eigenvalues for such operators. Up to extending coefficients to a field $L$
such that $k_{z}\subseteq L$, since $\mathscr{U}$ is an interior good
neighborhood, we have by Lemma 5

$\dim_{L}\left(\mathscr{H}_{!}^{\varepsilon}(\mathscr{V}_{\lambda}(L))[z]\right)=\dim_{L}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda}(L))[z]\right)=\sigma(\mathfrak{n}_{0}/\mathfrak{n}_{\mathscr{U}})$

for all $z$. We also have
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda}(L))[z]\simeq\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda}(L))_{\mathfrak{m}_{z}}$
for all $z$ (see the proof of [BH, Lemma 8.2.1]). The fiber of
$\mathscr{M}_{c}^{\varepsilon}(\mathscr{U})$ over $\lambda$ is
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda})_{\leq\nu}$, which is
isomorphic to $\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda})_{\leq\nu}$
by Theorem 3. Since $w:\mathscr{U}\rightarrow\Omega$ is étale at all $z$ of
degree $d_{\mathscr{U}}$, we have
$\dim_{L}\left((\mathscr{M}_{c}^{\varepsilon}(\mathscr{U})/\mathfrak{m}_{\lambda}\mathscr{M}_{c}^{\varepsilon}(\mathscr{U}))\otimes_{k_{\lambda}}L\right)=\sigma(\mathfrak{n}_{0}/\mathfrak{n}_{\mathscr{U}})\cdot
d_{\mathscr{U}}$ by (22). On the other hand,
$\dim_{L}\left((\mathscr{M}_{c}^{\varepsilon}(\mathscr{U})/\mathfrak{m}_{\lambda}\mathscr{M}_{c}^{\varepsilon}(\mathscr{U}))\otimes_{k_{\lambda}}L\right)=r_{\mathscr{U}}^{\varepsilon}$
by definition, concluding the proof. ∎

######  Definition 11.

A connected neighborhood $\mathscr{U}\subseteq\mathscr{X}^{\mathrm{fl}}$ of a
flat point $x\in\mathscr{X}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$ of tame
level $n_{x}$ is said to be _clean_ if:

  1. (1)

All classical points in $\mathscr{U}$ have constant tame level $n_{x}$;

  2. (2)

$r_{\mathscr{U}}^{\varepsilon}=\sigma\left(\mathfrak{n}_{0}/\mathfrak{n}_{x}\right)\cdot
d_{\mathscr{U}}$.

  3. (3)

$x$ is the only point in $\mathscr{U}$ above $w(x)$.

######  Lemma 8.

Interior very decent points where the weight map is flat have a clean
neighborhood.

###### Proof.

The intersection $\mathscr{U}$ between two connected neighborhoods choosen as
in Lemma 4 and Lemma 7 satisfies (1) and (2) in Definition 11. Since the
weight map is finite, up to shrinking $\mathscr{U}$ we see that also condition
(3) is satisfied. ∎

######  Lemma 9.

Non-critical points are flat.

###### Proof.

By [BDJ, Th. 2.20(ii)], one can choose affinoids
$\mathrm{Sp}(T)\subseteq\mathscr{X}$ and $\mathrm{Sp}(R)\subseteq\mathscr{W}$
in such a way that $\mathrm{Sp}(T)\rightarrow\mathrm{Sp}(R)$ is an
isomorphism. ∎

######  Corollary 10.

Interior very decent non-critical points have a clean neighborhood.

###### Proof.

Combine Lemma 8 and Lemma 9. ∎

######  Proposition 11.

Let $x$ be a very decent interior point of weight $\lambda=w(x)$ and tame
level $\mathfrak{n}_{0}$. Then the specialisation map
$\rho_{\lambda}:\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))[x]\rightarrow\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda}(k_{x}))[x]$
is an isomorphism of $1$-dimensional spaces if $w$ is unramified at $x$, and
is the trivial map if $w$ is ramified at $x$, in which case $x$ is critical.

###### Proof.

As recalled above, the source is $1$-dimensional by [BH, Theorem 1.5.3]. Since
$x$ is a classical interior point,
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{V}_{\lambda}(k_{x}))[x]\simeq\mathscr{H}_{!}^{\varepsilon}(\mathscr{V}_{\lambda}(k_{x}))[x]$
and the target is $1$-dimensional by Lemma 5. Moreover, the map
$\rho_{\lambda}:\mathscr{H}_{c}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\rightarrow\mathscr{H}_{c}(\mathscr{V}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}$
is surjective by Proposition 6, and
$\mathscr{H}_{c}(\mathscr{V}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\simeq\mathscr{H}_{c}(\mathscr{V}_{\lambda}(k_{x}))[x]$
by [BH, Lemma 8.2.1]. So,
$\rho_{\lambda}:\mathscr{H}_{c}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\twoheadrightarrow\mathscr{H}_{!}(\mathscr{V}_{\lambda}(k_{x}))[x]$
is surjective onto a $1$-dimensional $k_{x}$-vector space, and the source of
$\rho_{\lambda}$ contais the $1$-dimensional $k_{x}$-vector space
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))[x]$. Fix a clean
interior neighborhood $\mathscr{U}$ of $x$. Since $x$ is the only point of
$\mathscr{U}$ above $w(x)$, by (22) we have
$\dim_{k_{x}}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\right)=r_{\mathscr{U}}$.
We also have $r_{\mathscr{U}}=d_{\mathscr{U}}$ by Lemma 7. If $w$ is
unramified at $x$, then $d_{\mathscr{U}}=1$, so
$\dim_{k_{x}}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\right)=1$.
Therefore the surjective map $\rho_{\lambda}$ is an isomorphism, and we have
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}=\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))[x]$.
If $x$ is ramified and critical, then
$\rho_{\lambda}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))[x]\right)=0$
by [BH, Lemma 8.2.1]. It remains to show that non-critical points are
unramified. Indeed, $\rho_{\lambda}$ is an isomorphism by Theorem 3, so
$\dim_{k_{x}}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}\right)=1$,
and therefore $r_{\mathscr{U}}=1$, proving that $x$ is unramified and that we
have
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))_{\mathfrak{m}_{x}}=\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\lambda}(k_{x}))[x]$.
∎

######  Corollary 12.

The weight map is étale at very decent interior non-critical points of tame
level $\mathfrak{n}_{0}$.

###### Proof.

As recalled above, all decent points are smooth, hence flat, and interior non-
critical very decent points are unramifed by Proposition 11. ∎

######  Remark.

In Theorem 28 we will strength Corollary 12 by considering ramification
properties at other decent points.

###  4.11. The $\ell$-stratum middle degree eigenvariety

Fix an integer $\ell$. In this subsection we define a rigid Zariski closed
subset $\mathscr{W}_{\ell}\subseteq\mathscr{W}$ containing all classical
points $\lambda=(\lambda_{1},\lambda_{2})$ of $\mathscr{W}$ such that
$\sum_{i}\lambda_{2,i}=\ell$, and the rigid analytic space
$\mathscr{X}_{\ell}=\mathscr{X}\times_{\mathscr{W}}\mathscr{W}_{\ell}$. We
begin with the definition of $\mathscr{W}_{\ell}$.

######  Definition 12.

Let $\ell$ be an integer. The _$\ell$ -stratum_ $\mathscr{W}_{\ell}$ of
$\mathscr{W}$ is the rigid Zariski closure in $\mathscr{W}$ of the set
$\mathscr{W}_{\ell}^{\mathrm{cl}}$ of classical weights
$\lambda=(\lambda_{1},\lambda_{2})$ such that $\sum_{i}\lambda_{2,i}=\ell$.

We equip $\mathscr{W}_{\ell}$ with the structure of rigid analytic space
induced the closed immersion
$j_{\ell}:\mathscr{W}_{\ell}\hookrightarrow\mathscr{W}$; thus, a basis of open
subsets of $\mathscr{W}_{\ell}$ is given by the sets
$\mathscr{U}_{\ell}=\mathscr{W}_{\ell}\cap\mathscr{U}$ where $\mathscr{U}$ is
an open affinoid subdomain of $\mathscr{W}$. Any classical point $\lambda$
belongs to exactly one $\mathscr{W}_{\ell}$, so the union
$\cup_{\ell}\mathscr{W}_{\ell}^{\mathrm{cl}}$ is disjoint and equal to
$\mathscr{W}^{\mathrm{cl}}$. The rigid analytic closure of
$\cup_{\ell}\mathscr{W}_{\ell}^{\mathrm{cl}}$ is therefore equal to
$\mathscr{W}$, so the rigid analytic closure of
$\cup_{\ell}\mathscr{W}_{\ell}$ is also equal to $\mathscr{W}$.

######  Lemma 13.

$\mathscr{W}_{\ell_{1}}\cap\mathscr{W}_{\ell_{2}}=\emptyset$ if
$\ell_{1}\neq\ell_{2}$.

###### Proof.

Let $H$ be the subgroup of $T_{0}(\mathbb{Z}_{p})$ consisting of matrices of
the form $\bigl{(}\begin{smallmatrix}1&0\\\ 0&a\end{smallmatrix}\bigr{)}$ for
$a\in\mathbb{Z}_{p}^{\times}$. Embed diagonally $T_{0}(\mathbb{Z}_{p})$ in
$T(\mathbb{Z}_{p})$. Composing the map
$a\mapsto\bigl{(}\begin{smallmatrix}1&0\\\ 0&a\end{smallmatrix}\bigr{)}$ with
the diagonal embedding of $T_{0}(\mathbb{Z}_{p})$ in $T(\mathbb{Z}_{p})$ we
obtain a map $\delta:\mathbb{Z}_{p}^{\times}\rightarrow T(\mathbb{Z}_{p})$.
For any $\kappa:T(\mathbb{Z}_{p})\rightarrow\mathbb{Q}_{p}^{\times}$, we may
then consider the homomorphism
$\kappa\circ\delta:\mathbb{Z}_{p}^{\times}\rightarrow\mathbb{Q}_{p}^{\times}$.
If $\lambda\in\mathscr{W}_{\ell}^{\mathrm{cl}}$, then $\lambda\circ\delta$ is
the endomorphism of $\mathbb{Z}_{p}^{\times}$ given by $a\mapsto a^{\ell}$.
Therefore, for any $\kappa\in\mathscr{W}_{\ell}$, $\kappa\circ\delta$ is also
equal to the endomorphism $a\mapsto a^{\ell}$ of $\mathbb{Z}_{p}^{\times}$. If
$\kappa\in\mathscr{W}_{\ell_{1}}\cap\mathscr{W}_{\ell_{2}}$, then
$\kappa\circ\delta$ is equal to both $a\mapsto a^{\ell_{1}}$ and to $a\mapsto
a^{\ell_{2}}$, which implies that $\ell_{1}=\ell_{2}$. ∎

Define $\mathscr{X}_{\ell}=\mathscr{X}\times_{\mathscr{W}}\mathscr{W}_{\ell}$,
where the fiber product is taken in the category of rigid analytic varieties
with respect to the weight map $w:\mathscr{X}\rightarrow\mathscr{W}$ and the
rigid Zariski closed embedding
$j_{\ell}:\mathscr{W}_{\ell}\hookrightarrow\mathscr{W}$; thus, a basis of open
subsets of $\mathscr{X}_{\ell}$ for the rigid analytic topology is given by
$\mathrm{Sp}(T\hat{\otimes}_{R}S)$, where $\mathrm{Sp}(T)$ is an affinoid
subdomain of $\mathscr{X}$, $\mathrm{Sp}(R)$ is an open affinoid subdomain of
$\mathscr{W}$ and $\mathrm{Sp}(S)$ is an open affinoid subdomain of
$\mathscr{W}_{\ell}$ such that
$j_{\ell}(\mathrm{Sp}(S))\subseteq\mathrm{Sp}(R)$. Let
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ be the projection
map. Since $j_{\ell}:\mathscr{W}_{\ell}\hookrightarrow\mathscr{W}$ is a closed
immersion, the canonical map
$i_{\ell}:\mathscr{X}_{\ell}\hookrightarrow\mathscr{X}$ is also a closed
immersion ([Bos14, §4.2]). For any slope adapted pair $(\Omega,\nu)$, define
$\Omega_{\ell}=\Omega\cap\mathscr{W}_{\ell}$. Since $\mathscr{W}_{\ell}$ is
rigid Zariski closed in $\mathscr{W}$, $\Omega_{\ell}$ is rigid Zariski closed
in $\Omega$. Define
$\mathbb{T}_{\Omega_{\ell}}=\mathbb{T}_{\Omega,\nu}\otimes_{\mathscr{O}_{\mathscr{W}}(\Omega)}\mathscr{O}_{\mathscr{W}}(\Omega_{\ell})$.
Define the complex of coherent sheaves of
$\mathscr{O}_{\mathscr{X}_{\ell}}$-modules
$\mathscr{M}_{\ell}^{\bullet}=i_{\ell}^{*}(\mathscr{M}_{c}^{\bullet})$ on
$\mathscr{X}_{\ell}$ and the coherent sheaf of
$\mathscr{O}_{\mathscr{X}_{\ell}}$-modules
$\mathscr{M}_{\ell}=i_{\ell}^{*}(\mathscr{M}_{c})$ on $\mathscr{X}_{\ell}$. If
$\mathscr{U}=\mathrm{Sp}(T)\hat{\otimes}_{R}S$, then
$\mathscr{M}_{\ell}(\mathscr{U})=\mathscr{M}_{c}(\mathrm{Sp}(T))\hat{\otimes}_{R}S$.

######  Lemma 14.

If $(\Omega,\nu)$ is a slope adapted pair, then
$\mathscr{M}_{\ell}(\mathbb{T}_{\Omega_{\ell}})=H^{d}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu}$
and $H^{i}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu}=0$ if $i\neq d$.

###### Proof.

For the first statement, it is enough to show that
$H^{j}_{c}(Y_{K},\mathscr{D}_{\Omega})_{\leq\nu}\otimes_{\mathscr{O}_{\mathscr{W}}(\Omega)}\mathscr{O}_{\mathscr{W}}(\Omega_{\ell})\simeq
H^{j}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu}$. By [Han, Th. 3.3.1]
we have a convergent second quadrant spectral sequence (for $j\geq 0$ and
$i\leq 0$ integers)

(23)
$E^{i,j}_{2}=\mathrm{Tor}_{-i}^{\mathscr{O}_{\mathscr{W}}(\Omega)}\left(H^{j}_{c}(Y_{K},\mathscr{D}_{\Omega})_{\leq\nu},\mathscr{O}_{\mathscr{W}}(\Omega_{\ell})\right)\Longrightarrow
H^{i+j}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu}.$

Localising at the maximal ideal $\mathfrak{m}_{x}$ corresponding to a point
$x$, all term $E^{i,j}_{2}$ vanish by the definition of $\mathscr{X}$, except
the term $E_{2}^{0,d}$, which induces the sough-for isomorphism at any stalk,
and therefore the claimed canonical isomorphism of sheaves. On the other hand,
again localising the Tor exact sequence 23 at $\mathfrak{m}_{x}$ shows that
the highest degree for which the support of
$H^{i}(Y_{K},\mathscr{D}_{\Omega_{\ell}})$ is non empty is $i=d$. On the other
hand, by [Han, Th. 3.3.1] we also have convergent second quadrant Ext spectral
sequence (for $j\geq 0$ and $i\geq 0$ integers)

(24)
$E^{i,j}_{2}=\mathrm{Ext}^{i}_{\mathscr{O}_{\mathscr{W}}(\Omega)}\left(H_{j}^{\mathrm{BM}}(Y_{K},\mathscr{D}_{\Omega})_{\leq\nu},\mathscr{O}_{\mathscr{W}}(\Omega_{\ell})\right)\Longrightarrow
H^{i+j}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu}.$

Again localising at $\mathfrak{m}_{x}$, all terms $E^{i,j}_{2}$ vanish except
$E_{2}^{0,d}$, which shows that the lower degree for which the support of
$H^{i}(Y_{K},\mathscr{D}_{\Omega_{\ell}})$ is non empty is $i=d$. Therefore,
the support of $H^{i}(Y_{K},\mathscr{D}_{\Omega_{\ell}})$ is empty for $i\neq
d$, which implies the statement. ∎

Recall the algebra homomorphism
$\phi_{\mathscr{X}}:\mathbb{T}\rightarrow\mathscr{O}_{\mathscr{X}}(\mathscr{X})$
and denote
$\phi_{\mathscr{X}_{\ell}}:\mathbb{T}\rightarrow\mathscr{O}_{\mathscr{X}_{\ell}}(\mathscr{X}_{\ell})$
the algebra homomorphism obtained by composing $\phi_{\mathscr{X}}$ with the
canonical restriction map
$\mathscr{O}_{\mathscr{X}}(\mathscr{X})\rightarrow\mathscr{O}_{\mathscr{X}_{\ell}}(\mathscr{X}_{\ell})$.
Recall also the canonical maps
$\psi_{\Omega,\nu}:\mathbb{T}\rightarrow\mathrm{End}_{\mathbb{T}_{\Omega,\nu}}\left(\mathscr{M}_{c}^{\bullet}(\mathrm{Sp}(\mathbb{T}_{\Omega,\nu}))\right)$
for $(\Omega,\nu)$ a slope adapted pair. From
$\psi_{\Omega,\nu}:\mathbb{T}\rightarrow\mathrm{End}_{\mathbb{T}_{\Omega,\nu}}\left(\mathscr{M}_{c}(\mathrm{Sp}(\mathbb{T}_{\Omega,\nu}))\right)$
we obtain a map
$\psi_{\Omega_{\ell}}:\mathbb{T}\rightarrow\mathrm{End}_{\mathbb{T}_{\Omega_{\ell}}}\left(\mathscr{M}_{\ell}(\mathrm{Sp}(\mathbb{T}_{\Omega_{\ell}}))\right)$
which, by Lemma 14 gives rise to a map
$\psi_{\Omega_{\ell}}:\mathbb{T}\rightarrow\mathrm{End}_{\mathbb{T}_{\Omega_{\ell}}}\left(H^{d}_{c}(Y_{K},\mathscr{D}_{\Omega_{\ell}})_{\leq\nu})\right)$;
these maps glue together to get a map
$\psi_{\ell}:\mathbb{T}\rightarrow\mathrm{End}_{\mathscr{O}_{\mathscr{X}_{\ell}}}(\mathscr{M}_{\ell})$.
For each open subset $\mathscr{U}\subseteq\mathscr{X}_{\ell}$, let
$\mathbb{T}_{\mathscr{U}}$ be the image of $\psi_{\ell}$ in
$\mathrm{End}_{\mathscr{O}_{\mathscr{X}_{\ell}}}(\mathscr{M}_{\ell}(\mathscr{U}))$.

######  Definition 13.

Let $\ell$ be an integer. The _$\ell$ -stratum middle degree eigenvariety_ is
the rigid analytic space
$\mathscr{X}_{\ell}=\mathscr{X}\times_{\mathscr{W}}\mathscr{W}_{\ell}$
equipped with the canonical projection map
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$, the coherent sheaf
of coherent $\mathscr{O}_{\mathscr{X}_{\ell}}$-modules $\mathscr{M}_{\ell}$
and the map
$\psi_{\ell}:\mathbb{T}\rightarrow\mathrm{End}_{\mathscr{O}_{\mathscr{X}_{\ell}}}(\mathscr{M}_{\ell})$.

As before, for any point $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$,
write $\phi_{\mathscr{X}_{\ell},x}:\mathbb{T}\rightarrow k_{x}$ for the
composite composition of $\phi_{\mathscr{X}_{\ell}}$ with the canonical map
$\mathscr{O}_{\mathscr{X}_{\ell}}(\mathscr{X}_{\ell})\rightarrow\mathscr{O}_{\mathscr{X}_{\ell},x}\twoheadrightarrow{}k_{x}$,
where $\mathscr{O}_{\mathscr{X}_{\ell},x}$ is the stalk of
$\mathscr{O}_{\mathscr{X}_{\ell}}$ at $x$ and $k_{x}$ its residue field. We
say that $\phi_{\mathscr{X}_{\ell},x}$ is the _eigenpacket_ parameterized by
$x$ and denote again $\mathfrak{m}_{x}=\ker(\phi_{\mathscr{X}_{\ell},x})$ the
corresponding prime ideal. It follows from Lemma 14 that if
$x\in\Omega_{\ell}$ for some slope adapted pair $(\Omega,\nu)$ then the
localisations $\mathscr{M}_{\ell,\mathfrak{m}_{x}}^{\bullet}$ of
$\mathscr{M}_{\ell}^{\bullet}$ and the localisation
$\mathscr{M}_{c,\mathfrak{m}_{x}}^{\bullet}$ of $\mathscr{M}_{c}^{\bullet}$ at
$\mathfrak{m}_{x}$ are both isomorphic to the localisation to
$\left(H^{d}_{c}(Y_{K},\mathscr{D}_{\lambda})\right)_{\mathfrak{m}_{x}}$ of
$H^{d}_{c}(Y_{K},\mathscr{D}_{\lambda})$ at $\mathfrak{m}_{x}$, where
$\lambda=w_{\ell}(x)=w(x)$. Finally, we also denote
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})=\mathscr{M}_{\ell}(\mathscr{U})[\varepsilon]$
the $\mathbb{T}_{\mathscr{U}}$-submodule of $\mathscr{M}_{\ell}(\mathscr{U})$
given by the $\varepsilon$-eigenspace of $\mathscr{M}_{\ell}(\mathscr{U})$,
for any choice of sign $\varepsilon$; if
$\mathscr{U}=\mathrm{Sp}(T)\hat{\otimes}_{R}S$ as above, then
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})=\mathscr{M}_{c}^{\varepsilon}(\mathrm{Sp}(T))\hat{\otimes}_{R}S$.

######  Lemma 15.

Let $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$. The weight map
$w:\mathscr{X}\rightarrow\mathscr{W}$ is unramified at $i_{\ell}(x)$ if and
only if the weight map
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ is unamified at
$x$.

###### Proof.

Rigid Zariski closed immersions are unramified and composition of unramified
maps is unramified (see for example [Hub96, Prop. 1.6.7]), therefore
$\mathscr{X}_{\ell}\overset{i_{\ell}}{\hookrightarrow}\mathscr{X}\overset{w}{\rightarrow}\mathscr{W}$
is unramified at $x$ if and only if $w$ is unramified at $i_{\ell}(x)$, and
$\mathscr{X}_{\ell}\overset{w_{\ell}}{\rightarrow}\mathscr{W}_{\ell}\overset{j_{\ell}}{\hookrightarrow}\mathscr{W}$
is unramified at $x$ if and only if $w_{\ell}$ is unramified at $x$. Since
$w\circ i_{\ell}=j_{\ell}\circ w_{\ell}$, the statement follows. ∎

######  Definition 14.

A point $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ is said to be
_classical_ (reps. _critical_ , _non-critical_ , _decent_ , _very decent_ ,
_parabolic_ or _interior_) if its image $i_{\ell}(x)$ under the closed
embedding $i_{\ell}:\mathscr{X}_{\ell}\hookrightarrow\mathscr{X}$ is classical
(reps. critical, non-critical, decent, very decent, parabolic or interior). We
say that a point $x\in\mathscr{W}_{\ell}(\overline{\mathbb{Q}}_{p})$ is _flat_
if the weight map $w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$
is flat at $x$, and we denote $\mathscr{X}_{\ell}^{\mathrm{fl}}$ the open
analytic subvariety of $\mathscr{X}_{\ell}$ where the map $w_{\ell}$ is flat.

######  Definition 15.

Let $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ (resp.
$x\in\mathscr{X}_{\ell}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$). A _good_
(resp. _clean_) neoghborhood of $x$ is a neighborhood
$\mathscr{U}=\mathrm{Sp}(T\hat{\otimes}_{R}S)$ of $x$ contained in
$\mathscr{X}_{\ell}$ (resp. in $\mathscr{X}_{\ell}^{\mathrm{fl}}$) such that
$\mathrm{Sp}(S)\subseteq\Omega_{\ell}=\Omega\cap\mathscr{W}_{\ell}$ for some
slope adapted affinoid $\Omega=\mathrm{Sp}(R)\subseteq\mathscr{W}$ with
$\mathrm{Sp}(S)$ connected, and $\mathrm{Sp}(T)\rightarrow\Omega$ is a good
(resp. clean) neighborhood of $i_{\ell}(x)$.

######  Lemma 16.

Let $\mathscr{U}=\mathrm{Sp}(T\hat{\otimes}_{R}S)$ be a good neighborhood of
$x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$. Then
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is a finite projective
$S$-module. For each maximal ideal $\mathfrak{m}$ of $S$, the localisation
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})_{\mathfrak{m}}$ of
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ at $\mathfrak{m}$ is free of
rank $r^{\varepsilon}_{\mathscr{U}}=r^{\varepsilon}_{\mathrm{Sp}(T)}$ over
$S$, where $r^{\varepsilon}_{\mathrm{Sp}(T)}$ is the rank of the finite
projective $R$-module $\mathscr{M}_{c}(\mathrm{Sp}(T))$.

###### Proof.

As recalled in §4.7, $\mathscr{M}_{c}^{\varepsilon}(\mathrm{Sp}(T))$ is a
finite projective $R$-module of rank $r_{\mathrm{Sp}(T)}^{\varepsilon}$, so
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is a finite projective
$S$-module. For the ranks, fix a maximal ideal $\mathfrak{m}$ of $S$ and let
$\mathfrak{P}$ be the prime ideal of $R$ lying over $\mathfrak{m}$; then
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})_{\mathfrak{m}}\simeq\mathscr{M}_{c}^{\varepsilon}(\mathrm{Sp}(T))\otimes_{R_{\mathfrak{P}}}S_{\mathfrak{m}}$
is finite free over $S_{\mathfrak{m}}$ of rank
$r_{\mathrm{Sp}(T)}^{\varepsilon}$, and the statement follows. ∎

##  5\. Inner products for overconvergent cohomology groups

Fix an embedding
$\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$. Let
$\mathbb{C}_{p}$ be the completion of $\overline{\mathbb{Q}}_{p}$, and
$\mathscr{O}_{\mathbb{C}_{p}}$ its valuation ring; let finally $|\cdot|_{p}$
be the absolute value on $\mathbb{C}_{p}$ extending the $p$-adic absolute
value $|\cdot|_{p}$ on $\mathbb{Q}_{p}$ obtained from the normalized as
$|p|_{p}=1/p$. We will sometimes write $|\cdot|$ for $|\cdot|_{p}$. As before,
we take $\mathfrak{n}=\mathfrak{n}_{0}p$ with $(\mathfrak{n}_{0},(p))=(1)$. We
suppose throughout that $\mathfrak{n}$ is square-free.

###  5.1. Twisted Poincaré product for distributions

For an algebraic weight $\lambda$, recall the specialization map
$\rho_{\lambda}:H^{d}_{c}(Y_{K},\mathscr{D}_{\lambda}(L))\rightarrow
H^{d}_{c}(Y_{K},\mathscr{V}_{\lambda}(L))$ and the map
$\iota_{\lambda}:H^{d}_{c}(Y_{K},\mathscr{L}_{\lambda}(L))\rightarrow
H^{d}_{c}(Y_{K},\mathscr{A}_{\lambda}(L))$ induced by the inclusion of
polynomials into spaces of analytic functions, where $L$ is any field
extension of $\mathbb{Q}_{p}$. We also recall that we have natural pairings
$\langle\ ,\
\rangle_{\mathrm{can}}:\mathscr{V}_{\lambda}(L)\times\mathscr{L}_{\lambda}(L)\to
L,$ (as above $L$ is a field extension of $\mathbb{Q}_{p}$) and, for any
affinoid $\mathbb{Q}_{p}$-algebra $E$, we have a pairing $\langle\ ,\
\rangle_{\mathrm{can}}:\mathscr{D}_{\lambda}(E)\times\mathscr{A}_{\lambda}(E)\to
E$. We also considered pairings $\langle\,,\,\rangle$ and the twisted pairing
$[\,,]$ on $H^{d}_{c}(Y_{0}(\mathfrak{n}),\mathscr{V}_{\lambda}(L))$. We lift
this pairing to $H^{d}_{c}(Y_{K},\mathscr{V}_{\lambda}(L))$ and
$H^{d}_{c}(Y_{K},\mathscr{D}_{\lambda}(E))$ by

$[\Phi_{1},\Phi_{2}]=[\rho_{\lambda}(\Phi_{1}),\rho_{\lambda}(\Phi_{2})]=\langle\rho_{\lambda}(\Phi_{1}),\theta_{\lambda}\mathsf{W}_{\mathfrak{n}}\rho_{\lambda}(\Phi_{2})\rangle_{\mathrm{can}}=\langle\Phi_{1},\iota_{\lambda}\theta_{\lambda}\mathsf{W}_{\mathfrak{n}}\rho_{\lambda}(\Phi_{2})\rangle_{\mathrm{can}},$

for $\Phi_{1},\Phi_{2}\in H^{d}_{c}(Y_{K},\mathscr{V}_{\lambda}(L))$ or
$H^{d}_{c}(Y_{K},\mathscr{D}_{\lambda}(E))$, for any field $L/\mathbb{Q}_{p}$
and any affinoid $\mathbb{Q}_{p}$-algebra $E$. The pairing on
$H^{d}_{c}(Y_{K},\mathscr{V}_{\lambda}(L))$ is clearly compatible with the
pairing on $H^{d}_{c}(Y_{0}(\mathfrak{n}),\mathscr{V}_{\lambda}(L))$ via the
canonical map induced by the inclusion $K\subseteq K_{0}(\mathfrak{n}))$.

Let $E$ be a $\mathbb{Q}_{p}$-Banach algebra with norm $|\cdot|_{E}$ extending
the norm $|\cdot|_{p}$ of $\mathbb{Q}_{p}$. Extend the binomial operator by
the formula $\binom{x}{r}=\frac{x(x-1)\dots(x-r)}{r!}$ for each
$r\in\mathbb{N}$ and $x\in E$. For $x=(x_{1},\dots,x_{d})\in E^{d}$, and
$r=(r_{1},\dots,r_{d})\in\mathbb{N}^{d}$, we put
$\binom{x}{r}=\prod_{i=1}^{d}\binom{x_{i}}{r_{i}}$. Using the embedding
$X=\mathscr{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\subseteq\prod_{\sigma\in\Sigma_{p}}\mathscr{O}_{F,\sigma}$
and the fixed ordering of $\Sigma_{p}$, write $x\in X$ as
${x}=(x_{\tau})_{\tau}\in E^{\Sigma_{p}}$ and put
$\binom{{x}}{r}=\prod_{\tau}^{d}\binom{x_{\tau}}{r_{\tau}}$ for
$r\in\mathbb{N}^{\Sigma_{p}}$.

Let $\mathfrak{p}\mid p$ be a prime of $F$ above $p$. Recall the surjective
map $\varsigma:\Sigma_{p}\rightarrow\\{\mathfrak{p}\text{ prime of
}F:\mathfrak{p}\mid p\\}$; denote $F_{\mathfrak{p}}$ the completion of $F$ at
$\mathfrak{p}$, and by $\mathscr{O}_{\mathfrak{p}}$ its ring of algebraic
integers. We have an inclusion
$\mathscr{O}_{\mathfrak{p}}\subseteq\mathscr{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$
via the isomorphism
$X=\mathscr{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\simeq\prod_{\mathfrak{p}\mid
p}\mathscr{O}_{\mathfrak{p}}$ which takes $x\mapsto(1,\dots,1,x,1,\dots,1)$
(all elements equal to $1$ except that corresponding to $\mathfrak{p}$, which
is equal to $x$). By composition with this inclusion, any character
$\kappa:T(\mathbb{Z}_{p})\rightarrow E^{\times}$ can be restricted to a
character of $T_{0}(\mathscr{O}_{\mathfrak{p}})$ denoted
$\kappa_{\mathfrak{p}}:T_{0}(\mathscr{O}_{\mathfrak{p}})\rightarrow
E^{\times}$.

Fix now $\kappa\in\Omega(E)$, for a $\mathbb{Q}_{p}$-Banach algebra $E$.
Choose generators
$\gamma_{\mathfrak{p},1},\dots,\gamma_{\mathfrak{p},f_{\mathfrak{p}}}$ where
$f_{\mathfrak{p}}=[F_{\mathfrak{p}}:\mathbb{Q}_{p}]$ of the group of principal
units in $\mathscr{O}_{\mathfrak{p}}^{\times}$ as $\mathbb{Z}_{p}$-module. We
fix our choice as follows: for each $\mathfrak{p}$, fix a normal basis
$\\{\theta_{\mathfrak{p},i}\\}$ for the extension
$\mathscr{O}_{F,\mathfrak{p}}$ over $\mathbb{Z}_{p}$, as
$i=1,\dots,f_{\mathfrak{p}}$. We take
$\gamma_{\mathfrak{p},i}=1+\theta_{\mathfrak{p},i}p$. Each
$\sigma\in\Sigma_{\infty}$ corresponds to some $\mathfrak{p},i$, and we use
the notation $\gamma_{\sigma}$ interchangeably for $\gamma_{\mathfrak{p},i}$.
Define $t(\gamma_{\sigma})=\bigl{(}\begin{smallmatrix}\gamma_{\sigma}&0\\\
0&\gamma_{\sigma}^{-1}\end{smallmatrix}\bigr{)}.$ By continuity, $\kappa$
sends $t(\gamma_{\sigma})$ to a element in $x\in E$ such that $|x-1|_{E}<1$.
We can define the $p$-adic logarithm $\log_{E}$ using the usual power series:
since the norm on $E$ extends the norm of $\mathbb{Q}_{p}$, this series
converges when valued on $\kappa_{\mathfrak{p}}(t(\gamma_{\sigma}))$. For any
character $\kappa\in\mathscr{W}(E)$, any $\sigma\in\Sigma_{p}$ and any
$r_{\sigma}\in\mathbb{N}$, put

$\binom{\kappa_{\sigma}}{r_{\sigma}}=\binom{\log_{E}(\kappa(t(\gamma_{\sigma})))/\log(\gamma_{\sigma})}{r_{\sigma}}.$

Then, for $r=(r_{\sigma})\in\mathbb{N}^{\Sigma_{p}}$, put
$\binom{\kappa}{r}=\prod_{\sigma\in\Sigma_{p}}\binom{\kappa_{\sigma}}{r_{\sigma}}.$
When $\kappa=\lambda$ is a point on the weight space coming from an algebraic
character $\lambda$, it is an easy check to conclude that
$\binom{\kappa}{r}=\binom{n(\lambda)}{r}$.

Fix an integer $\ell$. Let $q_{i}$ and $a_{i}$ be the elements introduced in
§2.4 with respect to the level $K$. For $\kappa\in\mathscr{W}_{\ell}(E)$,
$r\in\mathbb{Z}^{\Sigma_{p}}$ and $e\in\mathscr{O}_{F}\subset X$, we consider
the function of $z\in X$ defined by:

$\psi_{\kappa,r}^{(i)}(z)=\kappa(t(1+q_{i}e{z}))\cdot
q_{i}^{\ell}\cdot\frac{z^{r}(z-e)^{r}}{(1+q_{i}e{z})^{r}}.$

Let $\star=c,\emptyset$. Introduce the notation
$\mathscr{H}_{\star}(\mathscr{D}_{\kappa}(E))=H_{\star}^{d}(Y_{K},\mathscr{D}_{\kappa}(E))$
similarly as before. Fix an element
$\varphi\in\mathscr{H}_{\star}(\mathscr{D}_{\kappa}(E))$, which we write as
$\varphi=(\varphi_{1},\dots,\varphi_{h})$ with $\varphi_{i}$ in the cohomology
group
$H_{\star}^{d}(\Gamma^{i}\backslash\mathbb{H}^{d},\mathscr{D}_{\kappa}(E))$.
For $\kappa\in\mathscr{W}_{\ell}(E)$, $e\in\mathscr{O}_{F}$, $z\in X$ and
$\underline{\gamma}\in C_{\bullet}(D_{\mathbb{A}})$, define the formal power
series in $d$ variables by

(25) $T_{\kappa,\varphi,\underline{\gamma},e}^{(i)}(z):=\kappa(-1)\sum_{r\geq
0}q_{i}^{r}\binom{{\kappa}}{r}\varphi_{j}(a_{i}\underline{\gamma})\psi_{\kappa,r}^{(i)}(z).$

Note that for any classical weight $\kappa=\lambda\in\mathscr{W}_{\ell}(L)$,
where $L/\mathbb{Q}_{p}$ is a field extension, we have
$q_{i}^{\lambda_{2}}=q_{i}^{\ell}$ and hence the above equation matches with
(6) using the map
$H_{c}^{d}(\Gamma^{i}\backslash\mathbb{H}^{d},\mathscr{D}_{\kappa}(L))\twoheadrightarrow
H_{!}^{d}(\Gamma^{i}\backslash\mathbb{H}^{d},\mathscr{D}_{\kappa}(L))$, and
the description of cohomology classes in
$H^{d}(\Gamma^{i}\backslash\mathbb{H}^{d},\mathscr{D}_{\kappa}(L))$ in terms
of group cohomology classes in $H^{d}(\Gamma^{i},\mathscr{D}_{\kappa}(L))$
(this justifies the slight abuse of notation in using the same symbol
$\underline{\gamma}$ for elements in $\Gamma^{i}$ and singular chains on
$\Gamma^{i}\backslash\mathbb{H}^{d}$).

The next lemma addresses the convergence property of this formal power series.
Define $\rho_{0}=p^{\frac{-p}{p-1}}.$

######  Lemma 17.

For any $e\in\mathscr{O}_{F}\subset X$, the formal power series
$T_{\kappa,\varphi,\underline{\gamma},e,\ell}^{(i)}(z)$ converges in the
closed polydisk $D_{\rho}(e):=\\{z\in
X\mid\max_{\sigma}|z_{\sigma}-e_{\sigma}|\leq\rho\\}$ for all positive real
numbers $\rho<\rho_{0}$.

###### Proof.

For each $\sigma\in\Sigma_{p}$, we have
$|\sigma(q_{i})|_{p}=|\sigma(p)|_{p}=p^{-1},$ so
$|\sigma(q_{i})^{r_{\sigma}}|_{p}=p^{-r_{\sigma}}$ for each integer
$r_{\sigma}$. Furthermore,
$\left|\binom{\kappa_{\sigma}}{r_{\sigma}}\right|_{p}=p^{\frac{r_{\sigma}p}{p-1}}$
for each $\sigma$ and each integer $r_{\sigma}$, and so
$|\sigma(q_{i})^{r_{\sigma}}|_{p}\cdot\left|\binom{\kappa_{\sigma}}{r_{\sigma}}\right|_{p}\leq
p^{\frac{r_{\sigma}}{p-1}}$ for each $\sigma$. Hence,
$|q_{i}^{r}\binom{{\kappa}}{r}|_{p}\leq p^{\frac{r}{p-1}}$. Note that the
function $\psi_{\kappa,r}^{(i)}(z)$ belongs to $\mathscr{A}_{s[\kappa]}(X,L)$,
for $s[\kappa]$ the convergence radius of $\kappa$. Hence,

$|\varphi_{j}(a_{i}\underline{\gamma}a_{i}^{-1})(\psi_{\kappa,r}^{(i)}(z))|_{p}\leq\max_{i=1,\dots,h}\\{|\\!|\varphi_{j}(a_{i}\underline{\gamma}a_{i}^{-1})|\\!|_{s[\kappa]}\\}.$

Since this operator norm is bounded, the formal power series for
$T_{\kappa,\varphi,\underline{\gamma},e,\ell}^{(i)}(z)$ converges in the
polydisk considered in the lemma. ∎

###  5.2. Twisted Poincaré product over affinoids

The task of this subsection is to extend the definition of twisted Poincaré
pairing to affinoid neighbourhoods of a classical algebraic weight $\lambda$.
For this, using the explicit description of the pairing given in the previous
section, we only need to extend the expression (25) to affinoid
neighbourhoods.

Fix as above an integer $\ell$. Let $\Omega=\mathrm{Sp}(E)$ be an affinoid
$\Omega\subseteq\mathscr{W}_{\ell}$ which is contained in
$\Omega^{\prime}_{\ell}=\Omega^{\prime}\cap\mathscr{W}_{\ell}$ for some slope
adapted $(\Omega^{\prime},\nu)$ with $\Omega^{\prime}\subseteq\mathscr{W}$.
Let $s[\kappa_{\Omega}]=s[\kappa_{\Omega^{\prime}}]$, where
$s[\kappa_{\Omega^{\prime}}]$ is the radius of analyticity of
$\kappa_{\Omega^{\prime}}$. Put
$\rho_{\Omega,0}=\min\left\\{p^{-s[\kappa_{\Omega}]},p^{-\frac{1}{p-1}}\right\\}.$
For $\kappa_{\Omega}$, we consider the power series
$T_{\kappa_{\Omega},\varphi,\underline{\gamma},e,\ell}^{(i)}(z)$ defined in
(25). The following lemma shows that these functions can be patched together
to form a locally analytic function on $X$.

######  Lemma 18.

For any $e\in\mathscr{O}_{F}$, the formal power series
$T_{\kappa_{\Omega},\varphi,\underline{\gamma},e,\ell}^{(i)}(z)$ is the
expansion around $e$ of a $E$-valued locally analytic function on $X$. We
denote this function by $T_{\Omega,\varphi,\underline{\gamma},\ell}^{(i)}(z)$,
which belongs to $\mathscr{A}_{\Omega,s}$ for all $s$ such that
$p^{-s}\leq\rho_{\Omega,0}$.

###### Proof.

We need to check that for all $e$ and $e^{\prime}$, for all integers $s$ as in
the lemma, and for all $z\in\mathbb{C}_{p}^{\Sigma_{p}}$ such that
$|z-e|_{p}\leq p^{-s}$ and $|z-e^{\prime}|_{p}\leq p^{-s}$, the values
$T_{\kappa_{\Omega},\varphi,\underline{\gamma},e,\ell}^{(i)}(z)$ and
$T_{\kappa_{\Omega},\varphi,\underline{\gamma},e^{\prime},\ell}^{(i)}(z)$ are
the same. Since $\Omega^{\mathrm{cl}}$ is rigid Zariski dense in $\Omega$, we
only need to check that these values are the same for all fixed classical
point $\kappa=\lambda=(\lambda_{1},\lambda_{2})$. Let
$\ell_{\lambda}=\sum_{i}\lambda_{i}$. Then by (6), we see that
$q_{i}^{\ell_{\lambda}-\ell}T_{\kappa_{\Omega},\varphi,\underline{\gamma},e,\ell}^{(i)}(z)=q_{i}^{\ell_{\lambda}-\ell}T_{\kappa_{\Omega},\varphi,\underline{\gamma},e^{\prime},\ell}^{(i)}(z)$,
and therefore also
$T_{\kappa_{\Omega},\varphi,\underline{\gamma},e,\ell}^{(i)}(z)=T_{\kappa_{\Omega},\varphi,\underline{\gamma},e^{\prime},\ell}^{(i)}(z)$.
The result follows.∎

For $\Omega=\mathrm{Sp}(E)\subseteq\mathscr{W}_{\ell}$ as above, define
$\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))=H^{d}_{c}(Y_{K},\mathscr{D}_{\Omega}(E))$
as before. For $\Phi\in\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))$, the above
lemma allows us to define the $E$-valued formal power series

(26)
$T_{\Phi,\underline{\gamma},\ell}^{(i)}(z)=T_{\Omega,\Phi,\underline{\gamma},\ell}^{(i)}(z).$

######  Lemma 19.

Let $\Phi$ be a cocycle representing a class
$[\Phi]\in\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))$ for some affinoid
algebra $E$. The function $\underline{\gamma}\mapsto
T^{(i)}_{\Phi,\underline{\gamma},\ell}$ defined in (26) is a $d$-cocycle with
values in $\mathscr{A}_{\Omega}(E)$. Moreover, the cohomology class
$[T_{\Phi,\ell}^{(i)}]$ of $T_{\Phi,\ell}^{(i)}$ is independent of the choice
of $\Phi$. Let $T_{\Phi,\ell}$ denote the corresponding cohomology class in
$\mathscr{H}_{\ell}(\mathscr{A}_{\Omega}(E))$.

###### Proof.

If $\Phi$ is a $d$-cocycle, then we need to check that
$d(T^{(i)}_{\Phi,\ell})=0$. However, since $d$ commutes with $T$, we have
$d(T^{(i)}_{\Phi,\ell})=T^{(i)}_{d\Phi,\ell}=0$. Here, $T^{(i)}_{d\Phi,\ell}$
is also defined for $d+1$-cochains with the same formula as above. If $\Phi$
and $\Phi^{\prime}$ are two cocycles representing the same class, then
$\Phi-\Phi^{\prime}=dG$ for some $d-1$-cochain $G$, and therefore
$T_{\Phi,\ell}^{(i)}-T_{\Phi^{\prime},\ell}^{(i)}=T_{dG,\ell}^{(i)}$. We can
define $T_{G,\ell}^{(i)}$ as above for the $d-1$-cochain $G$, and then it is a
straightforward computation to show that
$d\left(T_{G,\ell}^{(i)}\right)=T_{dG,\ell}^{(i)}$, completing the proof. ∎

Fix $\Omega=\mathrm{Sp}(E)\subseteq\mathscr{W}_{\ell}$ as above. Let
$P:H^{2d}_{c}(E)\to E$ denote the isomorphism given by Poincaré duality.
Define pairings

(27)
$[\,,\,]_{\Omega,\ell}:\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))\times\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))\longrightarrow
E$

by the formula
$[\Phi,\Psi]_{\Omega,\ell}=P(\langle\Phi,T_{\Psi,\ell}\rangle_{\mathrm{can}}),$
where the canonical pairing
$\langle\,,\,\rangle_{\mathrm{can}}:\mathscr{D}_{\Omega}(E)\times\mathscr{A}_{\Omega}(E)\to
E$ induces a pairing on cohomology, which is denoted by the same symbol. If we
write as above $\Phi=(\Phi_{i})_{i}$, where $\Phi_{i}\in
H^{d}_{c}(\Gamma^{i},\mathscr{D}_{\Omega}(E))$ are the components of $\Phi$,
then we have
$[\Phi,\Psi]_{\Omega,\ell}=\sum_{i}[\Phi_{i},\Psi_{i}]_{\Omega,\ell}$, where
for each $i$ we write

(28)
$[\Phi_{i},\Psi_{i}]_{\Omega,\ell}=P\left(\sum_{i}\langle\Phi_{i},T_{\Psi,\ell}^{(i)}\rangle_{\mathrm{can}}\right),$

$\langle,\rangle_{\mathrm{can}}:H^{d}_{c}(\Gamma^{i},\mathscr{D}_{\Omega}(E))\times
H^{d}_{c}(\Gamma^{i},\mathscr{A}_{\Omega}(E))\rightarrow E$ is the pairing
induced by the canonical pairing $\langle,\rangle_{\mathrm{can}}$, and $P$ is
as above Poincaré duality. Recall that $\Omega$ is contained in
$\Omega^{\prime}$ for some slope adapted affinoid $(\Omega^{\prime},\nu)$.
Denote

(29)
$[\,,\,]_{\Omega,\nu,\ell}:\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))_{\leq\nu}\times\mathscr{H}_{\ell}(\mathscr{D}_{\Omega}(E))_{\leq\nu}\longrightarrow
E$

the restriction of the pairing $[\,,\,]_{\Omega,\ell}$ to
$\mathscr{H}_{\ell}(\mathscr{D}_{\Omega})_{\leq\nu}$. In particular, for any
$\kappa\in\Omega$, we have a pairing

(30)
$[\,,\,]_{\kappa,\nu,\ell}:\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{\kappa}))_{\leq\nu}\times\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{\kappa}))_{\leq\nu}\longrightarrow
k_{\kappa}$

and for each good neighborhood
$\mathscr{U}=\mathrm{Sp}(T)\rightarrow\mathrm{Sp}(E)$ of a point
$x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ we have a pairing

(31)
$[,]_{\mathscr{U},\ell}:\mathscr{M}_{\ell}(\mathscr{U})\times\mathscr{M}_{\ell}(\mathscr{U})\longrightarrow
T.$

Restricting to $\varepsilon$-eigenspaces, we also obtain pairings

(32)
$[\,,\,]_{\kappa,\nu,\ell}^{\varepsilon}:\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\kappa}(k_{\kappa}))_{\leq\nu}\times\mathscr{H}_{c}^{-\varepsilon}(\mathscr{D}_{\kappa}(k_{\kappa}))_{\leq\nu}\longrightarrow
k_{\kappa},\qquad[,]_{\mathscr{U},\ell}^{\varepsilon}:\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})\times\mathscr{M}_{\ell}^{-\varepsilon}(\mathscr{U})\longrightarrow
T.$

Note that $[,]_{\mathscr{U},\ell}$ and $[,]_{\mathscr{U},\ell}^{\varepsilon}$
are $E$-bilinear $T$-equivariant pairings of finite $T$-modules.

###  5.3. Properties of the twisted pairing

We begin by reviewing some formal properties of the twisted Poincaré product
in (27).

######  Lemma 20.

The pairings $\eqref{eqn:twisted-pairing-for-affinoids}$, $\eqref{p1}$,
$\eqref{p2}$, $\eqref{p3}$, (32) commute with affinoid base change.

###### Proof.

This is a formal consequence of the fact that all operations we made to
defined the twisted Poincaré product over affinoids commute with affinoid base
change. ∎

######  Lemma 21.

Hecke operators are self-adjoint for the pairings $\eqref{eqn:twisted-pairing-
for-affinoids}$ $\eqref{p1}$, $\eqref{p2}$, $\eqref{p3}$, $\eqref{p4}$.

###### Proof.

The rigid analytic function $[\Phi,T\Psi]-[T\Phi,\Psi]$ in
$E=\mathscr{O}_{\mathscr{W}_{\ell}}(\Omega)$ is trivial when specialized at
any classical point, and therefore identically zero by rigid Zariski density
of classical points in $\mathscr{W}_{\ell}$. ∎

Let $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ and $\kappa=w(x)$. Let
$\mathscr{U}$ be a good neighborhood of $x$. The pairing
$[,]_{\kappa,\nu,\ell}$ induces a pairing

$[,]_{x,\ell}:\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{\leq\nu}[x]\times\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{\leq\nu}[x]\longrightarrow
k_{x}.$

######  Theorem 22.

Let $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ and $\kappa=w(x)$. Let
$\mathscr{U}$ be a good neighborhood of $x$. The pairing $[,]_{x,\ell}$ is
non-degenerate if one of the following conditions holds:

  1. (1)

$\kappa$ is not a classical weight;

  2. (2)

$x$ is a non-critical parabolic point.

###### Proof.

Fix $\Phi\in\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{\leq\nu}[x]$ such
that $[\Phi,\Psi]_{\kappa,\nu,\ell}=0$ for all
$\Psi\in\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{\leq\nu}$. Recall that
$\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))$ admits a slope decomposition
$\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))\simeq\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{\leq\nu}\oplus\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))_{>\nu}$.
Since $U_{p}$ is self-adjoint for the pairing $[,]_{\Omega,\nu,\ell}$, it is
also self-adjoint for $[,]_{x,\ell}$, and the slope decomposition implies that
$[\Phi,\Psi]_{\kappa}=0$ for all
$\Psi\in\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))$. It follows then that
$T_{\kappa,\Phi,\underline{\gamma},\ell}^{(i)}(z)=0$ for all $i$ and all
$\underline{\gamma}$, and therefore, using the notation introduced before,
$\sum_{r\geq
0}q_{i}^{r}\binom{{\kappa}}{r}\Phi_{j}(a_{i}\underline{\gamma})=0$ for all $i$
and all $\underline{\gamma}$. If $\kappa$ is as in (1), then
$\binom{{\kappa}}{r}\neq 0$, and this forces
$\Phi_{j}(a_{i}\underline{\gamma})=0$ for all $i$ and all
$\underline{\gamma}$, which implies $\Phi=0$. If $x$ is as in (2), then
$\mathscr{H}_{c}(\mathscr{D}_{\kappa}(k_{x}))[x]\simeq\mathscr{H}_{c}(\mathscr{V}_{\kappa}(k_{x}))[x]$
by Theorem 3, and
$\mathscr{H}_{c}(\mathscr{V}_{\kappa}(k_{x}))[x]\simeq\mathscr{H}_{!}(\mathscr{V}_{\kappa}(k_{x}))[x]$
because $x$ is parabolic (_cf._ Definition 9), and therefore the pairing
$[,]_{\kappa,\nu,\ell}$ coincides with Poincaré pairing
$\mathscr{H}_{!}(\mathscr{V}_{\kappa}(k_{x}))[x]\times\mathscr{H}_{!}(\mathscr{V}_{\kappa}(k_{x}))[x]\rightarrow
k_{x}$, which is perfect, and again this implies $\Phi=0$. ∎

###  5.4. Interpolation of classical adjoint $L$-values

In this section, we show that the construction above interpolates the twisted
Poincaré pairing at classical points.

As in the previous sections, let
$\lambda=(\lambda_{1},\lambda_{2})\in\Omega(L)$ be a classical algebraic
weight, and we fix a rational number $\nu\geq 0$ and assume that $\nu$ is a
small slope for $\lambda$. Define $\ell_{\lambda}=\sum_{i}\lambda_{2,i}$.

Let $\mathscr{U}$ be a good neighbourhood in the eigenvariety, such that
$\mathscr{M}_{c}^{\varepsilon}(\mathscr{U})$ is free of rank one over
$\mathscr{O}_{\mathscr{W}}(\mathscr{U})$. Choose a generator
$\Phi^{\varepsilon}_{\mathscr{U}}$ for
$\mathscr{M}_{c}^{\varepsilon}(\mathscr{U})$.

Let $x\in\mathscr{U}$ be a classical point of weight $\lambda$ associated to
the refinement $(\pi,\alpha)$. Then there are constants
$\vartheta_{x}^{\varepsilon}\in L$ such that

(33)
$\rho_{x}(\Phi_{\mathscr{U}}^{\varepsilon})=\vartheta_{x}^{\varepsilon}\cdot\phi^{\varepsilon,\circ}_{\pi,\alpha},$

where the refinement $(\phi,\alpha)$ and
$\phi^{\varepsilon,\circ}_{\pi,\alpha}$ are defined in §3.3.

Define

$L^{\mathrm{alg}}_{\alpha}(1,\pi,\mathrm{Ad}^{0})=e_{p}(\pi,\alpha)\cdot\frac{L^{(p)}(1,\pi,\mathrm{Ad}^{0})}{\Omega\cdot\Omega_{\mathfrak{n}}(\pi)\cdot\Omega_{\infty}(\pi)\cdot\Omega^{\varepsilon}(\pi)\cdot\Omega^{-\varepsilon}(\pi)},$

with notation as in §3.3. Note that this value, as well as the $p$-adic
periods $\vartheta_{x}^{\epsilon}$, depend (in the same way) on the chosen
rational class $\phi^{\varepsilon,\circ}_{\pi,\alpha}$; thus, a notation as
$L^{\mathrm{alg}}_{\alpha}(1,\pi,\mathrm{Ad}^{0},\phi^{\varepsilon,\circ}_{\pi,\alpha})$
would be more clear, but we prefer the shorter notation and keep in mind this
dependence. We also note that, in particular, at this stage one do not expects
any $p$-adic property of these periods $\vartheta_{x}^{\epsilon}$.

######  Theorem 23.

Let $\mathscr{U}$ be a good neighbourhood associated to a slope adapted
affinoid $(\Omega,\nu)$ and $\Phi_{\mathscr{U}}^{\varepsilon}$ be as above.
For a classical point $x\in\mathscr{U}$ associated to a refinement
$(\pi,\alpha)$, we have

$[\Phi^{\varepsilon}_{\mathscr{U}},\Phi^{-\varepsilon}_{\mathscr{U}}]_{\Omega,\nu,\ell_{\lambda}}\textrm{
evaluated at
$x$}=\vartheta_{x}^{\varepsilon}\cdot\vartheta_{x}^{-\varepsilon}\cdot[\phi^{\varepsilon,\circ}_{\pi,\alpha},\phi^{\varepsilon,\circ}_{\pi,\alpha}]=\vartheta_{x}^{\varepsilon}\cdot\vartheta_{x}^{-\varepsilon}\cdot
L^{\mathrm{alg}}(1,\pi,\mathrm{Ad}^{0}).$

###### Proof.

The pairing
$[\Phi^{\varepsilon}_{\mathscr{U}},\Phi^{-\varepsilon}_{\mathscr{U}}]_{\Omega,\nu,\ell_{\lambda}}$
evaluated at $x$ is the twisted Poincaré pairing in §2.4, and the result is an
immediate consequence of (33) and the results explained in §3.3. ∎

##  6\. The adjoint $p$-adic $L$-function and the ramification locus

We now use the twisted Poincaré product over affinoid to construct adjoint
$p$-adic $L$-functions. We first recall some properties of ideals of scalar
products.

###  6.1. Scalar products and ramification

We recall the basic properties of ideals associated with scalar products. For
proofs, see for example the self-contained (and much more rich) exposition in
[Bel, §VI.1].

Let $R$ be a Noetherian domain, with fraction field $K$, and $T$ a finite
$R$-algebra. Denote $I$ the kernel of the multiplication map
$T\otimes_{R}T\rightarrow T$. Let $T^{\vee}=\mathrm{Hom}_{R}(T,R)$ be the
$R$-linear dual of $T$. Let $M$ and $N$ be finite $T$-modules which are flat
as $R$-modules, and let $\phi:M\times N\rightarrow R$ be a $R$-bilinear
$T$-equivariant scalar product. Extend $\phi$ to a $T$-bilinear scalar product
$\phi_{T}:(M\otimes_{R}T)\times(N\otimes_{R}T)\rightarrow{T}$. The _ideal of
the scalar product_ is the ideal $\mathfrak{d}_{\phi}(T/R)$ of $T$ defined by

$\mathfrak{d}_{\phi}(T/R)=\phi((M\otimes_{R}T)[I],(N\otimes_{R}T)[I]),$

where, for $X=M$ or $N$, the $T$-module structure on $X\otimes_{R}T$ is that
induced from the action of $T$ on itself, and $(X\otimes_{R}T)[I]$ denotes the
$I$-torsion submodule of $X\otimes_{R}T$.

Let $m:T\otimes_{R}T\rightarrow T$ be the multiplication map which takes
$t_{1}\otimes t_{2}\mapsto t_{1}\cdot t_{2}$. Let $I$ be the kernel of $m$,
and let $(T\otimes_{R}T)[I]$ be the $I$-torsion submodule of $T\otimes_{R}T$.
We define the _Noether’s different_ to be the ideal
$\mathfrak{d}_{\mathrm{N}}(T/R)$ of $T$ which is the image of
$(T\otimes_{R}T)[I]$ by $m$.

Fix as above a $R$-bilinear product $\phi:M\times N\rightarrow R$. We will
need the following results concerning $\mathfrak{d}_{\phi}(T/R)$ and
$\mathfrak{d}_{\mathrm{N}}(T/R)$.

  1. (1)

A prime ideal $\mathfrak{p}\subseteq T$ is ramified over $R$ if and only if
$\mathfrak{p}\supseteq\mathfrak{d}_{\mathrm{N}}(T/R)$ ([AB59, Thm. 2.7]).

  2. (2)

$\mathfrak{d}_{\mathrm{N}}(T/R)\subseteq\mathfrak{d}_{\phi}(T/R)$, and the
equality holds if $\phi$ is non-degenerate and $M$ and $N$ are flat
$T$-modules of rank $1$ ([Bel, Prop. VI.1.9, Prop. VI.1.12]).

  3. (3)

If $\mathfrak{p}$ belongs to the support of $\mathfrak{d}_{\phi}(T/R)$, then
$\mathfrak{p}$ is ramified (this is immediate from (1) and the first part of
(2)).

  4. (4)

If $T/R$ is Gorenstein, $M$, $N$ are flat of rank $1$ over $T$, then
$\mathfrak{d}_{\phi}(T/R)$ and $\mathfrak{d}_{\mathrm{N}}(T/R)$ are locally
principal ([Bel, Corollary VI.1.12]).

  5. (5)

If $T/R$ is Gorenstein, $M$, $N$ are flat of rank $1$ over $T$, and $\phi$ is
non-degenerate, then $\mathfrak{d}_{\phi}(T/R)$ and
$\mathfrak{d}_{\mathrm{N}}(T/R)$ are equal and locally principal, and
$\mathfrak{p}$ belongs to the support of $\mathfrak{d}_{\phi}(T/R)$ if an and
only if $\mathfrak{p}$ is ramified (this is immediate from (2) and (4)).

###  6.2. $p$-adic adjoint $L$-function and ramification locus

We now apply the above discussion to the construction of the adjoint $p$-adic
$L$-function.

Fix an integer $\ell$. Let $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$
and let $\mathscr{U}=\mathrm{Sp}(T)$ be a good neighborhood of $x$. Write
$T=A\otimes_{B}R$ with $\mathrm{Sp}(A)$ a good neighborhood of $i_{\ell}(x)$
in $\mathscr{X}$, $\Omega=\mathrm{Sp}(B)$ with $(\Omega,\nu)$ a slope adapted
affinoid of $\mathscr{W}$, and $\mathrm{Sp}(R)\subseteq\Omega$ a connected
affinoid in $\mathscr{W}_{\ell}$. Since the modules
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ are finite flat over $R$
(Lemma 16), and $[,]_{\mathscr{U},\ell}^{\varepsilon}$ is $R$-bilinear and
$T$-equivariant (Lemma 21), we can define

$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}=\mathfrak{d}_{[,]_{\mathscr{U},\ell}^{\varepsilon}}(\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})/R)$

to be the ideal of the scalar product $[,]_{\mathscr{U},\ell}^{\varepsilon}$.
By Lemma 20, the formation of $\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}$
commutes with affinoid base change. Since good affinoids form an admissible
cover of $\mathscr{X}_{\ell}$, we can glue the ideals
$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}$ and thus introduce the _$\ell$
-strata adjoint $L$-sheaf $\mathscr{L}^{\mathrm{adj}}_{\ell}$_ is the ideal
sheaf on $\mathscr{X}_{\ell}$ defined on good affinoids $\mathscr{U}$ by
$\mathscr{L}^{\mathrm{adj}}_{\mathscr{U},\ell}$.

######  Theorem 24.

Fix any integer $\ell$. The ramification locus of the weight map
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ contains the closed
subscheme defined by the the adjoint $L$-ideal sheaf
$\mathscr{L}_{\ell}^{\mathrm{adj}}$.

###### Proof.

Write $\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}(x)$ for
$x\left(\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}\right)$. The condition
$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}(x)=0$ does not depend on the
choice of the good neighborhood $\mathscr{U}$ of $x$ because the formation of
$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}$ commutes with affinoid base
change (Lemma 20). To show Theorem 24 we need to show that, for a point
$x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$, if
$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}(x)=0$ for any good neighborhood
$\mathscr{U}$ of $x$, then the weight map
$w_{\ell}:\mathscr{W}_{\ell}\rightarrow\mathscr{W}_{\ell}$ is ramified at $x$.
This follows from (3) in §6.1.∎

######  Corollary 25.

Let $x\in\mathscr{X}(\overline{\mathbb{Q}}_{p})$ and assume that
$x=i_{\ell}(z)$ for some integer $\ell$ and some point
$z\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$. Let $\mathscr{U}$ be a
good neighborhood of $z$. If
$\mathscr{L}^{\mathrm{adj}}_{\mathscr{U},\ell}(z)=0$, then
$w:\mathscr{X}\rightarrow\mathscr{W}$ is ramified at $x$.

###### Proof.

Since $x=i_{\ell}(z)$, by Lemma 15 we see that
$w:\mathscr{X}\rightarrow\mathscr{W}$ is unramified at $x$ if and only if
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ is unramified at
$z$. By Theorem 24, if $L_{\mathscr{U},\ell}^{\mathrm{adj}}(z)=0$ then
$w_{\ell}$ is ramified at $z$, and the result follows.∎

###  6.3. Ramification at good points

We study now ramification properties of the middle degree eigenvariety at a
special class of points, called _good_ , and introduced in the following
definition.

######  Definition 16.

A point $x\in\mathscr{X}_{\ell}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$ is
said to be _good_ if there exists a clean neighborhood
$\mathscr{U}=\mathrm{Sp}(T)$ of $x$ in $\mathscr{X}_{\ell}$ such that, writing
$T=A\hat{\otimes}_{B}R$ as in Definition 15:

  1. (1)

$w:\mathrm{Sp}(T)\rightarrow\mathrm{Sp}(R)$ is flat and $T/R$ is Gorenstein;

  2. (2)

$\mathscr{M}^{\varepsilon}_{\ell}(\mathscr{U})$ is flat of rank one over $T$
for all $\varepsilon$.

  3. (3)

$\dim_{k_{x}}\left(\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\kappa}(k_{x}))\right)[x]=1$,
where $\kappa=w(x)$.

We call such a $\mathscr{U}$ a _smooth_ neighborhood of $x$.

######  Remark.

If $x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ is smooth, then, up to
shrinking $\mathscr{U}=\mathrm{Sp}(T)$ if necessary, $T$ is regular, and
therefore also Gorenstein.

######  Lemma 26.

Let $x\in\mathscr{X}_{\ell}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$ be a
smooth interior very decent point of tame level $\mathfrak{n}_{0}$. Suppose
that there exists a neighborhood $\mathscr{U}=\mathrm{Sp}(T)$ of $x$ such that
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is flat over $T$. Then $x$ is
good.

###### Proof.

Fix a clean neighborhood $\mathscr{U}=\mathrm{Sp}(T)$ of $x$, with
$T=A\hat{\otimes}_{B}R$ and $\mathrm{Sp}(A)$ a clean neighborhood of
$x^{\prime}=i_{\ell}(x)$. Since $x$ is smooth one can also assume, as recalled
above, that $T/R$ is regular, and therefore Gorenstein, proving (1) in
Definition 16. Condition (3) follows from Lemma 5. Since we know by assumption
that $\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is finite free over $T$,
to prove (2) we only need to show that its rank is equal to $1$. Denote
$r=\mathrm{rank}_{T}(\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U}))$ the rank
of $\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ as $T$-module. Recall that
$\mathscr{M}_{c}^{\varepsilon}(\mathrm{Sp}(A))$ is a finite flat $B$-module of
rank $r_{\mathrm{Sp}(A)}^{\varepsilon}$, and that
$r_{\mathrm{Sp}(A)}^{\varepsilon}=d_{\mathrm{Sp}(A)}$ by Lemma 7, where
$d_{\mathrm{Sp}(A)}$ is the degree of the finite flat map
$\mathrm{Sp}(A)\rightarrow\mathrm{Sp}(B)$. From Lemma 16 we see that
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is also a finite flat
$R$-module of rank
$\mathrm{rank}_{R}(\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U}))=r_{\mathscr{U}}^{\varepsilon}=r_{\mathrm{Sp}(A)}^{\varepsilon}$.
Since $B\rightarrow A$ is finite flat, $R=B\hat{\otimes}_{B}R\rightarrow
T=A\hat{\otimes}_{B}T$ is also finite flat and if $d=\mathrm{rank}_{R}(T)$ is
the rank of $T$ as $R$-module, then $d=d_{\mathrm{Sp}(A)}$. The rank
$r_{\mathscr{U}}^{\varepsilon}$ of
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ as $R$-module is then equal to
$r_{\mathscr{U}}^{\varepsilon}=r\cdot d=r\cdot d_{\mathrm{Sp}(A)}=r\cdot
r_{\mathrm{Sp}(A)}^{\varepsilon}$, and since
$r_{\mathscr{U}}^{\varepsilon}=r_{\mathrm{Sp}(A)}^{\varepsilon}$ we must have
$r=1$. ∎

######  Lemma 27.

Smooth interior very decent non-critical points of tame level
$\mathfrak{n}_{0}$ are good.

###### Proof.

By Lemma 26, we are reduced to show that there exists a clean neighborhood
$\mathscr{U}=\mathrm{Sp}(T)$ such that
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is finite flat over $T$. As in
Lemma 9, by [BDJ, Th. 2.20(ii)] one can choose affinoids
$\mathrm{Sp}(A)\subseteq\mathscr{X}^{\mathrm{fl}}$ and
$\mathrm{Sp}(B)\subseteq\mathscr{W}$ in such a way that
$\mathrm{Sp}(A)\rightarrow\mathrm{Sp}(B)$ is an isomorphism. Since
$\mathscr{M}_{c}^{\varepsilon}(\mathrm{Sp}(A))$ is a finite flat $B$-module,
it is also a finite flat $A$-module, and we conclude that
$\mathscr{M}_{\ell}^{\varepsilon}(\mathscr{U})$ is a finite flat $T$-module. ∎

######  Remark.

Interior very decent points of tame level $\mathfrak{n}_{0}$ in $\mathscr{X}$
are smooth by [BH, Theorem 6.6.3]. However, this does not guarantee that
interior very decent points in $\mathscr{X}_{\ell}$ are also smooth.

Using (4) in §6.1, we see that if $\mathscr{U}$ is a smooth neighborhood of a
good point $x$ as in Definition 16, then the ideal of the scalar product
$\mathscr{L}_{\mathscr{U},\ell}^{\mathrm{adj}}$ is principal, generated by
$L^{\mathrm{adj}}_{\mathscr{U},\ell}$. This justifies the following:

######  Definition 17.

If $\mathscr{U}$ is a smooth neighborhood of a good point
$x\in\mathscr{X}_{\ell}(\overline{\mathbb{Q}}_{p})$ as in Definition 16, and
$\ell$ is an integer, then the _$\ell$ -strata adjoint $p$-adic $L$-function_
is the element $L^{\mathrm{adj}}_{\mathscr{U},\ell}$.

If $x$ is a good point, then the $\ell$-strata adjoint $p$-adic $L$-function
is well defined up to units. For $x\in\mathscr{U}$, write
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)\in k_{x}$ for the value of
$L^{\mathrm{adj}}_{\mathscr{U},\ell}$ at the morphism defined by $x$. The
condition $L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)=0$ does not depend on the
choice of the smooth neighborhood $\mathscr{U}$ of $x$.

######  Theorem 28.

Fix an integer $\ell$ and let
$x\in\mathscr{X}_{\ell}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$ be a good
point. If one of the following conditions is satisfied

  1. (1)

$x$ is a very decent interior point of tame level $\mathfrak{n}_{0}$;

  2. (2)

$\kappa=w(x)$ is not a classical weight;

  3. (3)

$x$ is a non-critical parabolic point;

then $w$ is unramified at $x$ if and only if
$L_{\mathscr{U},\ell}^{\mathrm{adj}}(x)\neq 0$, where $\mathscr{U}$ is any
smooth neighborhood of $x$. Moreover, if $\lambda=w(x)$ is a classical weight,
and $x$ is not a classical point, then
$L_{\mathscr{U},\ell}^{\mathrm{adj}}(x)=0$. Finally, if $x$ is a very decent
interior non-critical point of tame level $\mathfrak{n}_{0}$, then
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)\neq 0$.

###### Proof.

If $\kappa=w(x)$ is not a classical weight as in (2), or $x$ is a non-critical
parabolic point as in (3), then the pairing in Theorem 22 is non-degenerate.
The equivalence between the conditions that $x$ is unramified and
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)\neq 0$ follows from (4) in §6.1.

Suppose now that $x$ is as in (1). Since we already discussed (3), we may
assume that $x$ is critical. Let $\Phi_{x}^{\varepsilon}$ be a generator of
$\mathscr{H}_{c}^{\varepsilon}(\mathscr{D}_{\kappa}(k_{x}))[x]$. Then
$[\Phi_{x}^{\varepsilon},\Phi_{x}^{-\varepsilon}]_{\kappa,\nu}^{\varepsilon}\neq
0$ if and only if $L_{\mathscr{U},\ell}^{\mathrm{adj}}(x)\neq 0$, and
$[\Phi_{x}^{\varepsilon},\Phi_{x}^{-\varepsilon}]_{\kappa,\nu}^{\varepsilon}=[\rho_{\kappa}(\Phi_{x}^{\varepsilon}),\rho_{\kappa}(\Phi_{x}^{-\varepsilon})]_{\kappa,\nu}^{\varepsilon}$.
The last pairing is between $1$-dimensional spaces, so it is non-zero if and
only if $\rho_{\kappa}(\Phi_{x}^{\pm\varepsilon})\neq 0$, which is equivalent
to the condition that $x$ is unramified by Proposition 11.

Suppose that $\kappa=w(x)$ is a classical weight, but $x$ is not a classical
point. Then $\mathscr{H}_{c}(\mathscr{V}(k_{x}))[x]=0$, so
$\rho_{\kappa}(\Phi_{x}^{\pm\varepsilon})=0$ and therefore
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)=0$.

Finally, if $x$ is a very decent interior non-critical point of tame level
$\mathfrak{n}_{0}$, the weight map $w:\mathscr{X}\rightarrow\mathscr{W}$ is
unramified at $i_{\ell}(x)$ by Corollary 12. It follows from Lemma 15 that
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ is also unramified
at $x$, and therefore $L^{\mathrm{adj}}_{\mathscr{U},\ell}(x)\neq 0$ by
Theorem 24. ∎

######  Corollary 29.

Let $x\in\mathscr{X}(\overline{\mathbb{Q}}_{p})$ and assume that
$x=i_{\ell}(z)$ for some integer $\ell$ and some good point
$z\in\mathscr{X}_{\ell}^{\mathrm{fl}}(\overline{\mathbb{Q}}_{p})$. If one of
the following conditions is satisfied

  1. (1)

$x$ is a very decent interior point of tame level $\mathfrak{n}_{0}$;

  2. (2)

$\kappa=w(x)$ is not a classical weight;

  3. (3)

$x$ is a non-critical parabolic point;

then $w$ is unramified at $x$ if and only if
$L_{\mathscr{U},\ell}^{\mathrm{adj}}(z)\neq 0$, where $\mathscr{U}$ is any
smooth neighborhood of $x$. Moreover, if $\lambda=w(x)$ is a classical weight,
and $x$ is not a classical point, then
$L_{\mathscr{U},\ell}^{\mathrm{adj}}(z)=0$. Finally, if $x$ is a very decent
interior non-critical point of tame level $\mathfrak{n}_{0}$, then
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(z)\neq 0$.

###### Proof.

For the first statement, since $x=i_{\ell}(z)$, by Lemma 15 we see that
$w:\mathscr{X}\rightarrow\mathscr{W}$ is unramified at $x$ if and only if
$w_{\ell}:\mathscr{X}_{\ell}\rightarrow\mathscr{W}_{\ell}$ is unramified at
$z$, which is the case, under the conditions on $x$ or $z$, if and only if
$L_{\mathscr{U},\ell}^{\mathrm{adj}}(z)=0$ by Theorem 28. For the second
statement, use again Theorem 28 and note that if $x$ is not classical, then
$z$ is not classical, and they have the same weight $w(x)=w_{\ell}(z)$. For
the third statement, if $x$ is a very decent interior non-critical point of
tame level $\mathfrak{n}_{0}$, then $w$ is unramified at $x$ by Lemma 12, and
by Lemma 15 the same is true for $w_{\ell}$ at $z$, so
$L^{\mathrm{adj}}_{\mathscr{U},\ell}(z)\neq 0$ by Theorem 28. ∎

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