_{x,s}$. Since $L(f,m)=0$, we know that $eE(\mathcal{F}_{x_0})=E(\mathcal{F}_{x_0})$, and thus $K_{x,s}$ specializes to $E(\mathcal{F}_{x_0})$. One can check that the slope of $K_{x,s}$ is not of Eisenstein type, hence $K_{x,s}$ is cuspidal. In this way we have constructed a point on the cuspidal eigenvariety.

Last Update: 05/01/2018. Copyright © 2015 - 2018, Chao Li.