$\mathrm{R}\Gamma((\pi_{HT,K_{p}}^{tor})^{-1}(\mathcal{U}),\mathcal{D}_{\nu_{A}}^{n-an}(-D))$
are concentrated in degree $0$.

###### Proof.

We only consider the case of the sheaf $\mathcal{V}_{\nu_{A}}^{n-an}(-D)$, the
other case is identical. For any compact open $K_{p}^{\prime}\subseteq K_{p}$,
let $\mathfrak{S}^{tor}_{K^{p}K_{p}^{\prime},\Sigma}$ be an integral toroidal
compactification, and let $\mathfrak{S}^{\star}_{K^{p}K_{p}^{\prime}}$ be an
integral minimal compactification (whose existence is given by the main
results of [MP19], see also section 4.4.31). Let $\mathfrak{U}$ be a formal
model for $\mathcal{U}$, realized as an open subset of the normalization of a
blow-up of $\mathfrak{FL}_{G,\mu}$. By theorem 4.4.36, for
$K_{p}^{\prime}\subseteq K_{p}$ small enough, we have normal formal models
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime},\mathfrak{U}}\rightarrow\mathfrak{S}^{\star}_{K^{p}K_{p}^{\prime}}$
for the map
$\pi_{HT,K_{p}^{\prime}}^{-1}(\mathcal{U})\rightarrow\mathcal{S}^{\star}_{K^{p}K_{p}^{\prime}}$
and
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime},\Sigma,\mathfrak{U}}\rightarrow\mathfrak{S}^{tor}_{K^{p}K_{p}^{\prime},\Sigma}$
for the map
$(\pi^{tor}_{HT,K_{p}^{\prime}})^{-1}(\mathcal{U})\rightarrow\mathcal{S}^{tor}_{K^{p}K_{p}^{\prime},\Sigma}$
and we have a map
$f_{K_{p}^{\prime}}:\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime},\Sigma,\mathfrak{U}}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime},\mathfrak{U}}$.

We may actually assume that $K_{p}^{\prime}$ is a subgroup of the pro
$p$-group $K_{p,m^{\prime},1}$. By proposition 4.6.15 for
$K_{p}^{\prime\prime}\subseteq K_{p}^{\prime}$ small enough, the torsor
$\mathcal{M}_{HT,n,K_{p,m^{\prime},1}}$ is trivial on the generic fiber of a
Zariski cover of
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}$.

We can construct a formal model for the Banach sheaf
$\mathcal{V}_{\nu_{A}}^{n-an}$ that we denote
$\mathfrak{V}_{\nu_{A}}^{n-an,+}$ over
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}$.
Indeed let
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}=\cup_{i}\mathfrak{V}_{K_{p}^{\prime\prime},i}$
be a Zariski covering with the property that the torsor
$\mathcal{M}_{HT,n,K_{p,m^{\prime},1}}$ is trivial over the generic fiber
$\mathcal{V}_{K_{p}^{\prime\prime},i}$ of
$\mathfrak{V}_{K_{p}^{\prime\prime},i}$. We fix such a trivialization. We can
construct the associated $1$-cocycle: over each intersection
$\mathfrak{V}_{K_{p}^{\prime\prime},i,j}=\mathfrak{V}_{K_{p}^{\prime\prime},i}\cap\mathfrak{V}_{K_{p}^{\prime\prime},j}$
we have an element $m_{i,j}\in
K_{p,m^{\prime},1,w,M_{\mu}}^{c}\mathcal{M}_{\mu,n}^{c}$ describing the change
of trivialization of the torsor $\mathcal{M}_{HT,n,K_{p,m^{\prime},1}}$. Over
$\mathfrak{V}_{K_{p}^{\prime\prime},i}$ we let
$\mathfrak{V}_{\nu_{A}}^{n-an,+}=V_{\kappa_{A}}^{n-an,+}\hat{\otimes}_{\mathcal{O}_{F}}\mathscr{O}_{\mathfrak{V}_{K_{p}^{\prime\prime},i}}$
and we glue these sheaves using multiplication by $m_{i,j}$ on each
intersection. The sheaf $\mathfrak{V}_{\nu_{A}}^{n-an,+}$ is a flat formal
Banach sheaf (see section 2.5.5). We claim that it is also small (see again
section 2.5.5). Indeed, it follows from lemma 6.2.6 that the representation
$V_{\kappa_{A}}^{n-an,+}\otimes_{A^{+}}A^{+}/A^{++}$ of
$K_{p,w,M_{\mu}}^{c}\mathcal{M}_{\mu,n}^{c}$ is an inductive limit of finite
free $A^{+}/A^{++}$-submodules $\mathrm{colim}_{i}V_{i}$, stable under
$K_{p,w,M_{\mu}}^{c}\mathcal{M}_{\mu,n}^{c}$, and such that over
$V_{i}/V_{i-1}$ the action factors over the quotient $K_{p,w,M_{\mu}}^{c,p}$
of $K_{p,w,M_{\mu}}^{c}$ by its maximal normal pro-$p$ subgroup. We find that
$K_{p,w,M_{\mu}}^{c,p}=wT^{c}(\mathbb{Z}_{p})w^{-1}/wT_{1}^{c}w^{-1}$ is a
finite abelian group. Moreover, the elements $m_{i,j}$ map trivially to
$K_{p,w,M_{\mu}}^{c,p}$.

It follows that the sheaf
$\mathfrak{V}_{\nu_{A}}^{n-an,+}\otimes_{A^{+}}A^{+}/A^{++}=\mathrm{colim}_{i}\mathscr{F}_{i}$
is an inductive limit of locally free sheaves of
$A^{+}/A^{++}\otimes\mathscr{O}_{\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}}$-modules
with the property that $\mathscr{F}_{i}/\mathscr{F}_{i-1}$ is the trivial
sheaf. We now fix $\mathcal{L}$ an ample line bundle over
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime\prime},\mathfrak{U}}$ (this last
formal scheme is quasi-projective, so $\mathcal{L}$ exists).

We claim that there exists $m\geq 0$ such that
$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K_{p}^{\prime\prime}}))$
(where $-D_{K^{\prime\prime}_{p}}$ is the boundary divisor at level
$K^{\prime\prime}_{p}$) is concentrated in degree $0$. The cohomology is
represented by a complex of $A^{+}$-modules which are completions of free
$A^{+}$-modules (take the Čech complex associated with the covering
$\cup\mathfrak{V}_{K_{p}^{\prime\prime},i}$). It follows from lemma 6.6.3
below that it suffices to prove that there exists $m\geq 0$ such that
$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathfrak{V}_{\nu_{A}}^{n-an,+}/A^{++}(-D_{K_{p}}))$
is concentrated in degree $0$. We are therefore reduced to prove that there
exists $m$ such that
$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}}))$
is concentrated in degree $0$. By the projection formula,

$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}}))=\mathrm{R}\Gamma(\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},\mathcal{L}^{m}\otimes\mathrm{R}(f_{K_{p}^{\prime\prime}})_{\star}\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}})).$

We first observe that
$\mathrm{R}(f_{K_{p}^{\prime\prime}})_{\star}\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}})$
is a sheaf concentrated in degree $0$ by theorem 4.4.37. Since $\mathcal{L}$
is very ample on
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}$ and
$(f_{K_{p}^{\prime\prime}})_{\star}\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}})$
is a coherent sheaf independant of $i$, it follows that there exists $m$ such
that
$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathscr{F}_{i}/\mathscr{F}_{i-1}(-D_{K^{\prime\prime}_{p}}))$
is concentrated in degree $0$ for all $i$.

Let
$L=\mathrm{H}^{0}(\mathfrak{S}^{\star,mod}_{K^{p}K_{p}^{\prime\prime},\mathfrak{U}},\mathcal{L}[1/p])$.
This is a rank $1$ projective
$\mathscr{O}_{\mathcal{S}^{\star}_{K^{p}K_{p}^{\prime\prime}}}((\pi_{HT,K_{p}^{\prime\prime}})^{-1}(\mathcal{U}))$-module.
We observe that

$f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes_{\mathscr{O}_{\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}}}\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}})[1/p]=L^{m}\otimes_{\mathscr{O}_{\pi_{HT,K_{p}^{\prime\prime}}^{-1}(\mathcal{U})}}\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}})[1/p].$

We deduce that

$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},f_{K_{p}^{\prime\prime}}^{\star}\mathcal{L}^{m}\otimes\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}})[1/p])=$
$L^{m}\otimes_{\mathscr{O}_{\pi_{HT,K_{p}^{\prime\prime}}^{-1}(\mathcal{U})}}\mathrm{R}\Gamma(\pi_{HT,K_{p}^{\prime\prime}}^{-1}(\mathcal{U}),\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}})[1/p])$

and therefore
$\mathrm{R}\Gamma(\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}},\mathfrak{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}})[1/p])$
is concentrated in degree $0$. Taking a Zariski covering of
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p}^{\prime\prime},\Sigma,\mathfrak{U}}$, we
deduce that the associated augmented Čech complex on the generic fiber is
exact. By theorem 2.5.9, affinoids are acyclic for Banach sheaves arisings as
generic fibers of flat small formal Banach sheaves. We deduce from the Čech to
cohomology spectral sequence that
$\mathrm{R}\Gamma((\pi_{HT,K_{p}^{\prime\prime}}^{tor})^{-1}(\mathcal{U}),\mathcal{V}_{\nu_{A}}^{n-an,+}(-D_{K^{\prime\prime}_{p}}))$
is concentrated in degree $0$. The lemma follows by taking the invariants
under $K_{p}/K_{p}^{\prime\prime}$.

∎

######  Lemma 6.6.3.

Let $A^{+}$ be a ring, $A^{+}+$ be an ideal of $A^{+}$. Assume that $A^{+}$ is
$A^{+}+$-adically complete and separated, Let $M_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}M_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}M_{2}$ be a complex of complete, separated, and torsion
free $A^{+}$-modules. Assume that

$M_{0}\otimes_{A^{+}}A^{+}/A^{++}\rightarrow
M_{1}\otimes_{A^{+}}A^{+}/A^{++}\rightarrow M_{2}\otimes_{A^{+}}A^{+}/A^{++}$

is exact. Then the complex $M_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}M_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}M_{2}$ is exact.

###### Proof.

It follows from the assumptions that
$\mathrm{Im}(f_{0})+(A^{++}M_{1}\cap\mathrm{Ker}(f_{1}))=\mathrm{Ker}(f_{1})$.
Since $M_{2}$ is torsion free, we deduce that
$(A^{++}M_{1}\cap\mathrm{Ker}(f_{1}))=A^{++}\mathrm{Ker}(f_{1})$. Let
$m\in\mathrm{Ker}(f_{1})$. By successive approximation, we can construct a
sequence of elements $m_{n}\in M_{0}$ for $n\geq 0$, with
$m_{n}-m_{n+1}\in(A^{++})^{n}M_{0}$ and
$f_{0}(m_{n})-m\in\mathrm{(}A^{++})^{n+1}M_{1}$. The sequence $m_{n}$
converges to $m_{\infty}$ and $f_{0}(m_{\infty})=m$. ∎

######  Lemma 6.6.4.

Lemma 6.6.2 holds without assuming that $(G,X)$ is an Hodge type Shimura
datum.

###### Proof.

We consider a diagram of Shimura datum with $(G,X)$ of abelian type and
$(G_{1},X_{1})$ of Hodge type as in section 4.4.41:

We may assume that the various morphisms between the groups $G_{1},B_{1},B,G$
over $\mathbb{Q}$ extend to morphisms over $\mathcal{O}_{F}$. We will prove
that lemma 6.6.2 holds first for $B_{1}$ and second for $G$. We add
decorations $H$ or $(H,X_{H})$ to objects depending on the Shimura datum
$(H,X_{H})$ in order to make the distinction between the Shimura data. Let
$\nu_{A}:T(B_{1})(\mathbb{Z}_{p})\rightarrow A^{\times}$ be a continuous,
$n$-analytic character. It induces a character
$\nu^{\prime}_{A}:T(G_{1})(\mathbb{Z}_{p})\rightarrow A^{\times}$. We consider
the morphism of compact open subgroups
$K_{p}=K_{p,m^{\prime},0}(B_{1})\rightarrow
K^{\prime}_{p}=K_{p,m^{\prime},0}(G_{1})$ and we deduce a finite morphism of
Shimura varieties:

$f:\mathcal{S}^{tor}(B_{1},X_{B_{1}})_{K,\Sigma}\rightarrow\mathcal{S}^{tor}(G_{1},X_{G_{1}})_{K^{\prime},\Sigma}$

with $K=K^{p}K_{p}$ and $K^{\prime}=(K^{\prime})^{p}K^{\prime}_{p}$ and by
restriction (see theorem 4.4.45, lemma 3.5.6) we have a finite morphism:

$f:(\pi_{HT,K_{p}}^{tor})^{-1}(]C_{w,k}[_{n,n}K_{p})\rightarrow(\pi_{HT,K^{\prime}_{p}}^{tor})^{-1}(]C_{w,k}[_{n,n}K^{\prime}_{p})$

We claim that there is an isomorphism

$f^{\star}\mathcal{V}_{\nu^{\prime}_{A}}^{n-an}=\mathcal{V}_{\nu_{A}}^{n-an}.$

This follows from remark 4.6.6 and remark 4.6.13, observing that the torsor
comming from the Shimura variety $(T,X_{T})$ is trivial. We insist that the
above formula does not describe the Hecke action at the level of $B_{1}$, but
this is irrelevant for us now. We can the find another compact
$K^{\prime\prime}\subseteq K^{\prime}$ such that there is a finite,
generically finite étale morphism
$f:\mathcal{S}^{tor,0}(G_{1},X_{G_{1}})_{K^{\prime\prime},\Sigma}\rightarrow\mathcal{S}^{tor,0}(B_{1},X_{B_{1}})_{K,\Sigma}$
with group $\Delta$. We now deduce that
$\mathrm{R}\Gamma((\pi_{HT,K_{p}}^{tor})^{-1}(\mathcal{U})^{0},\mathcal{V}_{\nu_{A}}^{n-an}(-D))$
is the direct factor of invariants under $\Delta$ of
$\mathrm{R}\Gamma((\pi_{HT,K^{\prime\prime}_{p}}^{tor})^{-1}(\mathcal{U})^{0},\mathcal{V}_{\nu^{\prime}_{A}}^{n-an}(-D))$.
We wrote $(\pi_{HT,K_{p}}^{tor})^{-1}(\mathcal{U})^{0}$ for
$\mathcal{S}^{tor,0}(B_{1},X_{B_{1}})_{K,\Sigma}\cap(\pi_{HT,K_{p}}^{tor})^{-1}(\mathcal{U})$
and similarly for $G_{1}$. The lemma is thus proven for $B_{1}$. We observe
that the lemma holds for any refinement of the cone decomposition $\Sigma$ by
the projection formula. We now deduce the lemma for $G$. We reset the
notations. Let $\nu_{A}:T(G)(\mathbb{Z}_{p})\rightarrow A^{\times}$ be a
continuous, $n$-analytic character. It induces a character
$\nu^{\prime}_{A}:T(B_{1})(\mathbb{Z}_{p})\rightarrow A^{\times}$. Let
$K=K^{p}K_{p}$ be a compact open subgroup of $G(\mathbb{A}_{f})$. We can find
a compact open subgroup $K^{\prime}\subseteq B_{1}(\mathbb{A}_{f})$ such that
there is a finite, generically finite étale morphism
$f:\mathcal{S}^{tor,0}(B_{1},X_{B_{1}})_{K^{\prime},\Sigma}\rightarrow\mathcal{S}^{tor,0}(G,X_{G})_{K,\Sigma}$
with group $\Delta(K,K^{\prime})$. This map induces a finite map :
$f:(\pi_{HT,K^{\prime}_{p}}^{tor})^{-1}(]C_{w,k}[_{n,n}K^{\prime}_{p})^{0}\rightarrow(\pi_{HT,K_{p}}^{tor})^{-1}(]C_{w,k}[_{n,n}K_{p})^{0}$.
We find that

$\mathcal{V}^{n-an}_{\nu_{A}}=(f_{\star}\mathcal{V}^{n-an}_{\nu^{\prime}_{A}})^{\Delta(K,K^{\prime})}.$

We deduce that
$\mathrm{R}\Gamma((\pi_{HT,K_{p}}^{tor})^{-1}(\mathcal{U})^{0},\mathcal{V}_{\nu_{A}}^{n-an}(-D))$
is the direct factor of invariants under $\Delta(K,K^{\prime})$ of
$\mathrm{R}\Gamma((\pi_{HT,K^{\prime}_{p}}^{tor})^{-1}(\mathcal{U})^{0},\mathcal{V}_{\nu^{\prime}_{A}}^{n-an}(-D))$.
The lemma is thus proven for $G$. ∎

###  6.7. Cup products and Serre duality

We now consider cup-products on locally analytic overconvergent cohomology.

######  Theorem 6.7.1.

For all $w\in{}^{M}W$ and weights $\nu_{A}$ there is a pairing:

$\langle,\rangle:\mathrm{H}^{i}_{w,an}(K_{p},\nu_{A},cusp)^{\pm,fs}\times\mathrm{H}^{d-i}_{w,an}(K_{p},\nu_{A})^{\mp,fs}\rightarrow
A$

For any $t\in T(\mathbb{Q}_{p})$ we have $\langle
t-,-\rangle=\langle-,t^{-1}-\rangle$. Let $\nu=\nu_{alg}\chi$ be a locally
algebraic weight so that $\kappa_{alg}=-w_{0,M}w\nu_{alg}-(w_{0,M}w\rho+\rho)$
is $M$-dominant. The above pairing induces a pairing between the spectral
sequences:

$\langle,\rangle_{p,q,r}:\mathbf{E}_{w,r}^{p,q}(K^{p},\kappa_{alg},\chi,cusp)^{\pm}\times\mathbf{E}^{-p,d-q}_{w,r}(K^{p},\kappa,\chi^{-1})^{\mp}\rightarrow
F$

On the abutment of the spectral sequence the pairing
$\langle,\rangle_{p,q,\infty}$ is induced by the pairing of theorem 5.7.1:

$\mathrm{H}^{p+q}_{w}(K^{p},\kappa,\chi,cusp)^{\pm,fs}\times\mathrm{H}^{d-p-q}_{w}(K^{p},-2\rho_{nc}-w_{0,M}\kappa,\chi^{-1})^{\mp,fs}\rightarrow
F.$

###### Proof.

We construct the pairing

$\langle,\rangle:\mathrm{H}^{i}_{w,an}(K_{p},\nu_{A},cusp)^{+,fs}\times\mathrm{H}^{d-i}_{w,an}(K_{p},\nu_{A})^{-,fs}\rightarrow
F.$

We can realize $\mathrm{R}\Gamma_{w,an}(K^{p},\nu_{A},cusp)^{+,fs}$ as the the
finite slope part of

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{s,\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))$

for $s>>0$ and $m^{\prime}>>s$ by definition 6.4.7 and theorem 6.4.8.
Similarly we can realize $\mathrm{R}\Gamma_{w,an}(K^{p},\nu_{A})^{-,fs}$ as
the finite slope part of

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s+1},s-1}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{-1,s-1}K_{p,m^{\prime},0}),\mathcal{D}^{n-an}_{\nu_{A}}).$

Moreover, by construction we have a pairing
$\mathcal{V}_{\nu_{A}}^{n-an}(-D)\times\mathcal{D}_{\nu_{A}}^{n-an}\to\mathcal{V}_{-2\rho_{nc}}(-D)\otimes
A$. We have a cup-product by proposition 2.2.1:

$\mathrm{H}^{i}_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{s,\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))\times$
$\mathrm{H}^{d-i}_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s+1},s-1}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{-1,s-1}K_{p,m^{\prime},0}),\mathcal{D}^{n-an}_{\nu_{A}})$
$\rightarrow\mathrm{H}^{d}_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s+1},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,s-1}K_{p,m^{\prime},0}),\mathcal{V}_{-2\rho_{nc}}(-D)\otimes
A)$

and there is a trace map (by theorem 2.7.1):

$\mathrm{H}^{d}_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s+1},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,s-1}K_{p,m^{\prime},0}),\mathcal{V}_{-2\rho_{nc}}(-D)\otimes
A)\rightarrow A.$

This pairing intertwines the actions of $\mathcal{H}_{p,m,0}^{+}$ and
$\mathcal{H}_{p,m,0}^{-}$. It is straightforward (and painful) to check that
the induced pairing

$\langle,\rangle:\mathrm{H}^{i}_{w,an}(K_{p},\nu_{A},cusp)^{+,fs}\times\mathrm{H}^{d-i}_{w,an}(K_{p},\nu_{A})^{-,fs}\rightarrow
A$

is independent of choices.

The rest of the theorem follows from the functoriality of the trace map.

∎

######  Theorem 6.7.2.

Let $w\in{}^{M}W$, let $A$ be a field and fix a weight $\nu_{A}$. The pairing
$\langle,\rangle:\mathrm{H}^{i}_{w,an}(K_{p},\nu_{A},cusp)^{\pm,fs}\times\mathrm{H}^{d-i}_{w,an}(K_{p},\nu_{A})^{\mp,fs}\rightarrow
A$ is non-degenerate.

###### Proof.

We follow closely the proof of theorem 5.7.2. Our strategy is to express the
cohomologies as cones of cohomologies (possibly with compact support) of
dagger spaces and to use the duality for affinoid dagger spaces to produce
dual complexes whose finite slope part computes the cohomology. We can realize
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu_{A},cusp)^{+,fs}$ as the the finite slope
part of

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{s,\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))$

for $s>>0$ and $m^{\prime}>>s$ by definition 6.4.7 and theorem 6.4.8

We now introduce the following closed subspaces of $\mathcal{FL}_{G,\mu}$:

$]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}=\cap_{\epsilon>0}]C_{w,k}[_{s-1-\epsilon,\overline{s}},~{}\textrm{and}~{}]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}=]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}\setminus]C_{w,k}[_{s,\overline{s}}.$

Both are the closure of quasi-compact open subspaces of
$\mathcal{FL}_{G,\mu}$. We have an exact triangle:

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s-2,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))\rightarrow$
$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s-2,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))$
$\rightarrow\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{s,\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

We observe that
$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s-2,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))$
and
$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s-2,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D))$
are the cohomology with compact support of the dagger spaces
$(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0})$
and
$(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}K_{p,m^{\prime},0})$
respectively.

For the $-$ theory, we see that for any $\epsilon>0$, the cohomology

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s},s-\epsilon}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{-1,s-\epsilon}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})$

is quasi-isomorphic to the cone of

$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{s}^{\prime},s-\epsilon}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1]\rightarrow$
$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},s-\epsilon}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1]$

We point out that in this argument, in order to apply duality theorems, we
prefer to use as coefficients
$(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}}$ rather
than $\mathcal{D}^{n-an}_{\nu_{A}}$. There is a map
$\mathcal{D}^{n-an}_{\nu_{A}}\rightarrow(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}}$.
When we pass to the inverse limit on $n$, these maps induce an isomorphism.
Since $A$ is a field,
$(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}}$ is a
projective Banach sheaf.

Passing to the limit over $\epsilon$, we deduce easily that the finite slope
part of the cone of

$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{s}^{\prime},\overline{s}}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1]\rightarrow$
$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1]$

computes $\mathrm{R}\Gamma_{w,an}(K^{p},\nu_{A})^{-,fs}$.

We can consider a finite covering $\cup_{i}U_{i}^{\dagger}$ of
$(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{s-1}^{\prime},\overline{s}}K_{p,m^{\prime},0})$
by affinoid dagger spaces (or equivalently by closed spaces which are the
closure of quasi-compact open spaces) and we let $\cup_{j}V_{j}^{\dagger}$ be
a covering of
$(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0})$
by affinoid dagger spaces, refining
$\cup_{i}(U_{i}^{\dagger}\cap(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},\overline{s}}K_{p,m^{\prime},0}))$.

We let
$A^{\bullet}=\check{C}_{c}(\\{U_{i}^{\dagger}\\},\mathcal{V}^{n-an}_{\nu_{A}}(-D))$
and
$B^{\bullet}=\check{C}_{c}(\\{V_{i}^{\dagger}\\},\mathcal{V}^{n-an}_{\nu_{A}}(-D))$
be the Čhech cohomology complex with compact support. There is a natural map
$B^{\bullet}\rightarrow A^{\bullet}$ and we let $C^{\bullet}$ be the cone of
this map which represents

$\mathrm{R}\Gamma_{(\pi_{HT,K_{p,m^{\prime},0}}^{tor})^{-1}(]C_{w,k}[_{s,\overline{s}}K_{p,m^{\prime},0})}((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{s,-1}K_{p,m^{\prime},0}),\mathcal{V}^{n-an}_{\nu_{A}}(-D)).$

Let $t\in T^{++}$. As explained in lemma 5.2.6 , we can lift $t$ to a compact
endomorphism of $C^{\bullet}$.

We let $D^{\bullet}$ be the complex which is the strict dual of $C^{\bullet}$.
By the duality for affinoid spaces, this complex computes the cone of

$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{s}^{\prime},s-\epsilon}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1]\rightarrow$
$\mathrm{R}\Gamma((\pi^{tor}_{HT,K_{p,m^{\prime},0}})^{-1}(]C_{w,k}[_{\overline{[s-1,s]}^{\prime},s-\epsilon}K_{p,m^{\prime},0}),(\mathcal{V}^{n-an}_{\nu_{A}})^{\vee}\otimes\mathcal{V}_{-2\rho_{nc}})[-1].$

For any slope $h$, taking the slope $\leq h$ part yields a perfect pairing
between $C^{\bullet,\leq h}$ and $D^{\bullet,\leq h}$ and passing to
cohomology gives the theorem.

∎

Using duality we can deduce an improved vanishing theorem.

######  Theorem 6.7.3.

Let $w\in{}^{M}W$ and let $\nu_{A}$ be a weight. The cohomology
$\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A},cusp)^{\pm,fs}$ is concentrated in degree
$[0,\ell_{\pm}(w)]$ and the cohomology
$\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A})^{\pm,fs}$ is concentrated in degree
$[\ell_{\pm}(w),d]$. Moreover, if $A$ is a domain,
$\mathrm{H}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu_{A})^{\pm,fs}$ is a torsion free
sheaf of $\mathscr{O}_{\mathrm{Spa}(A,A^{+})}$-modules.

###### Proof.

The case of $\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A},cusp)^{\pm,fs}$ is theorem
6.6.1. The case of $\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A})^{\pm,fs}$ when $A$ is
a field follows from the duality theorem 6.7.2. The case of general
coefficients can be deduced as follows. First observe that we can suppose that
$A$ is reduced as the universal weight space is reduced. Let us fix a compact
operator $t\in T^{\pm}$. For any point $x\in\mathrm{Spa}(A,A^{+})$ and any
slope $h$, we can find a slope $h^{\prime}\geq h$ and an open
$\mathrm{Spa}(B,B^{+})\hookrightarrow\mathrm{Spa}(A,A^{+})$ such that
$\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A})|_{\mathrm{Spa}(B,B)^{+}}$ has slope
$h^{\prime}$ decomposition for $t$, and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A})(\mathrm{Spa}(B,B)^{+})^{\pm,\leq
h^{\prime}}$ is computed by a perfect complex $C^{\bullet}$ of $B$-modules.
Since we know that for any $x\in\mathrm{Spa}(B,B^{+})$,
$C^{\bullet}\otimes_{B}k(x)$ has cohomology concentrated in the range
$[\ell_{\pm}(w),d]$, we deduce that $C^{\bullet}$ is quasi-isomorphic to
$\tau_{\geq\ell_{\pm}(w)}C^{\bullet}$, and this truncated complex is a perfect
complex (by an application of [Mum08], lemma 2 on p. 49). It follows that its
cohomology in degree $\ell_{\pm}(w)$ is torsion free if $A$ is a domain. ∎

###  6.8. Slope estimates and control theorem

We now formulate a conjecture regarding the slopes of the locally analytic
overconvergent cohomologies. This is of course consistent with conjecture
5.9.2.

######  Conjecture 6.8.1.

Fix $w\in{}^{M}W$, and let
$\nu:T^{c}(\mathbb{Z}_{p})\rightarrow\mathbb{C}_{p}^{\times}$ be a continuous
character.

  1. (1)

For any character $\lambda$ of $T^{+}$ on
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu)^{+,fs}$ or
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu,cusp)^{+,fs}$, we have $v(\lambda)\geq 0$.

  2. (2)

For any character $\lambda$ of $T^{-}$ on
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu)^{-,fs}$ or
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu,cusp)^{-,fs}$, we have $v(\lambda)\leq 0$.

######  Remark 6.8.2.

The inequalities in conjecture 6.8.1 are compatible with those of conjecture
5.9.2 due to the way we have renormalized the Hecke operators acting on
locally analytic cohomology. Similarly the slightly weaker bounds we prove in
theorem 6.8.3 are compatible with those of theorem 5.9.6.

We can prove a bound which is slightly weaker than the conjecture.

######  Theorem 6.8.3.

Fix $w\in{}^{M}W$, and let
$\nu:T(\mathbb{Z}_{p})\rightarrow\mathbb{C}_{p}^{\times}$ be a continuous
character.

  1. (1)

For any character $\lambda$ of $T^{+}$ on
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu)^{+,fs}$ and
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu,cusp)^{+,fs}$, we have
$v(\lambda)\geq-w^{-1}w_{0,M}\rho-\rho$ and
$v(\lambda)\geq-w^{-1}w_{0,M}\rho-\rho+w^{-1}2\rho_{nc}$

  2. (2)

For any character $\lambda$ of $T^{-}$ on
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu)^{-,fs}$ and
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu,cusp)^{-,fs}$, we have
$v(\lambda)\leq-w^{-1}\rho+\rho$ and
$v(\lambda)\leq-w^{-1}\rho+w^{-1}2\rho_{nc}$.

###### Proof.

This is similar to the proof of theorem 5.9.6 and left to the reader. ∎

######  Corollary 6.8.4.

Fix $w\in{}^{M}W$, and let $\nu=\nu_{alg}\chi$ be a locally algebraic
character. Let $\kappa_{alg}=-w_{0,M}w\nu_{alg}-(w_{0,M}w\rho+\rho)$ and
suppose $\kappa_{alg}\in X^{\star}(T)^{M,+}$. Then the morphisms

$\mathrm{R}\Gamma_{w}(K^{p},\kappa_{alg},\chi)^{+,sss_{M,w}(\kappa_{alg})}\rightarrow\mathrm{R}\Gamma_{w,an}(K^{p},\nu)(-\nu_{alg})^{+,sss_{M,w}(\kappa_{alg})}$
$\mathrm{R}\Gamma_{w,an}(K^{p},\nu)(\nu_{alg})^{-,sss_{M,w}(-w_{0,M}\kappa_{alg}-2\rho_{nc})}\rightarrow\mathrm{R}\Gamma_{w}(K^{p},-w_{0,M}\kappa-2\rho_{nc},\chi^{-1})^{-,sss_{M,w}(-w_{0,M}\kappa_{alg}-2\rho_{nc})}$

and the corresponding morphisms for cuspidal cohomology are all quasi-
isomorphisms.

###### Proof.

This follows from the spectral sequence of theorem 6.5.5, together with the
slope bounds of theorem 6.8.3. ∎

######  Remark 6.8.5.

If we assume conjectures 5.9.2 and 6.8.1, we can replace the $sss_{M,w}$
condition by the $ss_{M,w}$ condition in the above corollary.

###  6.9. Eigenvarieties

Consider the Iwasawa algebra $\mathbb{Z}_{p}[[T^{c}(\mathbb{Z}_{p})]]$ and the
weight space

$\mathcal{W}=\mathrm{Spa}(\mathbb{Z}_{p}[[T^{c}(\mathbb{Z}_{p})]],\mathbb{Z}_{p}[[T^{c}(\mathbb{Z}_{p})]])\times_{\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})}\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p}).$

For a $(A,A^{+})$ a complete Tate $(F,\mathcal{O}_{F})$-algebra,

$\mathrm{Hom}(\mathrm{Spa}(A,A^{+}),\mathcal{W})=\\{\textrm{Continuous
characters}~{}\nu_{A}:T^{c}(\mathbb{Z}_{p})\rightarrow A^{\times}\\}.$

Also let $\widehat{T}$ be the analytic adic space of characters of
$T(\mathbb{Q}_{p})$, whose restriction to $T(\mathbb{Z}_{p})$ factor through
$T^{c}(\mathbb{Z}_{p})$. If we fix a splitting $\xi$ for the map
$T(\mathbb{Q}_{p})\rightarrow T(\mathbb{Q}_{p})/T(\mathbb{Z}_{p})$, and we fix
an isomorphism $T(\mathbb{Q}_{p})/T(\mathbb{Z}_{p})\simeq\mathbb{Z}^{r}$ (for
$r=\mathrm{rank}(X^{\star}(T^{d}))$), then we have an isomorphism
$\widehat{T}\simeq\mathcal{W}\times(\mathbb{G}_{m}^{an})^{r}$, where the map
sends a character $\lambda$ to
$(\lambda|_{T^{c}(\mathbb{Z}_{p})},\lambda(\xi(e_{1})),\cdots,\lambda(\xi(e_{r})))$
(for the canonical basis $e_{1},\cdots,e_{r}$ of $\mathbb{Z}^{r}$). We also
observe that there is a natural map $T\rightarrow\mathscr{O}_{\widehat{T}}$.

Let $\mathrm{Spa}(A,A^{+})\subset\mathcal{W}$ be an affinoid open and let
$\nu_{A}^{un}:T^{c}(\mathbb{Z}_{p})\rightarrow A^{\times}$ be the universal
character.

For each $w\in{}^{M}W$ we have sheaves over $\mathrm{Spa}(A,A^{+})$:
$\oplus_{i}\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A}^{un})^{\pm,fs}$ and
$\oplus_{i}\mathrm{H}^{i}_{w,an}(K^{p},\nu_{A}^{un},cusp)^{\pm,fs}$. Gluing
these sheaves for an affinoid covering of $\mathcal{W}$ we deduce that there
are sheaves of $\mathscr{O}_{\mathcal{W}}$-modules
$\oplus_{i=\ell_{\pm}(w)}^{d}\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{\pm,fs}$
and
$\oplus_{i=0}^{\ell_{\pm}(w)}\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{\pm,fs}$,
admitting slope decomposition.

######  Remark 6.9.1.

Our weight variable $\nu$ is morally a translate of the infinitesimal
character. We recall how to switch between the weight of our coherent
cohomologies and the infinitesimal character. Let $\nu=\nu^{alg}\chi$ be a
locally algebraic weight. Then
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{+,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)$
and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{+,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)$
are related to classical or overconvergent cohomology in weight $\kappa_{alg}$
where $\nu_{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$. On the other hand,
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{-,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)$
and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{-,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)$
are related to classical or overconvergent cohomology in weight
$\kappa_{alg}^{\vee}=-w_{0,M}\kappa_{alg}-2\rho_{nc}$ and
$\nu_{alg}=w^{-1}(\kappa_{alg}^{\vee}+\rho)-\rho$.

We also have cup-products:

$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{\pm,fs}\otimes\mathrm{H}^{d-i}_{w,an}(K^{p},\nu^{un},cup)^{\mp,fs}\rightarrow\mathscr{O}_{\mathcal{W}}.$

We modify the action of $T(\mathbb{Q}_{p})$ on
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{-,fs}$ and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{-,fs}$, by composing it with the
inverse map $t\mapsto t^{-1}$. With this modification, the operators $t\in
T(\mathbb{Q}_{p})$ are self adjoint for the pairing. The action of
$T(\mathbb{Q}_{p})$ on the various sheaves
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{\pm,fs}$ and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{\pm,fs}$ over $\mathcal{W}$
produces a spectral variety, which is a Zariski closed subset
$\mathcal{Z}\hookrightarrow\widehat{T}$ (see section 6.1). We have coherent
sheaves $\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$ and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\pm,fs}$ of
$\mathscr{O}_{\widehat{T}}$-modules supported on $\mathcal{Z}$ and
$\mathscr{O}_{\mathcal{Z}}$ acts faithfully on the direct sum of all these
sheaves. Moreover, if $\pi:\widehat{T}\rightarrow\mathcal{W}$ is the
projection to the weight space,
$\pi_{\star}\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}=\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})^{\pm,fs}$
and
$\pi_{\star}\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\pm,fs}=\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)^{\pm,fs}$.

We assume that $K^{p}=\prod K_{\ell}$ and we let $S^{\prime}$ be the set of
primes $\ell\neq p$ such that $K_{\ell}$ is not hyperspecial. We let
$S=S^{\prime}\cup\\{p\\}$. Let
$\mathcal{H}^{S}=\mathcal{C}^{\infty}_{c}(G(\mathbb{A}_{f}^{S})//K^{S},\mathbb{Q})$
be the spherical Hecke algebra at places away from $S$. We let $\mathcal{E}$
be the finite adic space over $\mathcal{Z}$ whose algebra is the coherent
$\mathscr{O}_{\mathcal{Z}}$-algebra generated by the operators
$h\in\mathcal{H}^{S}$ acting on the sum of the sheaves
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$ and
$\mathrm{H}^{i}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\pm,fs}$, for all
$i\in\mathbb{Z}$, all $w\in{}^{M}W$, and all choices of $+$ or $-$.

A (classical rigid analytic) point of $\mathcal{Z}$ corresponds to a character
$\lambda_{p}:T(\mathbb{Q}_{p})\to\mathbb{C}_{p}^{\times}$. We can attach to
$\lambda_{p}$ the weight
$\nu=\lambda_{p}|_{T^{c}(\mathbb{Z}_{p})}:T^{c}(\mathbb{Z}_{p})\to\mathbb{C}_{p}^{\times}$.
When the weight $\nu=\nu_{alg}\chi$ is locally algebraic, then we let
$\lambda_{p}^{sm}=\lambda_{p}\nu_{alg}^{-1}$ with $\nu_{alg}$ viewed as a
character of $T(\mathbb{Q}_{p})$. Then $\lambda_{p}^{sm}$ factors through a
character of $T(\mathbb{Q}_{p})/T_{b}$ where $b$ is the conductor of $\chi$.

######  Remark 6.9.2.

The superscript ${sm}$ stands for smooth, because the character
$\lambda_{p}^{sm}$ is the smooth character attached to $\lambda_{p}$. Remark
that the Hecke action on classical cohomology produces smooth characters. This
is visible in point $(1)$ of theorem 6.9.3 below.

A point of $\mathcal{E}$ corresponds to a pair $(\lambda_{p},\lambda^{S})$
where $\lambda^{S}:\mathcal{H}^{S}\to\mathbb{C}_{p}$ is a character.

######  Theorem 6.9.3.

The eigenvariety $\pi:\mathcal{E}\rightarrow\mathcal{W}$ is locally quasi-
finite and partially proper. It carries graded coherent sheaves

$\bigoplus_{w\in{}^{M}W,k\in\mathbb{Z}}(\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{+,fs}\oplus\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{-,fs}$
$\oplus\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{+,fs}\oplus\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{-,fs})$

and they satisfy the following properties:

  1. (1)

(Any classical, finite slope eigenclass gives a point of the eigenvariety) For
any $\kappa_{alg}\in X^{\star}(T^{c})^{M,+}$, finite order character
$\chi:T^{c}(\mathbb{Z}_{p})\to\overline{F}^{\times}$, and any system of Hecke
eigenvalues $(\lambda^{sm}_{p},\lambda^{S})$ occurring in
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$ (resp.
$\mathrm{H}^{i}(K^{p},\kappa_{alg}^{\vee},\chi^{-1})^{-,fs}$,
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi,cusp)^{+,fs}$, or
$\mathrm{H}^{i}(K^{p},\kappa_{alg}^{\vee},\chi^{-1},cusp)^{-,fs}$) there is a
$w=w_{M}w^{M}\in W$, so that if $\nu=\nu_{alg}\chi$ with
$\nu_{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$, then
$(\lambda^{sm}_{p}\nu_{alg},\lambda^{S})$ is a point of the eigenvariety
$\mathcal{E}$ which lies in the support of
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w^{M},an}(K^{p},\nu^{un})_{\mathcal{Z}}^{+,fs}$
(resp.
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w^{M},an}(K^{p},\nu^{un})_{\mathcal{Z}}^{-,fs}$,
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w^{M},an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{+,fs}$,
or
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w^{M},an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{-,fs}$).

  2. (2)

(Small slope points of the eigenvariety in regular, locally algebraic weights
are classical) Conversely if $\nu=\nu_{alg}\chi$ is a locally algebraic weight
with $\nu_{alg}\in X^{\star}(T)^{+}$, and $(\lambda_{p},\lambda^{S})$ is a
point of $\mathcal{E}$ in the support of
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{+,fs}$
(resp.
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{-,fs}$,
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{+,fs}$,
or
$\oplus_{k\in\mathbb{Z}}\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{-,fs}$)
for some $w\in{}^{M}W$, and if $\lambda_{p}^{sm}=\lambda_{p}\nu_{alg}^{-1}$
satisfies $+,sss_{w}(\nu)$ then $(\lambda^{sm}_{p},\lambda^{S})$ occurs in
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$ (resp.
$\mathrm{H}^{i}(K^{p},\kappa^{\vee}_{alg},\chi^{-1})^{-,fs}$,
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi,cusp)^{+,fs}$, or
$\mathrm{H}^{i}(K^{p},\kappa^{\vee}_{alg},\chi^{-1},cusp)^{-,fs}$) for
$\kappa_{alg}=-w_{0,M}w(\nu_{alg}+\rho)-\rho$.

  3. (3)

(Serre duality interpolates over the eigenvariety) We have pairings:

$\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}\otimes\mathrm{H}^{d-k}_{w,an}(K^{p},\nu^{un},cusp)_{\mathcal{Z}}^{\mp,fs}\rightarrow\pi^{-1}\mathscr{O}_{\mathcal{E}}.$

and these pairings are compatible with Serre duality under the classicality
theorem.

###### Proof.

We only prove the first point for
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$. The other cases are similar.
There is a succession of three spectral sequences, having as input the sheaves
$\mathrm{H}^{k}_{w,an}(K^{p},\nu^{un})^{+,fs}$ (for varying $k$ and $w$) and
finally converging to the classical cohomology
$\mathrm{H}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$. The first spectral sequence
is the Tor spectral sequence of theorem 6.5.2, from the cohomology sheaf over
$\mathcal{W}$ to the locally analytic overconvergent cohomology in a given
weight. The second spectral sequence is the BGG spectral sequence of theorem
6.5.5, from locally analytic overconvergent cohomology to overconvergent
cohomology. The third spectral sequence is the spectral sequence of theorem
5.5.1, from overconvergent to classical cohomology. Therefore, starting from a
classical class, we can lift it successively to the $E_{1}$ terms of the last
two spectral sequences (for suitable choices of $w_{M}$ and $w^{M}$) and then
to a class in a suitable
$\mathrm{Tor}^{\mathscr{O}_{\mathcal{W}}}_{-p}(\mathrm{H}^{q}_{w,an}(K^{p},\nu^{un})^{+,fs},k(\nu))$
for $\nu=\nu_{alg}\chi$. Therefore, $(\lambda^{sm}_{p}\nu_{alg},\lambda^{S})$
is in the support of $\mathrm{H}^{q}_{w,an}(K^{p},\nu^{un})^{+,fs}$.

We now prove point $(2)$. Let $(\lambda_{p},\lambda^{S})$ be a point in the
support of $\oplus_{k\in\mathbb{Z}}H^{k}_{w,an}(K^{p},\nu^{un})^{+,fs}$ over a
weight $\nu$. Then one sees that there is a corresponding class in
$\oplus_{k}\mathrm{H}^{k}_{w,an}(K^{p},\nu)^{+,fs}$. Indeed, take $k$ maximal
such that $(\lambda_{p},\lambda^{S})$ is in the support of
$\oplus_{k\in\mathbb{Z}}H^{k}_{w,an}(K^{p},\nu^{un})^{+,fs}$. The Tor spectral
sequence implies that $(\lambda_{p},\lambda^{S})$ is in the support of
$\mathrm{H}^{k}_{w,an}(K^{p},\nu)^{+,fs}$. Then we conclude by the control
theorems (corollary 6.8.4 and theorem 5.12.3):

$\oplus_{k}\mathrm{H}^{k}_{w,an}(K^{p},\nu)^{+,sss_{w}(\nu)}=\oplus_{k}\mathrm{H}^{k}(K^{p},-w_{0,M}w(\nu_{alg}+\rho)-\rho,\chi)^{+,sss_{w}(\nu)}.$

The last point follows from the functoriality of the pairing. ∎

We now define certain components of the eigenvariety of maximal dimension and
show that they contain all finite slope interior cohomology classes. We first
define sheaves:

$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})^{\pm,fs}=\mathrm{Im}({\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un},cusp)^{\pm,fs}\rightarrow{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})^{\pm,fs}).$

######  Proposition 6.9.4.

  1. (1)

The sheaves
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})^{\pm,fs}$ are
torsion free sheaves of $\mathscr{O}_{\mathcal{W}}$-modules.

  2. (2)

For any $\nu:T^{c}(\mathbb{Z}_{p})\to\mathbb{C}_{p}^{\times}$, the map
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})^{\pm,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)\rightarrow\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu)^{\pm,fs}$
is surjective.

  3. (3)

For any $\kappa_{alg}\in X^{\star}(T^{c})^{M_{\mu},+}$ and
$\chi:T^{c}(\mathbb{Z}_{p})\rightarrow\overline{F}^{\times}$ a finite order
character, let $\nu=\nu^{alg}\chi$ with
$\nu^{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$. Then the map

$\overline{\mathrm{H}}^{\ell_{+}(w)}_{w}(K^{p},\kappa,\chi)^{+,fs}\rightarrow\overline{\mathrm{H}}^{\ell_{+}(w)}_{w,an}(K^{p},\nu)^{+,fs}$

is injective and the map

$\overline{\mathrm{H}}^{\ell_{-}(w)}_{w,an}(K^{p},\nu)^{-,fs}\rightarrow\overline{\mathrm{H}}^{\ell_{-}(w)}_{w}(K^{p},-w_{0,M}\kappa-2\rho_{nc},\chi^{-1})^{-,fs}$

is surjective.

  4. (4)

There is a pairing

$\overline{\mathrm{H}}^{\ell_{+}(w)}_{w,an}(K^{p},\nu^{un})^{+,fs}\times\overline{\mathrm{H}}^{\ell_{-}(w)}_{w,an}(K^{p},\nu^{un})^{-,fs}\rightarrow\mathscr{O}_{\mathcal{W}}$

compatible with the preceding maps and the pairings

$\overline{\mathrm{H}}^{\ell_{+}(w)}_{w,an}(K^{p},\nu)^{+,fs}\times\overline{\mathrm{H}}^{\ell_{-}(w)}_{w,an}(K^{p},\nu)^{-,fs}\rightarrow
k(\nu)$

and

$\overline{\mathrm{H}}^{\ell_{+}(w)}_{w}(K^{p},\kappa,\chi)^{+,fs}\times\overline{\mathrm{H}}^{\ell_{-}(w)}_{w}(K^{p},-w_{0,M}\kappa-2\rho_{nc},\chi^{-1})^{-,fs}\rightarrow
F.$

###### Proof.

The first point follows from theorem 6.7.3. The vanishing theorem and Tor
spectral sequence imply that
${\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un},cusp)^{\pm,fs}\otimes_{\mathscr{O}_{\mathcal{W}}}k(\nu)\rightarrow{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu,cusp)^{\pm,fs}$
is an isomorphism, which gives the second point. The vanishing theorems and
the BGG spectral sequence imply that
${\mathrm{H}}^{\ell_{+}(w)}_{w}(K^{p},\kappa,\chi)^{+,fs}\rightarrow{\mathrm{H}}^{\ell_{+}(w)}_{w,an}(K^{p},\nu)^{+,fs}$
is injective and
${\mathrm{H}}^{\ell_{-}(w)}_{w,an}(K^{p},\nu,cusp)^{-,fs}\rightarrow\overline{\mathrm{H}}^{\ell_{-}(w)}_{w}(K^{p},-w_{0,M}\kappa-2\rho_{nc},\chi^{-1},cusp)^{-,fs}$
is surjective, which gives the third point follows. The last point is clear. ∎

######  Remark 6.9.5.

There is an asymmetry between the $-$ and $+$ theories in point (3). This
asymmetry results from our choice to develop the $+$ theory using locally
analytic induction sheaves and the $-$ theory using their dual distribution
sheaves.

We let $\mathcal{Z}^{!}_{w}$, $\mathcal{E}^{!}_{w}$ be the supports of
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$
and
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu^{un})_{\mathcal{Z}}^{\pm,fs}$.
We let $\mathcal{Z}^{!}$ and $\mathcal{E}^{!}$ be the union of the
$\mathcal{Z}^{!}_{w}$ and $\mathcal{E}^{!}_{w}$.

######  Theorem 6.9.6.

The following holds:

  1. (1)

The eigenvariety $\mathcal{E}^{!}$ and the spectral variety $\mathcal{Z}^{!}$
are either empty or equidimensional of dimension $\dim\mathcal{W}$. The same
is true of the $\mathcal{E}^{!}_{w}$ and $\mathcal{Z}^{!}_{w}$ for each
$w\in{}^{M}W$.

  2. (2)

For all $w\in{}^{M}W$, $\kappa_{alg}\in X^{\star}(T^{c})^{M,+}$, and
$\chi:T^{c}(\mathbb{Z}_{p})\rightarrow\overline{F}^{\times}$ a finite order
character, let $\nu_{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$. Any
eigenclass in
$\overline{\mathrm{H}}^{\ell_{-}(w)}_{w}(K^{p},\kappa^{\vee}_{alg},\chi^{-1})^{-,fs}$
or $\overline{\mathrm{H}}^{\ell_{+}(w)}_{w}(K^{p},\kappa_{alg},\chi)^{+,fs}$
gives a point of $\mathcal{E}_{w}^{!}$ of weight $\nu=\nu_{alg}\chi$.

  3. (3)

Let $\kappa_{alg}\in X^{\star}(T^{c})^{M,+}$ and
$\chi:T^{c}(\mathbb{Z}_{p})\to\overline{F}^{\times}$ be a finite order
character. For any eigenclass $c$ in
$\overline{\mathrm{H}}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$ or
$\overline{\mathrm{H}}^{i}(K^{p},\kappa_{alg}^{\vee},\chi^{-1})^{-,fs}$ there
is a $w\in{}^{M}W$ with $\ell_{\pm}(w)=i$ so that $c$ gives a point of
$\mathcal{E}_{w}^{!}$ of weight $\nu=\nu_{alg}\chi$ where
$\nu_{alg}=-w^{-1}w_{0,M}(\kappa_{alg}+\rho)-\rho$.

###### Proof.

This first two points follow from proposition 6.9.4. The third point follows
from the second point and corollary 5.8.1. ∎

This theorem in particular implies that any finite slope system of Hecke
eigenvalues in classical or overconvergent interior cohomology admits an
analytic deformation of maximum dimension.

In light of point (3) of the theorem, it is natural to ask: given an
eigenclass $c$ occurring in
$\overline{\mathrm{H}}^{i}(K^{p},\kappa_{alg},\chi)^{+,fs}$, what is the set
${}^{M}W(c)$ of $w\in{}^{M}W$ with $\ell_{+}(w)=i$ for which $c$ occurs in
$\overline{\mathrm{H}}^{i}_{w}(K^{p},\kappa_{alg},\chi)$? It is natural to
compare ${}^{M}W(c)$ to the set $C(\kappa_{alg})^{+}$, i.e. the set of
$w\in{}^{M}W$ for which $\nu_{alg}+\rho\in X^{\star}(T^{c})^{+}$.

######  Proposition 6.9.7.

Assume that the eigenclass $c$ satisfies $+,ss^{M}(\kappa_{alg})$. Then
${}^{M}W(c)\subseteq C(\kappa)^{+}$.

###### Proof.

By theorem 6.10.1 (2) below,
$\overline{\mathrm{H}}_{w}^{i}(K^{p},\kappa_{alg},\chi)^{+,ss^{M}(\kappa)}=0$
unless $w\in C(\kappa)^{+}$. ∎

In particular if $C(\kappa)^{+}$ has only one element of length $i$ (for
example if $\kappa+\rho$ is regular) then ${}^{M}W(c)$ is determined under
this small slope hypothesis.

######  Example 6.9.8.

We give a basic counter-example to the proposition if we drop the small slope
hypothesis. We assume the Shimura variety is compact, so that the constant
function $1\in H^{0}(S_{K},\mathscr{O})$ defines a finite slope interior
cohomology class. Then ${}^{M}W(1)=\\{Id\\}$, and the corresponding point of
the eigenvariety $\mathcal{E}$ is in weight
$\nu_{alg}=-w_{0,M}\rho-\rho=-2\rho_{nc}$, and so in particular
$\nu_{alg}+\rho$ is not dominant. On the other hand we have
$C(0)^{+}=\\{w_{0}^{M}\\}$.

######  Remark 6.9.9.

If $c$ is a cohomology class represented by an automorphic representation
$\pi$ which is tempered at $\infty$ and contributes to coherent cohomology in
weight $\kappa$, then $\pi_{\infty}$ is a limit of discrete series which is
described by the pair consisting of its infinitesimal character $-\kappa-\rho$
and a chamber $wX^{\star}(T)^{+}_{\mathbb{Q}}\subseteq
X^{\star}(T)^{M,+}_{\mathbb{Q}}$ for $w\in C(\kappa)^{+}$ (see [Har90a], sect.
3.3). We can ask if this class lifts to a point in $\mathcal{E}_{w}^{!}$.

###  6.10. Improved slope bounds for interior cohomology and applications

Using the interior eigenvariety we are able to prove that the conjectured
slopes bounds 6.8.1, 5.9.2, and 5.10.7 hold for interior cohomology. The idea
is that classical points in regular weight satisfy the correct slope bound by
corollary 5.10.12, and these points are dense in the interior eigenvariety.

######  Theorem 6.10.1.

  1. (1)

Fix $w\in{}^{M}W$ and a locally algebraic weight $\nu=\nu_{alg}\chi$ for
$\nu_{alg}\in X^{\star}(T^{c})$ and
$\chi:T^{c}(\mathbb{Z}_{p})\to\overline{F}^{\times}$ a finite order character.
For any character $\lambda$ of $T^{\pm}$ on
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w,an}(K^{p},\nu)^{\pm,fs}$ we have
$v(\lambda)\geq 0$ in the $+$ case and $v(\lambda)\leq 0$ in the $-$ case.

  2. (2)

Fix $w\in{}^{M}W$, $\kappa\in X^{\star}(T^{c})^{M,+}$, and a finite order
character $\chi:T^{c}(\mathbb{Z}_{p})\to F^{\times}$. For any character
$\lambda$ of $T^{\pm}$ on
$\overline{\mathrm{H}}^{\ell_{\pm}(w)}_{w}(K^{p},\kappa,\chi)^{\pm,fs}$ we
have $v(\lambda)\geq w^{-1}w_{0,M}(\kappa+\rho)+\rho$ in the $+$ case and
$v(\lambda)\leq w^{-1}(\kappa+\rho)-\rho$ in the $-$ case.

  3. (3)

Fix $\kappa\in X^{\star}(T^{c})^{M,+}$ and a finite order character
$\chi:T^{c}(\mathbb{Z}_{p})\to\overline{F}^{\times}$. Let
$\nu=-w^{-1}w_{0,M}(\kappa+\rho)-\rho$ for any $w\in C(\kappa)^{+}$. Then for
any character $\lambda$ of $T\pm$ on
$\overline{\mathrm{H}}^{i}(K^{p},\kappa,\chi)^{\pm,fs}$ we have
$v(\lambda)\geq-\nu$ in the $+$ case and $v(\lambda)\leq-w_{0}\nu$ in the $-$
case.

###### Proof.

In the $-$ case, the first point implies the second point by theorem 6.9.6
(3). In the $+$ case, one repeats the constructions of this section,
exchanging the roles of the sheaves of locally analytic functions and
distributions. The second point implies the third point as in the proof of
proposition 5.10.9 using corollary 5.8.1.

We now prove the first point. The eigensystem $\lambda$ gives a point
$(\nu,\lambda_{p},\lambda^{S})$ of $\mathcal{E}^{!}$ (where
$\lambda_{p}=\lambda$ in the $+$ case and $\lambda^{t}$ in the $-$ case.) We
can find another point
$(\nu^{\prime},\lambda_{p}^{\prime},\lambda^{{}^{\prime}S})$ which satisfies
$v(\lambda^{\prime}_{p})=v(\lambda_{p})$ and
$\nu^{\prime}=\nu^{\prime}_{alg}\chi$ is locally algebraic, and
$\nu^{\prime}_{alg}\in X^{\star}(T^{c})^{+}$ is sufficiently large so that
$v(\lambda^{\prime}_{p})$ satisfies $+,sss(\nu)$. Then this point is classical
by theorem 6.9.3 (2), and so the slope bound is satisfied by corollary
5.10.12. ∎

As a consequence we deduce a vanishing theorem for interior cohomology which
improves on theorem 5.12.5.

######  Theorem 6.10.2.

Let $\kappa\in X^{\star}(T^{c})^{M,+}$ and let
$\chi:T^{c}(\mathbb{Z}_{p})\rightarrow\overline{F}^{\times}$ be a finite order
character. We have that
$\overline{\mathrm{H}}^{i}(K^{p},\kappa,\chi)^{ss^{M}(\kappa)}$ is supported
in the range $[\ell_{\min}(\kappa),\ell_{\max}(\kappa)]$.

###### Proof.

Theorem 6.10.1 (2) implies that the complex
$\overline{\mathcal{C}ous}(K^{p},\kappa,\chi)^{\pm,ss^{M}(\kappa)}$ is
concentrated in the range $[\ell_{\min}(\kappa),\ell_{\max}(\kappa)]$, which
implies the theorem by corollary 5.8.1 ∎

As another application we are able to improve our classicality theorem for
overconvergent cuspforms.

######  Theorem 6.10.3.

Let $\kappa\in X^{\star}(T^{c})^{M,+}$ satisfy $C(\kappa)^{+}=\\{1\\}$, and
let $\chi:T^{c}(\mathbb{Z}_{p})\to\overline{F}^{\times}$ be a finite order
character. Then
$\mathrm{H}^{0}_{Id}(K^{p},\kappa,\chi,cusp)^{+,ss^{M}(\kappa)}\to\mathrm{H}^{0}_{Id}(K^{p},\kappa,\chi)^{+,ss^{M}(\kappa)}$
factors through $\mathrm{H}^{0}(K^{p},\kappa,\chi)^{+,ss^{M}(\kappa)}$.

###### Proof.

We have a commutative diagram

where the bottom row is exact by proposition 5.7.4. After passing to the
$+,ss^{M}(\kappa)$ part of this diagram, the right vertical arrow is 0 by
theorem 6.10.2. ∎

######  Remark 6.10.4.

Theorem 6.10.3 says that a small slope overconvergent cuspform comes from a
classical eigenform. Although we have not proved it, this eigenform should
also be cuspidal (for instance this would follow from conjeture 5.9.2). We
note that it is possible for an overconvergent cuspform which does not have
small slope to come from a non cuspidal eigenform. A basic example is the non-
ordinary $p$-stabilization of an Eisenstein series for $\mathrm{GL}_{2}$.

###  6.11. Application: construction of Galois representations and local-
global compatibility $p$

Let $L$ be a totally real or CM number field. The class of regular, algebraic,
essentially (conjugate) self dual, cuspidal automorphic representations for
$\mathrm{GL}_{n}/L$ have been studied intensively. In particular, one can
attach to them compatible systems of Galois representations which satisfy many
of the expected properties (see, e.g., [CH13], [BLGGT14]):

######  Theorem 6.11.1 (Bellaiche, Caraiani, Chenevier, Clozel, Harris,
Kottwitz, Labesse, Shin, Taylor, …).

Let $\pi$ be a regular, algebraic, essentially (conjugate) self dual, cuspidal
automorphic representation of $\mathrm{GL}_{n}/L$ with
$\pi^{c}=\pi^{\vee}\otimes\chi$ and infinitesimal character
$\lambda=((\lambda_{1,\tau},\ldots,\lambda_{n,\tau})_{\tau\in\mathrm{Hom}(L,\mathbb{C})})$
with $\lambda_{1,\tau}>\cdots>\lambda_{n,\tau}$. Then for each isomorphism
$\iota:\overline{\mathbb{Q}}_{p}\simeq\mathbb{C}$ there is a continuous Galois
representation
$\rho_{\pi,\iota}:G_{L}\rightarrow\mathrm{GL}_{n}(\overline{\mathbb{Q}}_{p})$
such that:

  1. (1)

$\rho_{\pi,\iota}^{c}\simeq\rho_{\pi,\iota}^{\vee}\otimes\epsilon_{p}^{1-n}\otimes\chi_{\iota}$
where $\chi_{\iota}$ is the $p$-adic realization of $\chi$ and $\epsilon_{p}$
is the cyclotomic character.

  2. (2)

$\rho_{\pi,\iota}$ is pure.

  3. (3)

$\rho_{\pi,\iota}$ is de Rham at all places dividing $p$, with
$\iota^{-1}\circ\tau$-Hodge-Tate weights:
$(-\lambda_{n,\tau}+\frac{n-1}{2},\cdots,-\lambda_{1,\tau}+\frac{n-1}{2})$.

  4. (4)

For all finite place $v$ one has

$\iota\mathrm{WD}(\rho_{\pi,\iota}|_{G_{F_{v}}})^{F-ss}=\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

The Galois representations of this theorem are usually found in the étale
cohomology of unitary Shimura varieties, while in the remaining cases they are
constructed by $p$-adic interpolation.

In [FP19], section 9, we defined a certain class of cuspidal automorphic
representations for the group $\mathrm{GL}_{n}/L$ which (when $L$ is CM)
realize in the coherent cohomology of unitary Shimura varieties, by weakening
the condition of regular to weakly regular and odd. To these weakly regular,
odd, algebraic, essentially (conjugate) self dual, cuspidal automorphic
representations, one can still attach compatible system of Galois
representations ([Box15], [GK19], [PS16]), but at present many of their
expected properties are not known. The techniques of this paper allows for a
new construction of the Galois representation via analytic families. The
advantage of this construction is that we can prove some instances of local-
global compatibility at $p$, using results of Kisin [Kis03] on the
interpolation of crystalline periods in analytic families, as in the work of
Jorza and Mok [Jor12], [Mok14].

######  Theorem 6.11.2.

Let $\pi$ be a weakly regular, odd, algebraic, essentially (conjugate) self
dual, cuspidal automorphic representation of $\mathrm{GL}_{n}/L$ with
$\pi^{c}=\pi^{\vee}\otimes\chi$ and infinitesimal character
$\lambda=((\lambda_{1,\tau},\ldots,\lambda_{n,\tau})_{\tau\in\mathrm{Hom}(L,\mathbb{C})})$
with $\lambda_{1,\tau}\geq\cdots\geq\lambda_{n,\tau}$. Then for each
isomorphism $\iota:\overline{\mathbb{Q}}_{p}\simeq\mathbb{C}$ there is a
continuous Galois representation
$\rho_{\pi,\iota}:G_{L}\rightarrow\mathrm{GL}_{n}(\overline{\mathbb{Q}}_{p})$
such that:

  1. (1)

$\rho_{\pi,\iota}^{c}\simeq\rho^{\vee}\otimes\epsilon_{p}^{1-n}\otimes\chi_{\iota}$
where $\chi_{\iota}$ is the $p$-adic realization of $\chi$.

  2. (2)

$\rho_{\pi,\iota}$ is unramified at all finite places $v\nmid p$ for which
$\pi_{v}$ is unramified and one has

$\iota
WD(\rho_{\pi,\iota}|_{G_{L_{v}}})^{F-ss}=\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

  3. (3)

$\rho_{\pi,\iota}$ has generalized $\iota^{-1}\circ\tau$-Hodge–Tate weights
$(-\lambda_{n,\tau}+\frac{n-1}{2},\cdots,-\lambda_{1,\tau}+\frac{n-1}{2})$.

  4. (4)

Let $v\mid p$ be a place of $L$ and assume that $\pi_{v}$ is a regular
principal series (i.e. the Jacquet module $(\pi_{v})_{U}$ is a direct sum of
$n!$ distinct characters of $T(L_{v})$). Then $\rho_{\pi,\iota}|_{G_{L_{v}}}$
is potentially crystalline and

$\iota
WD(\rho_{\pi,\iota}|_{G_{L_{v}}})^{F-ss}=\mathrm{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

###### Proof.

The existence of a Galois representation satisfying points (1) and (2) is
already known (see [FP19], theorem 9.10). In order to prove points (3) and
(4), by base change we may assume that $L$ is CM with maximal totally real
subfield $F$, all the primes above $p$ in $F$ split in $L$, and for all primes
$v\mid p$ of $L$, $\pi_{v}$ is a constituent of a principal series
representation (and so has finite slope in the sense of section 4.3).

Let $\mathrm{GU}(n)/F$ be the quasi-split unitary similitude group in $n$
variables, and let $G\subset\mathrm{Res}_{F/\mathbb{Q}}$ be the subgroup where
the similitude factor lands in
$\mathbb{G}_{m}\subseteq\mathrm{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}$. The group
$G$ admits a PEL Shimura datum $(G,X)$, so that $\pi$ realizes in the interior
coherent cohomology of the corresponding Shimura variety, as explained in the
proof of theorem 9.11 of [FP19] (we note that this is conditional on the main
results of [Mok15]).

The result now follows from theorems 6.9.6 and 6.11.1, using the results of
[Kis03] as explained in proof of theorem 4.1 of [Jor12]. ∎

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