# Higher Hida and Coleman Theories on the modular curve

George Boxer<EMAIL_ADDRESS>Unité de Mathématiques pures et
appliquées, Ecole normale supérieure de Lyon, 46 allée d’Italie, 69 364 Lyon
Cedex 07, France  and  Vincent Pilloni<EMAIL_ADDRESS>Unité de
Mathématiques pures et appliquées, Ecole normale supérieure de Lyon, 46 allée
d’Italie, 69 364 Lyon Cedex 07, France

###### Abstract.

We construct Hida and Coleman theories for the degree $0$ and $1$ cohomology
of automorphic line bundles on the modular curve and we define a $p$-adic
duality pairing between the theories in degree $0$ and $1$.

##  1\. Introduction

In the 80’s, Hida introduced an ordinary projector on modular forms and he
constructed $p$-adic families of ordinary modular forms [Hid86]. In the 90’s,
Coleman developed the finite slope theory [Col97] and Coleman and Mazur
constructed the eigencurve [CM98]. These theories have now been extended to
higher dimensional Shimura varieties.

Hida and Coleman theories combine two ideas. The first one is to restrict
modular forms from the full modular curve to the ordinary locus and its
neighborhoods. The additional structure on the universal $p$-divisible group
on and near the ordinary locus (the canonical subgroups) is used to
$p$-adically interpolate the sheaves of modular forms, and their cohomology.
The second idea is to use the Hecke operators at $p$ to detect when a section
of the sheaf of modular forms defined on (a neighborhood of) the ordinary
locus comes from a classical modular form.

Until recently, Hida and Coleman theories had only been developed for degree
$0$ coherent cohomology groups. In the recent works [Pil17], [BCGP18] we began
to develop them further in order to study the higher coherent cohomology of
automorphic vector bundle on the Shimura varieties for the group
$\mathrm{GSp}_{4}$, and we are now convinced that Hida and Coleman theories
should exist in all cohomological degrees for all Shimura varieties.

The purpose of the current work is to confirm this prediction in the simple
setting of modular curves and to construct Hida and Coleman theories for the
degree $1$ cohomology groups. We actually construct in parallel the theories
for degree $0$ and degree $1$ cohomology, as this sheds some new light on the
usual degree $0$ theory. We also prove a $p$-adic Serre duality, which gives a
perfect pairing between the theories in cohomological degree $0$ and $1$, but
our constructions are independent of this pairing.

Let us describe the results we prove. Let
$X\rightarrow\operatorname{Spec}\mathbb{Z}_{p}$ be the compactified modular
curve of level $\Gamma_{1}(N)$, where $N\geq 3$ is an integer prime to $p$,
and let $D$ be the boundary divisor. Let
$X_{1}\rightarrow\operatorname{Spec}\mathbb{F}_{p}$ be the special fiber and
$X_{1}^{ord}$ be the ordinary locus.

######  Theorem 1.1 (Hida’s control theorem).

There is a Hecke operator $T_{p}$ acting on the cohomology groups
$\mathrm{R}\Gamma(X_{1},\omega^{k})$,
$\mathrm{R}\Gamma_{c}({X}_{1}^{ord},\omega^{k})$ and
$\mathrm{R}\Gamma({X}_{1}^{ord},\omega^{k})$, and an associated ordinary
projector $e(T_{p})$. Moreover, we have quasi-isomorphisms

$e(T_{p})\mathrm{R}\Gamma(X_{1},\omega^{k})\rightarrow
e(T_{p})\mathrm{R}\Gamma(X_{1}^{ord},\omega^{k})~{}~{}~{}~{}\textrm{if $k\geq
3$}$

and

$e(T_{p})\mathrm{R}\Gamma_{c}(X^{ord}_{1},\omega^{k})\rightarrow
e(T_{p})\mathrm{R}\Gamma(X_{1},\omega^{k})~{}~{}~{}~{}\textrm{if $k\leq-1$}.$

The proof of this theorem relies on a local analysis of the cohomological
correspondence $T_{p}$ at supersingular points. The above theorem may also be
viewed as a vanishing result: $e(T_{p})\mathrm{R}\Gamma(X_{1},\omega^{k})$ is
concentrated in degree $0$ if $k\geq 3$, and degree $1$ if $k\leq-1$, because
the ordinary locus is affine. Of course, this vanishing result holds true even
without the ordinary projector, from the Riemann-Roch theorem and the Kodaira-
Spencer isomorphism, but the argument above is independent and generalizes.

Let $\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$ be the Iwaswa algebra.
Each integer $k\in\mathbb{Z}$ defines a character of
$\mathbb{Z}_{p}^{\times}$, and a morphism
$k:\Lambda\rightarrow\mathbb{Z}_{p}$.

######  Theorem 1.2.

There are two projective $\Lambda$-modules $M$ and $N$ carrying an action of
the Hecke algebra of level prime to $p$, and there are canonical, Hecke-
equivariant isomorphisms for all $k\geq 3$:

  1. (1)

$M\otimes_{\Lambda,k}\mathbb{Z}_{p}=e(T_{p})\mathrm{H}^{0}(X,\omega^{k})$,

  2. (2)

$N\otimes_{\Lambda,k}\mathbb{Z}_{p}=e(T_{p})\mathrm{H}^{1}(X,\omega^{2-k}(-D))$,

Moreover, there is a perfect pairing $M\times N\rightarrow\Lambda$ which
interpolates the classical Serre duality pairing.

The modules $M$ and $N$ are obtained by considering the ordinary factor of the
cohomology and cohomology with compact support of the ordinary locus, with
values in an interpolation sheaf of $\Lambda$-modules.

Let $X_{0}(p)\rightarrow\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ be
the adic modular curve of level $\Gamma_{1}(N)\cap\Gamma_{0}(p)$. We have two
quasi-compact opens $X_{0}(p)^{m}$ and $X_{0}(p)^{et}$ inside $X_{0}(p)$ which
are respectively the loci where the universal subgroup of order $p$ has
multiplicative and étale reduction. We let $X_{0}(p)^{m,\dagger}$ and
$X_{0}(p)^{et,\dagger}$ be the corresponding dagger spaces (these are the
inductive limit over all strict neighborhoods of $X_{0}(p)^{m}$ and
$X_{0}(p)^{et}$).

######  Theorem 1.3 (Coleman’s classicality theorem).

For all $k\in\mathbb{Z}$, there is a well defined Hecke operator $U_{p}$ which
is compact and has non negative slopes on
$\mathrm{H}^{i}(X_{0}(p),\omega^{k})$,
$\mathrm{H}^{i}(X_{0}(p)^{m,\dagger},\omega^{k})$ and
$\mathrm{H}^{i}_{c}(X_{0}(p)^{et,\dagger},\omega^{k})$. Moreover, the natural
maps (where the superscript $<\star$ means slope less than $\star$ for
$U_{p}$):

  1. (1)

$\mathrm{H}^{i}(X_{0}(p),\omega^{k})^{<k-1}\rightarrow\mathrm{H}^{i}(X_{0}(p)^{m,\dagger},\omega^{k})^{<k-1}$,

  2. (2)

$\mathrm{H}^{i}_{c}(X_{0}(p)^{et,\dagger},\omega^{k})^{<1-k}\rightarrow\mathrm{H}^{i}(X_{0}(p),\omega^{k})^{<1-k}$

are isomorphisms.

The proof of this theorem is based on some simple estimates for the operator
$U_{p}$ on the ordinary locus, reminiscent of [Kas06]. We can again derive
from this theorem a vanishing theorem for the small slope classical cohomology
(without appealing to the Riemann-Roch theorem).

Coleman and Mazur constructed the eigencurve $\mathcal{C}$ of tame level
$\Gamma_{1}(N)$. It carries a weight morphism
$w:\mathcal{C}\rightarrow\mathcal{W}$ where $\mathcal{W}$ is the analytic adic
space over $\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ associated with
the Iwasawa algebra $\Lambda$.

######  Theorem 1.4.

The eigencurve carries two coherent sheaves $\mathcal{M}$ and $\mathcal{N}$
interpolating the degree $0$ and $1$ finite slope cohomology. For any
$k\in\mathbb{Z}$, we have

  1. (1)

$\mathcal{M}_{k}^{<k-1}=\mathrm{H}^{0}(X_{0}(p),\omega^{k})^{<k-1}$,

  2. (2)

$\mathcal{N}_{k}^{<k-1}=\mathrm{H}^{1}(X_{0}(p),\omega^{2-k}(-D))^{<k-1}$,

and there is a perfect pairing between $\mathcal{M}$ and $\mathcal{N}$,
interpolating the usual Serre duality pairing.

###  1.1. Acknowledgements

We would like to thank Frank Calegari and Toby Gee for conversations about the
topics in this paper, and David Loeffler for his comments. This research was
supported by the ERC-2018-COG-818856-HiCoShiVa, and the first author was
supported in part by the NSF postdoctoral fellowship DMS-1503047.

##  2\. Preliminaries

###  2.1. Finite flat cohomological correspondences

####  2.1.1. The functors $f_{\star}$, $f^{\star}$ and $f^{!}$

Let $f:X\rightarrow Y$ be a finite flat morphism of schemes. We have a functor
$f_{\star}:Coh(X)\rightarrow Coh(Y)$. It induces an equivalence of categories
between $Coh(X)$ and the category of coherent sheaves of
$f_{\star}\mathscr{O}_{X}$-modules over $Y$.

The functor $f_{\star}$ has a left adjoint $f^{\star}:Coh(Y)\rightarrow
Coh(X)$, given by
$\mathscr{F}\mapsto\mathscr{F}\otimes_{\mathscr{O}_{Y}}f_{\star}\mathscr{O}_{X}$,
as well as a right adjoint $f^{!}:Coh(Y)\rightarrow Coh(X)$ defined by
$f^{!}\mathscr{F}=\underline{\mathrm{Hom}}_{\mathscr{O}_{Y}}(f_{\star}\mathscr{O}_{X},\mathscr{F})$.
For any $\mathscr{F}\in Coh(X)$, we have an isomorphism
$f^{!}\mathscr{F}=f^{!}\mathscr{O}_{X}\otimes_{\mathscr{O}_{Y}}\mathscr{F}$.

A finite flat morphism $f:X\rightarrow Y$ has a trace map
$\mathrm{tr}_{f}:f_{\star}\mathscr{O}_{X}\rightarrow\mathscr{O}_{Y}$. This
trace is by definition a global section of $f^{!}\mathscr{O}_{Y}$ or
equivalently a morphism $f^{\star}\mathscr{O}_{Y}\rightarrow
f^{!}\mathscr{O}_{X}$. It follows that the trace map provides a natural
transformation $f^{\star}\Rightarrow f^{!}$.

A finite flat morphism $f:X\rightarrow Y$ is called Gorenstein if
$f^{!}\mathscr{O}_{X}$ is an invertible sheaf. If $f:X\rightarrow Y$ if a
local complete intersection morphism, then it is Gorenstein (see [Eis95, Cor.
21.19]).

####  2.1.2. Cohomological correspondences

######  Definition 2.1.

A finite flat correspondence over a scheme $X$ is a scheme $C$ equipped with
two finite flat morphisms $X\stackrel{{\scriptstyle
p_{2}}}{{\leftarrow}}C\stackrel{{\scriptstyle p_{1}}}{{\rightarrow}}X$.

######  Definition 2.2.

Let $\mathscr{F}$ be a coherent sheaf on $X$. A finite flat cohomological
correspondence for $\mathscr{F}$ is the data of a pair $(C,T)$ consisting of a
finite flat correspondence $C$ and a map
$T:p_{2}^{\star}\mathscr{F}\rightarrow p_{1}^{!}\mathscr{F}$.

Given a finite flat cohomological correspondence $(C,T)$ on $\mathscr{F}$, we
get a morphism in cohomology that we also denote by $T$,

$T:\mathrm{R}\Gamma(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma(C,p_{2}^{\star}\mathscr{F})\stackrel{{\scriptstyle
T}}{{\rightarrow}}\mathrm{R}\Gamma(C,p_{1}^{!}\mathscr{F})\rightarrow\mathrm{R}\Gamma(X,\mathscr{F}).$

####  2.1.3. Composition

It is possible to compose finite flat cohomological correspondences. Assume we
have a diagram of finite flat correspondences $C$ and $D$:

We can form $E=C\times_{p_{1},X,q_{2}}D$:

If we are given $p_{2}^{\star}\mathscr{F}\rightarrow p_{1}^{!}\mathscr{F}$ and
$q_{2}^{\star}\mathscr{F}\rightarrow q_{1}^{!}\mathscr{F}$, we can consider
the following maps:

$t_{2}^{\star}p_{2}^{\star}\mathscr{F}\rightarrow
t_{2}^{\star}p_{1}^{!}\mathscr{F}~{}~{}~{}\textrm{and}~{}~{}~{}t_{1}^{!}q_{2}^{\star}\mathscr{F}\rightarrow
t_{1}^{!}q_{1}^{!}\mathscr{F}$

moreover we claim that there is a natural transformation
$t_{2}^{\star}p_{1}^{!}\Rightarrow t_{1}^{!}q_{2}^{\star}$ so that we can
compose and get the cohomological correspondence:

$t_{2}^{\star}p_{2}^{\star}\mathscr{F}\rightarrow
t_{1}^{!}q_{1}^{!}\mathscr{F}.$

It remains to check the claim. Equivalently (by adjunction) it suffices to
find a natural transformation
$(t_{1})_{\star}t_{2}^{\star}p_{1}^{!}\Rightarrow t_{1}^{!}q_{2}^{\star}$. We
have a base change isomorphism
$(t_{1})_{\star}t_{2}^{\star}=(q_{2})^{\star}(p_{1})_{\star}$, and by
adjunction $(p_{1})_{\star}p_{1}^{!}\Rightarrow\mathrm{Id}$, there is a
natural transformation $(q_{2})^{\star}(p_{1})_{\star}p_{1}^{!}\Rightarrow
q_{2}^{\star}$.

####  2.1.4. Restriction

Let $X$ is a scheme and $Y\hookrightarrow X$ is a closed subscheme defined by
an ideal $\mathscr{I}\hookrightarrow\mathscr{O}_{X}$. For any quasi-coherent
sheaf $\mathscr{F}$ over $X$, we let

$\underline{\Gamma}_{Y}(\mathscr{F})=\mathrm{Ker}(\mathscr{F}\rightarrow\underline{\mathrm{Hom}}(\mathscr{I},\mathscr{F}))$

be the subsheaf of sections with support in $Y$.

######  Proposition 2.3.

Consider a commutative diagram of schemes:

where $i$ is a closed immersion and $g$, $f$ are finite flat morphisms. Let
$\mathscr{F}$ be a coherent sheaf on $Z$. Then
$i_{\star}g^{!}\mathscr{F}=\underline{\Gamma}_{Y}f^{!}\mathscr{F}$.

###### Proof.

Let us denote by $\mathscr{I}$ the ideal sheaf of $Y$ in $X$. This follows
from the exact sequence:

$0\rightarrow\underline{\mathrm{Hom}}(g_{\star}\mathscr{O}_{Y},\mathscr{F})\rightarrow\underline{\mathrm{Hom}}(f_{\star}\mathscr{O}_{X},\mathscr{F})\rightarrow\underline{\mathrm{Hom}}(f_{\star}\mathscr{I},\mathscr{F}).$

∎

###  2.2. Local finiteness

Let $R$ be an artinian local ring with finite residue field.

######  Lemma 2.4.

Suppose $M$ is a finite type $R$-module and $T\in\operatorname{End}_{R}(M)$ is
an endomorphism of $M$. The sequence
$(T^{n!})_{n\in\mathbb{N}}\in(\mathrm{End}_{R}(M))^{\mathbb{N}}$ is eventually
constant and converges to an idempotent denoted $e(T)\in\mathrm{End}_{R}(M)$.

###### Proof.

We have a decreasing sequence of submodules $T^{n!}(M)$ of $M$ which is
eventually stationary since $M$ is artinian. Let $M_{0}$ be the limit. For all
$n$ large enough $T^{n!}$ induces a permutation of the finite set $M_{0}$, and
therefore, for all $n$ large enough, this permutation is trivial. It follows
that for all $n$ large enough, $T^{n!}$ is a projector from $M$ to $M_{0}$. ∎

######  Definition 2.5.

Let $M$ be an $R$-module and let $T\in\mathrm{End}_{R}(M)$. We say that $T$ is
locally finite on $M$ if $M$ is a union of finite type $R$-modules which are
stable under $T$.

######  Remark 2.6.

This condition is equivalent to the claim that for any finite type submodule
$V\subseteq M$, $\sum_{n\geq 0}T^{n}V$ is a finite type submodule. It is also
equivalent to the claim that for some $n\geq 1$,
$T^{n}V\subseteq\sum_{i=0}^{n-1}T^{i}V$.

######  Proposition 2.7.

If $M$ is an $R$-module and $T$ is locally finite on $M$, there is an
idempotent $e(T)$ attached to $T$ such that $T$ is an isomorphism on $e(T)M$
and $T$ is locally nilpotent on $(1-e(T))M$.

By locally nilpotent, we mean that for each $m\in(1-e(T))M$, there exists $N$
such that $T^{N}m=0$.

###### Proof.

This follows from lemma 2.4. ∎

######  Lemma 2.8.

If $f:M\rightarrow N$ is an $R$-linear morphism and $T$ is an $R$-linear
operator acting equivariantly and locally finitely on $M$ and $N$, then
$f(e(T)M)\subseteq e(T)N$ and $f((1-e(T))M)\subseteq(1-e(T))N$.

######  Proposition 2.9.

Let $0\rightarrow M\rightarrow N\rightarrow L\rightarrow 0$ be a short exact
sequence of $R$-modules and let $T$ be an $R$-linear operator acting
equivariantly on $M,N$ and $L$.

  1. (1)

$T$ is locally finite on $N$ if and only if $T$ is locally finite on $M$ and
$L$.

  2. (2)

If $T$ is locally finite on $M,N$, and $L$, the sequence $0\rightarrow
e(T)M\rightarrow e(T)N\rightarrow e(T)L\rightarrow 0$ is exact.

###### Proof.

We prove the first point. The direct implication is trivial. Let us prove the
reverse implication. Let $V\subset N$ be a finite submodule. Since $T$ is
locally finite on $L$, we deduce that there exists $n\geq 1$ and a finite
submodule $W\subset M$ such that:

$T^{n}(V)\subseteq\sum_{i=0}^{n-1}T^{i}V+W.$

Since $T$ is locally finite on $M$, there exists $m\geq 1$ such that:

$T^{m}(W)\subseteq\sum_{i=0}^{m-1}T^{i}W.$

We finally conclude that $T^{n+m}(V+W)\subseteq\sum_{i=0}^{n+m-1}T^{i}(V+W)$,
so that $\sum_{i=0}^{n+m-1}T^{i}(V+W)$ is a finite $R$-module, stable by $T$,
containing $V$. The second point follows easily. ∎

We will actually need to work with certain topological $R$-modules that we
call profinite $R$-modules and are in a certain sense duals of (discrete)
$R$-modules. A profinite $R$-module $M$ is a topological $R$-module which is
homeomorphic to a cofiltered limit $\lim_{i\in I}M_{i}$ of finite $R$-modules.

In other words, a topological $R$-module $M$ is a profinite $R$-module if it
is separated and complete and if there is a basis of neighborhoods of $0\in
M$, $\\{N_{i}\\}_{i\in{I}}$ consisting of cofinite submodules. In such a case,
we have $M=\lim_{i}M/N_{i}$.

######  Definition 2.10.

Let $M$ be a profinite $R$-module, and let $T\in\mathrm{End}_{R}(M)$ be a
continuous endomorphism. We say that $T$ is locally finite on $M$ if $M$ has a
basis of neighborhoods of $0$ consisting of submodules $\\{N_{i}\\}$ such that
$T(N_{i})\subseteq N_{i}$.

######  Remark 2.11.

We observe that the first condition in the definition is equivalent to the
condition that for any open submodule $V\subseteq M$, the submodule
$\cap_{n\geq 0}T^{-n}V$ is open. Since $M/\cap_{n\geq 0}T^{-n}V$ if a finite
module and therefore an artinian module, we deduce that the condition is also
equivalent to the condition that $\cap_{i=0}^{n-1}T^{-i}V\subseteq T^{-n}V$
for some $N\geq 1$.

######  Proposition 2.12.

If $M$ is a profinite $R$-module and $T$ is locally finite on $M$, there is an
idempotent $e(T)$ attached to $T$ such that $T$ is an isomorphism on $e(T)M$
and $T$ is topologically nilpotent on $(1-e(T))M$.

By topologically nilpotent, we mean that for each $m\in(1-e(T))M$,
$T^{N}m\rightarrow 0$ when $N\rightarrow+\infty$.

###### Proof.

This follows from lemma 2.4. ∎

######  Lemma 2.13.

If $f:M\rightarrow N$ is a continuous $R$-linear morphism and $T$ is a locally
finite operator acting equivariantly on $M$ and $N$, then $f(e(T)M)\subseteq
e(T)N$ and $f((1-e(T))M)\subseteq(1-e(T))N$.

We say that an exact sequence of profinite $R$-modules $0\rightarrow
M\stackrel{{\scriptstyle d}}{{\rightarrow}}N\stackrel{{\scriptstyle
s}}{{\rightarrow}}L\rightarrow 0$ is strict if $s$ is is open. We remark that
this forces $s$ to be continuous, and $M$ to be a closed subspace of $N$ (so
that $d$ is continuous and closed).

######  Proposition 2.14.

Let $0\rightarrow M\rightarrow N\rightarrow L\rightarrow 0$ be a strict short
exact sequence of profinite $R$-modules and let $T$ be a continuous $R$-linear
operator acting equivariantly on $M,N$ and $L$.

  1. (1)

$T$ is locally finite on $N$ if and only if $T$ is locally finite on $M$ and
$L$.

  2. (2)

If $T$ is locally finite on $M,N$, and $L$, the sequence $0\rightarrow
e(T)M\rightarrow e(T)N\rightarrow e(T)L\rightarrow 0$ is exact.

###### Proof.

We prove the first point. The direct implication is trivial. Let us prove the
reverse implication. Let $V\subset N$ be an open submodule. Since $T$ is
locally finite on $M$, we deduce that there is $n\geq 1$ such that
$\cap_{i=0}^{n-1}T^{-i}(V\cap M)\subseteq T^{-n}(V\cap M)$. It follows that:

$(T^{-n}V+M)\bigcap\cap_{i=0}^{n-1}T^{-i}(V)\subseteq T^{-n}(V).$

To shorten notations, let us denote by $W=T^{-n}V+M$. Since $W$ is open in
$N$, its image $\overline{W}$ in $L$ is open. Since $T$ is locally finite on
$L$, there is $m\geq 0$ such that $\cap_{i=0}^{m-1}T^{-i}(\overline{W})\subset
T^{-m}(\overline{W})$. We deduce that $\cap_{i=0}^{m-1}T^{-i}({W})\subset
T^{-m}({W})$. It follows that:

$\displaystyle T^{-(n+m)}(V\cap W)$ $\displaystyle=$ $\displaystyle
T^{-m}(T^{-n}V)\cap T^{-(n+m)}(W)$ $\displaystyle\supseteq$ $\displaystyle
T^{-m}(W\bigcap\cap_{i=0}^{n-1}T^{-i}(V))\bigcap\cap_{i=0}^{m-1}T^{-i}W$
$\displaystyle\supseteq$ $\displaystyle
T^{-m}(W)\bigcap\cap_{i=0}^{m+n-1}T^{-i}(V)\bigcap\cap_{i=0}^{m-1}T^{-i}W$
$\displaystyle\supseteq$
$\displaystyle\cap_{i=0}^{m+n-1}T^{-i}(V)\bigcap\cap_{i=0}^{m-1}T^{-i}W$
$\displaystyle\supseteq$ $\displaystyle\cap_{i=0}^{m+n-1}T^{-i}(V\cap W)$

Therefore, $\cap_{i=0}^{m+n-1}T^{-i}(V\cap W)$ is an open submodule of $V$
which is stable under $T$. The rest of the properties are easy. ∎

##  3\. The cohomological correspondence $T_{p}$

Let $N$ be an integer $N\geq 3$. Let $p$ be a prime number. Let
$X\rightarrow\operatorname{Spec}\mathbb{Z}_{p}$ be the compactified modular
curve of level $\Gamma_{1}(N)$ ([DR73]). This is a proper smooth relative
curve. Denote by $D$ the boundary divisor, and by $E\rightarrow X$ the semi-
abelian scheme which extends the universal elliptic curve and denote by $e$
the unit section. We let $\omega_{E}=e^{\star}\Omega^{1}_{E/X}$. For any
$k\in\mathbb{Z}$, we denote by $\omega^{k}=\omega_{E}^{\otimes k}$.

###  3.1. The cohomological correspondences $T_{p}$

We denote by $p_{1},p_{2}:X_{0}(p)\rightarrow X$ the Hecke correspondence
which parametrizes an isogeny $\pi:p_{1}^{\star}E\rightarrow p_{2}^{\star}E$
of degree $p$. We denote by $D_{0}(p)$ the boundary divisor in $X_{0}(p)$
(which is reduced, so that $D_{0}(p)=(p_{i}^{-1}D)^{red}$). We let
$\pi_{k}:p_{2}^{\star}\omega^{k}\dashrightarrow p_{1}^{\star}\omega^{k}$ the
rational map which is deduced from the pull-back map on differentials
$p_{2}^{\star}\omega_{E}\rightarrow p_{1}^{\star}\omega_{E}$ (this map is well
defined if $k\geq 0$, and is an isomorphism over $\mathbb{Q}_{p}$ for all
$k$). We also denote by $\pi_{k}^{-1}:p_{1}^{\star}\omega^{k}\dashrightarrow
p_{2}^{\star}\omega^{k}$ the inverse of $\pi_{k}$. We also have a dual isogeny
$\pi^{\vee}:p_{2}^{\star}E\rightarrow p_{1}^{\star}E$ and we denote by
$\pi^{\vee}_{k}:p_{1}^{\star}\omega^{k}\dashrightarrow
p_{2}^{\star}\omega^{k}$ the rational map which is deduced from the pull-back
map on differentials $p_{1}^{\star}\omega_{E}\rightarrow
p_{2}^{\star}\omega_{E}$. We also denote by
$(\pi^{\vee}_{k})^{-1}:p_{1}^{\star}\omega^{k}\dashrightarrow
p_{2}^{\star}\omega^{k}$ the inverse of $\pi^{\vee}_{k}$. We have the
following formula relating $\pi_{k}$ and $\pi^{\vee}_{k}$:

$\pi_{k}\circ\pi^{\vee}_{k}=p^{k}\mathrm{Id}.$

We have natural trace maps
$\mathrm{tr}_{p_{1}}:\mathscr{O}_{X_{0}(p)}\rightarrow
p_{1}^{!}\mathscr{O}_{X}$ and
$\mathrm{tr}_{p_{2}}:\mathscr{O}_{X_{0}(p)}\rightarrow
p_{2}^{!}\mathscr{O}_{X}$. Note that since $p_{1}$ and $p_{2}$ are local
complete intersection morphisms, $p_{1}^{!}\mathscr{O}_{X}$ and
$p_{2}^{!}\mathscr{O}_{X}$ are invertible sheaves. We can restrict the map
$\mathrm{tr}_{p_{i}}$ to $\mathscr{O}_{X_{0}(\ell)}(-D_{0}(p))$ and:

######  Lemma 3.1.

We have factorizations :
$\mathrm{tr}_{p_{1}}:\mathscr{O}_{X_{0}(p)}(-D_{0}(p))\rightarrow
p_{1}^{!}(\mathscr{O}_{X}(-D))$ and
$\mathrm{tr}_{p_{2}}:\mathscr{O}_{X_{0}(p)}(-D_{0}(p))\rightarrow
p_{2}^{!}(\mathscr{O}_{X}(-D))$.

###### Proof.

The boundary divisors in $X$ and $X_{0}(p)$ are reduced. The lemma boils down
to the statement that the trace of function vanishing along the boundary on
$X_{0}(p)$ vanishes on the boundary on $X$. ∎

We have a “naive” (i.e. unnormalized) cohomological correspondence:

$T^{naive}_{p,k}:p_{2}^{\star}\omega^{k}\dashrightarrow p_{1}^{!}\omega^{k}$

which is defined by taking the tensor product of the map
$p_{2}^{\star}\omega^{k}\dashrightarrow p_{1}^{\star}\omega^{k}$ and the map
$\mathrm{tr}_{p_{1}}:\mathscr{O}_{X_{0}(\ell)}\rightarrow
p_{1}^{!}\mathscr{O}_{X}$, and similarly a map
$T^{naive}_{p,k}:p_{2}^{\star}(\omega^{k}(-D))\dashrightarrow
p_{1}^{!}(\omega^{k}(-D))$. We finally let
$T_{p,k}=p^{-\inf\\{1,k\\}}T^{naive}_{p,k}$.

######  Proposition 3.2.

$T_{p,k}$ is a cohomological correspondence
$p_{2}^{\star}\omega^{k}\rightarrow p_{1}^{!}\omega^{k}$.

###### Proof.

The map $T_{p,k}$ is a rational map between invertible sheaves over the
regular scheme $X_{0}(p)$. To check that it is well defined, it is enough to
work in codimension $1$. Since the map is well defined over $\mathbb{Q}_{p}$,
we can thus localize at a generic points $\xi$ of the special fiber and we are
left to prove that it is well defined locally at these points. There are two
types of generic points corresponding to the possibility that the isogeny
$p_{1}^{\star}E\rightarrow p_{2}^{\star}E$ is either multiplicative or étale.
Let us first assume that $\xi$ is étale. The differential map
$(p_{2}^{\star}\omega)_{\xi}\tilde{\rightarrow}(p_{1}^{\star}\omega)_{\xi}$ is
an isomorphism and the map
$(tr_{p_{1}})_{\xi}:(p_{1}^{\star}\mathscr{O}_{X})_{\xi}\rightarrow(p_{1}^{!}\mathscr{O}_{X})_{\xi}$
factors into an isomorphism:
$(tr_{p_{1}})_{\xi}:(p_{1}^{\star}\mathscr{O}_{X})_{\xi}\tilde{\rightarrow}p(p_{1}^{!}\mathscr{O}_{X})_{\xi}$.
It follows that
$(T^{naive}_{p,k})_{\xi}:(p_{2}^{\star}\omega^{k})_{\xi}\tilde{\rightarrow}p(p_{1}^{!}\omega^{k})_{\xi}$.
Let us next assume that $\xi$ is multiplicative. The differential map
$(p_{2}^{\star}\omega)_{\xi}\tilde{\rightarrow}(p_{1}^{\star}\omega)_{\xi}$
factors into an isomorphism
$(p_{2}^{\star}\omega)_{\xi}\tilde{\rightarrow}p(p_{1}^{!}\omega)_{\xi}$ and
the map
$(tr_{p_{1}})_{\xi}:(p_{1}^{\star}\mathscr{O}_{X})_{\xi}\tilde{\rightarrow}(p_{1}^{!}\mathscr{O}_{X})_{\xi}$
is an isomorphism. It follows that
$(T^{naive}_{p,k})_{\xi}:(p_{2}^{\star}\omega^{k})_{\xi}\tilde{\rightarrow}p^{k}(p_{1}^{!}\omega^{k})_{\xi}$.
We deduce that $T_{p,k}$ is indeed a well defined map and that it is optimally
integral. ∎

When the weight is clear, we often write $T_{p}$. The map $T_{p}$ induces a
map on cohomology:

$T_{p}\in\mathrm{End}\mathrm{R\Gamma}(X,\omega^{k})~{}~{}~{}\textrm{and}~{}~{}~{}\mathrm{End}\mathrm{R\Gamma}(X,\omega^{k}(-D))$

obtained by composing the maps:

$\mathrm{R}\Gamma(X,\omega^{k})\stackrel{{\scriptstyle
p_{2}^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p),p_{2}^{\star}\omega^{k})\stackrel{{\scriptstyle
T_{p}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p),p_{1}^{!}\omega^{k})\stackrel{{\scriptstyle\mathrm{tr}_{p_{1}}}}{{\rightarrow}}\mathrm{R}\Gamma(X,\omega^{k})$

and similarly for cuspidal cohomology.

######  Remark 3.3.

The proposition 3.2 is a particular instance of constructions performed in
[FP19], where the problem of constructing Hecke operators on the integral
coherent cohomology of more general Shimura varieties is considered.

######  Remark 3.4.

One can check that our map $T_{p,k}$ has the following effect on
$q$-expansions (of given Nebentypus
$\chi:\mathbb{Z}/N\mathbb{Z}^{\times}\rightarrow\overline{\mathbb{Z}}_{p}^{\times}$):
it maps $\sum a_{n}q^{n}$ to $\sum a_{np}q^{n}+p^{k-1}\chi(p)\sum a_{n}q^{np}$
if $k\geq 1$ and to $p^{1-k}\sum a_{np}q^{n}+\chi(p)\sum a_{n}q^{np}$ if
$k\leq 1$.

######  Remark 3.5.

Our normalization is consistant with standard conjectures on the existence and
properties of Galois representations associated to automorphic forms ([BG14]).
The cohomology groups $\mathrm{H}^{i}(X,\omega^{k})\otimes\mathbb{C}$ can be
computed using automorphic forms and for any $\pi=\pi_{\infty}\otimes\pi_{f}$
contributing to the cohomology, we find that the infinitesimal character of
$\pi_{\infty}$ is given by $(t_{1},t_{2})\mapsto
t_{1}^{\frac{1}{2}}t_{2}^{k-\frac{1}{2}}$ and is indeed $C$-algebraic. By the
Satake isomorphism, we know that
$T^{naive}_{p}|\pi_{p}=p^{1/2}\mathrm{Trace}(\mathrm{Frob}^{-1}_{p}|\mathrm{rec}(\pi_{p}))$
(because $T_{p}^{naive}$ corresponds to the co-character $t\mapsto(1,t^{-1})$
[FP19, Rem. 5.6]!). It is convenient to introduce the twist
$\pi\otimes|.|^{-\frac{1}{2}}$ which is $L$-algebraic, for which we find that
the infinitesimal character of $\pi_{\infty}\otimes|.|^{-\frac{1}{2}}$ is
$(t_{1},t_{2})\mapsto t_{2}^{k-1}$. We make the following normalizations: the
Hodge-Tate weight of the cyclotomic character is $-1$, and we normalize the
reciprocity law by using geometric Frobenii. With these conventions, the Hodge
cocharacter is $t\mapsto(1,t^{1-k})$ and the corresponding Hodge polygon has
slopes $1-k$ and $0$. We find that
$T^{naive}_{p}|\pi_{p}=p\mathrm{Trace}\big{(}\mathrm{Frob}^{-1}_{p}|\mathrm{rec}(\pi_{p}\otimes|.|^{-\frac{1}{2}})\big{)}$.
The Katz-Mazur inequality predicts that the Newton polygon (which has slopes
the $p$-adic valuations of two eigenvalues of $\mathrm{Frob}_{p}$) is above
the Hodge polygon with same ending and inital point, from which we find that
$v(T^{naive}_{p}|\pi_{p})\geq\inf\\{1,k\\}$ and that $T_{p}$ is indeed
optimally integral.

###  3.2. Duality

We let $D_{\mathbb{Z}_{p}}=\mathrm{RHom}(\cdot,\mathbb{Z}_{p})$ be the
dualizing functor on the category of bounded complexes of finite type
$\mathbb{Z}_{p}$-modules [Har66, V]. We denote by
$f:X\rightarrow\mathbb{Z}_{p}$, and by
$f^{!}\mathbb{Z}_{p}=\omega_{X/\mathbb{Z}_{p}}$. This is a dualizing complex
on $X$ [Har66, V, Thm. 8.3]. We recall that
$\omega_{X/\mathbb{Z}_{p}}=\Omega^{1}_{X/\mathbb{Z}_{p}}[1]$ (see [Har66,
III.8]). We denote by $g:X_{0}(p)\rightarrow\mathbb{Z}_{p}$, and by
$\omega_{X_{0}(p)/\mathbb{Z}_{p}}=g^{!}\mathbb{Z}_{p}=p_{1}^{!}\omega_{X/\mathbb{Z}_{p}}$.
This is also a dualizing complex on $X_{0}(p)$. We also recall that
$\omega_{X_{0}(p)/\mathbb{Z}_{p}}=\Omega^{1}_{X_{0}(p)/\mathbb{Z}_{p}}(\mathrm{log}(SS))[1]$
where $SS$ is the reduced closed subscheme of supersingular points in
$X_{0}(p)$. We let $D_{X}=\mathrm{RHom}(\cdot,\omega_{X/\mathbb{Z}_{p}})$ and
$D_{X_{0}(p)}=\mathrm{RHom}(\cdot,\omega_{X_{0}(p)/\mathbb{Z}_{p}})$ the
corresponding dualizing functors on the derived category of (say) bounded
complexes of coherent sheaves on $X$ and $X_{0}(p)$. When the context is clear
we only write $D$ for the dualizing functor.

We have the following Serre duality isomorphism ([Har66, III, Thm 11.1]):

$D(\mathrm{R}f_{\star}\omega^{k})=\mathrm{R}f_{\star}D(\omega^{k})$

and similarly for cuspidal cohomology.

We now want to understand how this duality isomorphism behaves with respect to
Hecke operators. The Hecke operator

$\mathrm{R}\Gamma(X,\omega^{k})\stackrel{{\scriptstyle
p_{2}^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p),p_{2}^{\star}\omega^{k})\stackrel{{\scriptstyle
T_{p}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p),p_{1}^{!}\omega^{k})\stackrel{{\scriptstyle\mathrm{tr}_{p_{1}}}}{{\rightarrow}}\mathrm{R}\Gamma(X,\omega^{k}).$

dualizes to an operator:

$D(\mathrm{R}\Gamma(X,\omega^{k}))\stackrel{{\scriptstyle
p_{1}^{\star}}}{{\rightarrow}}D(\mathrm{R}\Gamma(X_{0}(p),p_{1}^{!}\omega^{k}))\stackrel{{\scriptstyle
D(T_{p})}}{{\rightarrow}}D(\mathrm{R}\Gamma(X_{0}(p),p_{2}^{\star}\omega^{k}))\stackrel{{\scriptstyle\mathrm{tr}_{p_{2}}}}{{\rightarrow}}D(\mathrm{R}\Gamma(X,\omega^{k})).$

We have

$D(\mathrm{R}\Gamma(X_{0}(p),p_{1}^{!}\omega^{k}))=\mathrm{R}\Gamma((X_{0}(p),p_{1}^{\star}D(\omega^{k})),$
$D(\mathrm{R}\Gamma(X_{0}(p),p_{2}^{\star}\omega^{k}))=\mathrm{R}\Gamma((X_{0}(p),p_{2}^{!}D(\omega^{k})),$

by [Har66, III, Thm 11.1 and V, Prop. 8.5], and it remains to understand
$D(T_{p}):p_{1}^{\star}D(\omega^{k})\rightarrow p_{2}^{!}D(\omega^{k})$. We
first recall that we have the Kodaira-Spencer isomorphism over $X$ [Kat73,
A.1.3.17]

$KS:\omega^{2}(-D)\rightarrow\Omega^{1}_{X/\mathbb{Z}_{p}}.$

We consider the correspondence $T_{p}:p_{2}^{\star}\omega^{k}\rightarrow
p_{1}^{!}\omega^{k}$. Applying the functor $D_{X_{0}(p)}$ yields a map
$D(T_{p}):p_{1}^{\star}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})\rightarrow
p_{2}^{!}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})$. If we use the
Kodaira-Spencer isomorphism on both sides and make a $[-1]$-shift, we obtain a
map: $D(T_{p}):p_{1}^{\star}(\omega^{-k+2}(-D))\rightarrow
p_{2}^{!}(\omega^{-k+2}(-D))$. The correspondence $X_{0}(p)$ is isomorphic to
its transpose, by the involution sending the isogeny
$p_{1}^{\star}E\rightarrow p_{2}^{\star}E$ to the dual isogeny. We can
therefore think of $D(T_{p})$ as a map
$p_{2}^{\star}(\omega^{-k+2}(-D))\rightarrow p_{1}^{!}(\omega^{-k+2}(-D))$ by
applying this isomorphism. The main result of this section is the following:

######  Proposition 3.6.

We have $D(T_{p})=T_{p}$ as maps $p_{2}^{\star}(\omega^{-k+2}(-D))\to
p_{1}^{!}(\omega_{-k+2}(-D))$.

######  Lemma 3.7.

The following diagram is commutative:

###### Proof.

Over $X_{0}(p)$ we have a map
$\pi^{\star}:p_{2}^{\star}(\mathcal{H}^{1}_{dR}(E/X),\nabla)\rightarrow
p_{1}^{\star}(\mathcal{H}^{1}_{dR}(E/X),\nabla)$ which induces a commutative
diagram:

or equivalently

It remains to observe that $\pi_{1}^{\vee}\pi_{1}=p$.

∎

######  Lemma 3.8.

$D(T_{p}^{naive})=p^{k-1}T^{naive}_{p}$.

###### Proof.

The dual of the map $p_{2}^{\star}\omega^{k}\dashrightarrow
p_{1}^{\star}\omega^{k}\rightarrow p_{1}^{!}\omega^{k}$ writes
$p_{1}^{\star}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})\rightarrow
p_{1}^{!}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})\dashrightarrow
p_{2}^{!}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})$. The first map
$p_{1}^{\star}\omega^{-k}\otimes
p_{1}^{\star}\omega_{X/\mathbb{Z}_{p}}\rightarrow
p_{1}^{\star}\omega^{-k}\otimes\omega_{X_{0}(p)/\mathbb{Z}_{p}}$ is just
$1\otimes\mathrm{tr}_{p_{1}}$.

The second map
$p_{1}^{!}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})\rightarrow
p_{2}^{!}(\omega^{-k}\otimes\omega_{X/\mathbb{Z}_{p}})$ writes also
$p_{1}^{\star}\omega^{-k}\otimes\omega_{X_{0}(p)/\mathbb{Z}_{p}}\rightarrow
p_{2}^{\star}\omega^{-k}\otimes\omega_{X_{0}(p)/\mathbb{Z}_{p}}$ and is
$\pi_{-k}^{-1}\otimes 1$.

On the other hand,
$\omega_{X_{0}(p)/\mathbb{Z}_{p}}=p_{1}^{!}\mathscr{O}_{X}\otimes
p_{1}^{\star}\omega_{X/\mathbb{Z}_{p}}=p_{1}^{!}\mathscr{O}_{X}\otimes
p_{1}^{\star}\omega_{2}(-D)[1]$ using Kodaira-Spencer and similarly
$\omega_{X_{0}(p)/\mathbb{Z}_{p}}=p_{2}^{!}\mathscr{O}_{X}\otimes
p_{2}^{\star}\omega_{X/\mathbb{Z}_{p}}=p_{2}^{!}\mathscr{O}_{X}\otimes
p_{2}^{\star}\omega_{2}(-D)[1]$ using again Kodaira-Spencer. The identity map:
$p_{1}^{!}\mathscr{O}_{X}\otimes p_{1}^{\star}\omega_{2}(-D)\rightarrow
p_{2}^{!}\mathscr{O}_{X}\otimes p_{2}^{\star}\omega_{2}(-D)$ decomposes into :

$\mathrm{tr}_{p_{2}}\mathrm{tr}_{p_{1}}^{-1}\otimes(\pi^{\vee}_{1}\otimes(\pi_{1})^{-1})$

according to lemma 3.7. We get that
$D(T_{\ell}^{naive}):p_{1}^{\star}(\omega^{-k+2}(-D))\dashrightarrow
p_{2}^{!}(\omega^{-k+2}(-D))$ is
$\mathrm{tr}_{p_{2}}\mathrm{tr}_{p_{1}}^{-1}\circ\mathrm{tr}_{p_{1}}\otimes(\pi^{\vee}_{1}\otimes(\pi_{1})^{-1})\otimes(\pi_{-k})^{-1}$.
It remains to observe that $\pi_{1-k}\pi^{\vee}_{1-k}=p^{1-k}$. ∎

###### Proof of Proposition 3.6.

This follows from the identity:

$-\inf\\{1,k\\}+k-1=-\inf\\{1,2-k\\}.$

∎

##  4\. Higher Hida theory

###  4.1. The mod $p$ theory

Let $\mathfrak{X}$ be the $p$-adic completion of $X$ and
$X_{n}\rightarrow\operatorname{Spec}\mathbb{Z}/p^{n}\mathbb{Z}$ be the scheme
obtained by reduction modulo $p^{n}$. Let $X_{n}^{ord}$ be the ordinary locus
in $X_{n}$ and $\mathfrak{X}^{ord}$ the ordinary locus in $\mathfrak{X}$.

We recall that $X_{0}(p)_{1}=X_{0}(p)_{1}^{F}\cup X_{0}(p)_{1}^{V}$ is the
union of the Frobenius and Verschiebung correspondences. We let $p_{i}^{F}$
and $p_{i}^{V}$ be the restrictions of the projections $p_{i}$ to these
components. The projection $p^{V}_{2}:X_{0}(p)_{1}^{V}\rightarrow X_{1}$ is an
isomorphism (and $X_{0}(p)_{1}^{V}$ parametrizes the Verschiebung isogeny
$(p^{V}_{1})^{\star}E\simeq(p^{V}_{2})^{\star}E^{(p)}\rightarrow(p^{V}_{2})^{\star}E$).
The projection $p^{F}_{1}:X_{0}(p)_{1}^{F}\rightarrow X_{1}$ is an isomorphism
(and $X_{0}(p)_{1}^{F}$ parametrizes the Frobenius isogeny
$(p^{F}_{1})^{\star}E\rightarrow(p^{F}_{2})^{\star}E\simeq(p^{F}_{1})^{\star}E^{(p)}$).
We denote by $i^{F}$ and $i^{V}$ the inclusions
$X_{0}(p)_{1}^{F}\hookrightarrow X_{0}(p)_{1}$ and
$X_{0}(p)_{1}^{V}\hookrightarrow X_{0}(p)_{1}$.

######  Lemma 4.1.

If $k\geq 2$, we have a factorization:

If $k\leq 0$, we have a factorization:

###### Proof.

By proposition 2.3, this amounts to check that the cohomological
correspondence $T_{p}:p_{2}^{\star}\omega^{k}\rightarrow p_{1}^{!}\omega^{k}$
vanishes at any generic point of multiplicative type in $X_{0}(p)_{1}$ if
$k\geq 2$, and at any generic point of étale type in $X_{0}(p)_{1}$ if $k\leq
0$. This follows from the normalization of the correspondence as explained in
the proof of proposition 3.2. ∎

######  Remark 4.2.

We can informally rephrase this lemma by saying that we have congruences:
$T_{p}=U_{p}\mod p$ if $k\geq 2$ and $T_{p}=\mathrm{Frob}\mod p$ if $k\leq 0$,
see remark 3.4.

######  Proposition 4.3.

For all $k\geq 2$, the cohomological correspondence $T_{p}$ induces a map:

$p_{2}^{\star}(\omega^{k}((np+k-2)SS))\rightarrow p_{1}^{!}(\omega^{k}(nSS)).$

For all $k\leq 0$, the cohomological correspondence $T_{p}$ induces a map:

$p_{2}^{\star}(\omega^{k}(-nSS))\rightarrow p_{1}^{!}(\omega^{k}((-np+k)SS)).$

###### Proof.

We first prove the first claim (when $k\geq 2$). The cohomological
correspondence is supported on $X_{0}(p)_{1}^{V}$. The map $p^{V}_{1}$ is
totally ramified of degree $p$ and the map $p_{2}^{V}$ is an isomorphism. It
follows that we have an equality of divisors
$(p^{V}_{1})^{\star}(SS)=p(p^{V}_{2})^{\star}(SS)$. Let us simply denote by
$SS=(p^{V}_{2})^{\star}(SS)$ (this is the reduced supersingular locus
divisor).

In particular, we deduce that the map
$(p^{V}_{2})^{\star}\omega^{2}\rightarrow(p^{V}_{1})^{!}\omega^{2}$ induces
(after twisting by $\mathscr{O}_{X_{0}(p)^{V}_{1}}(npSS)$ a morphism:
$(p^{V}_{2})^{\star}(\omega^{2}(npSS))\rightarrow(p^{V}_{1})^{!}(\omega^{2}(nSS))$.

This proves the claim for $k=2$. For $k\geq 3$, we remark that the
cohomological correspondence
$(p^{V}_{2})^{\star}\omega^{k}\rightarrow(p^{V}_{1})^{!}\omega^{k}$ is the
tensor product of the map
$(p^{V}_{2})^{\star}\omega^{2}\rightarrow(p^{V}_{1})^{!}\omega^{2}$ and the
map
$(p^{V}_{2})^{\star}\omega^{k-2}\rightarrow(p^{V}_{1})^{\star}\omega^{k-2}$.
But $(p^{V}_{1})^{\star}\omega_{E}\simeq((p^{V}_{2})^{\star}\omega_{E})^{p}$
and the differential of the isogeny
$(p^{V}_{2})^{\star}\omega_{E}\rightarrow(p^{V}_{1})^{\star}\omega_{E}$
identifies with the Hasse invariant and induces an isomorphism:
$(p^{V}_{2})^{\star}\omega_{E}(SS)\rightarrow(p^{V}_{1})^{\star}\omega_{E}$.
We deduce that there is therefore a map:
$p_{2}^{\star}(\omega^{k}((np+k-2)SS))\rightarrow p_{1}^{!}(\omega^{k}(nSS))$.

We now prove the second claim (for $k\leq 0$). The cohomological
correspondence is supported on $X_{0}(p)_{1}^{F}$. The map $p^{F}_{2}$ is
totally ramified of degree $p$ and the map $p_{1}^{F}$ is an isomorphism. It
follows that we have an equality of divisors
$(p^{F}_{2})^{\star}(SS)=p(p^{F}_{1})^{\star}(SS)$. Let us simply denote by
$SS=(p^{F}_{1})^{\star}(SS)$ (this is the reduced supersingular locus
divisor).

In particular, we deduce that the map
$(p^{F}_{2})^{\star}\mathscr{O}_{X}\rightarrow(p^{F}_{1})^{!}\mathscr{O}_{X}$
induces (after twisting by $\mathscr{O}_{X_{0}(p)^{V}_{1}}(-npSS)$ an
morphism:
$(p^{F}_{2})^{\star}\mathscr{O}_{X}(-nSS)\rightarrow(p^{F}_{1})^{!}\mathscr{O}_{X}(-npSS)$.

This proves the claim for $k=0$. For $k\leq-1$, we remark that the
cohomological correspondence
$(p^{F}_{2})^{\star}\omega^{k}\rightarrow(p^{F}_{1})^{!}\omega^{k}$ is the
tensor product of the map
$(p^{F}_{2})^{\star}\mathscr{O}_{X}\rightarrow(p^{F}_{1})^{!}\mathscr{O}_{X}$
(the cohomological correspondence for $k=0$) and a map
$(p^{F}_{2})^{\star}\omega^{k}\rightarrow(p^{F}_{1})^{\star}\omega^{k}$ that
we now describe. Informally, this map is deduced from the differential of the
isogeny $(p_{1}^{F})^{\star}E\rightarrow(p_{2}^{F})^{\star}E$, after
normalizing by a factor $p^{-1}$ (one can give sense of this over the formal
scheme ordinary locus). Equivalently, it is deduced from the differential of
the isogeny of the dual map
$(p_{2}^{F})^{\star}E\rightarrow(p_{1}^{F})^{\star}E$ (the Verschiebung map).

We observe that
$(p^{F}_{2})^{\star}\omega_{E}\simeq((p^{F}_{1})^{\star}\omega_{E})^{p}$ and
there is a natural isomorphism:
$(p^{F}_{2})^{\star}\omega_{E}\stackrel{{\scriptstyle(p^{F}_{1})^{\star}Ha^{-1}}}{{\longrightarrow}}(p^{F}_{1})^{\star}\omega_{E}(SS)$,
and therefore, for all $k\leq 0$, an isomorphism:
$(p^{F}_{2})^{\star}\omega^{k}\stackrel{{\scriptstyle(p^{F}_{1})^{\star}Ha^{k}}}{{\longrightarrow}}(p^{F}_{1})^{\star}\omega^{k}(-kSS)$
which factors the map
$(p^{F}_{2})^{\star}\omega^{k}\stackrel{{\scriptstyle(p^{F}_{1})^{\star}Ha^{k}}}{{\rightarrow}}(p^{F}_{1})^{\star}\omega^{k}$.
We deduce that there is therefore a map:
$p_{2}^{\star}(\omega^{k}(-nSS))\rightarrow p_{1}^{!}(\omega^{k}((-np+k)SS))$.
∎

######  Corollary 4.4.

  1. (1)

The $T_{p}$ operator acts on $\mathrm{R}\Gamma(X_{1},\omega^{k}(nSS))$ for all
$n\geq 0$ and $k\geq 2$, and the maps
$\mathrm{R}\Gamma(X_{1},\omega^{k}(nSS))\rightarrow\mathrm{R}\Gamma(X_{1},\omega^{k}(n^{\prime}SS))$
are equivariant for $0\leq n\leq n^{\prime}$,

  2. (2)

We have commutative diagrams for all $n\geq 0$ and $k\geq 2$:

  3. (3)

The $T_{p}$ operator acts on $\mathrm{R}\Gamma(X_{1},\omega^{k}(nSS))$ for all
$n\leq 0$ and $k\leq 0$, and the maps
$\mathrm{R}\Gamma(X_{1},\omega^{k}(nSS))\rightarrow\mathrm{R}\Gamma(X_{1},\omega^{k}(n^{\prime}SS))$
are equivariant for $0\geq n^{\prime}\geq n$,

  4. (4)

We have commutative diagrams for all $n\leq 0$ and $k\leq 0$:

For any $k$, we define as usual
$\mathrm{H}^{i}_{c}(X^{ord}_{1},\omega^{k})=\lim_{n}\mathrm{H}^{i}(X_{1},\omega^{k}(-nSS))$
following [Har72]. This is a profinite $\mathbb{F}_{p}$-vector space. We also
recall that
$\mathrm{H}^{i}(X^{ord}_{1},\omega^{k})=\mathrm{colim}_{n}\mathrm{H}^{i}(X_{1},\omega^{k}(nSS))$.

######  Corollary 4.5.

  1. (1)

If $k\geq 2$, $T_{p}$ is locally finite on
$\mathrm{H}^{i}(X^{ord}_{1},\omega^{k})$.

  2. (2)

If $k\leq 0$, $T_{p}$ is locally finite on
$\mathrm{H}^{i}_{c}(X^{ord}_{1},\omega^{k})$.

  3. (3)

If $k\geq 3$, we have
$e(T_{p})\mathrm{H}^{i}(X^{ord}_{1},\omega^{k})=e(T_{p})\mathrm{H}^{i}(X_{1},\omega^{k})$.

  4. (4)

If $k=2$, we have
$e(T_{p})\mathrm{H}^{i}(X^{ord}_{1},\omega^{2})=e(T_{p})\mathrm{H}^{i}(X_{1},\omega^{2}(SS))$.

  5. (5)

If $k\leq-1$, we have
$e(T_{p})\mathrm{H}^{i}_{c}(X^{ord}_{1},\omega^{k})=e(T_{p})\mathrm{H}^{i}(X_{1},\omega^{k})$.

  6. (6)

If $k=0$, we have
$e(T_{p})\mathrm{H}^{i}_{c}(X^{ord}_{1},\mathscr{O}_{X})=e(T_{p})\mathrm{H}^{i}(X_{1},\mathscr{O}_{X}(-SS))$.

######  Corollary 4.6.

  1. (1)

If $k\leq-1$, $e(T_{p})\mathrm{R}\Gamma(X_{1},\omega^{k})$ is concentrated in
degree $1$,

  2. (2)

If $k\geq 3$, $e(T_{p})\mathrm{R}\Gamma(X_{1},\omega^{k})$ is concentrated in
degree $0$.

###### Proof.

This follows from $\mathrm{H}^{1}(X^{ord}_{1},\omega^{k})=0$ and
$\mathrm{H}^{0}_{c}(X^{ord}_{1},\omega^{k})=0$ because $X^{ord}_{1}$ is affine
∎

######  Remark 4.7.

Of course, this last corollary is obvious from the Riemann-Roch theorem, but
the given proof is not using this theorem.

###  4.2. The $p$-adic theory

####  4.2.1. The Igusa tower

Recall that the principal $\mathbb{G}_{m}$-torsor $\omega_{E}$ has a
$\mathbb{Z}_{p}^{\times}$-reduction (in the pro-étale topology) over
$\mathfrak{X}^{ord}$ given by $T_{p}(E)^{et}$ and the Hodge-Tate map:

$\mathrm{HT}:T_{p}(E)^{et}\rightarrow\omega_{E}$

which induces an isomorphism:

$\mathrm{HT}\otimes
1:T_{p}(E)^{et}\otimes_{\mathbb{Z}_{p}}\mathscr{O}_{\mathfrak{X}^{ord}}\rightarrow\omega_{E}.$

We can form
$\pi:\mathfrak{IG}=\mathrm{Isom}(\mathbb{Z}_{p},T_{p}(E)^{et})\rightarrow\mathfrak{X}^{ord}$,
the Igusa tower. This is a $p$-adic formal scheme, and a pro-finite étale
cover of $\mathfrak{X}^{ord}$.

Let $\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$. Let
$\kappa^{un}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda^{\times}$ be the
universal character. For any $k\in\mathbb{Z}$, we have an algebraic character
$\mathbb{Z}_{p}^{\times}\rightarrow\mathbb{Z}_{p}^{\times}$, given by
$x\mapsto x^{k}$ and we also denote by $k:\Lambda\rightarrow\mathbb{Z}_{p}$
the corresponding algebra morphism. We let
$\omega^{\kappa^{un}}=(\mathscr{O}_{\mathfrak{IG}}\hat{\otimes}\Lambda)^{\mathbb{Z}_{p}^{\times}}$
where the invariants are taken for the diagonal action.

######  Lemma 4.8.

The sheaf $\omega^{\kappa^{un}}$ is an invertible sheaf of
$\Lambda\hat{\otimes}\mathcal{O}_{\mathfrak{X}^{ord}}$-modules and for any
$k\in\mathbb{Z}$, there is a canonical isomorphism of invertible sheaves over
$\mathfrak{X}^{ord}$:

$\omega^{k}\rightarrow\omega^{\kappa^{un}}\otimes_{\Lambda,k}\mathbb{Z}_{p}.$

####  4.2.2. Cohomology of the ordinary locus

We may now consider the following $\Lambda$-modules:

$\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$

and

$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}}).$

Let us give the definition of the second module. Let $\mathfrak{m}_{\Lambda}$
be the kernel of the reduction map
$\Lambda\rightarrow\mathbb{F}_{p}[\mathbb{F}_{p}^{\times}]$. We first define
$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})=\mathrm{H}^{1}_{c}(X_{n}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})$
as follows: we can take any extension of
$\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}$ to a coherent sheaf
$\mathscr{F}$ over $X_{n}$ (this means that if $j:X_{n}^{ord}\rightarrow
X_{n}$ is the inclusion, then
$j^{\star}\mathscr{F}=\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}$) and
we let
$\mathrm{H}^{1}_{c}(X_{n}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})=\lim_{l}\mathrm{H}^{1}(X_{n},\mathscr{I}^{l}\mathscr{F})$
for $\mathscr{I}$ the ideal of the complement of $X_{n}^{ord}$ in $X_{n}$. We
remark that this is a profinite $\Lambda/(\mathfrak{m}_{\Lambda})^{n}$-module.
The pro-finite module
$\mathrm{H}^{1}_{c}(X_{n}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})$
is well-defined (it does not depend on the choice of $\mathscr{F}$ as a
topological module) following [Har72, §2]. We then define
$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}}):=\lim_{n}\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})$.

####  4.2.3. $U_{p}$ and Frobenius

We now define a $U_{p}$-operator and Frobenius operator on these cohomologies.

There is a Frobenius morphism
$F:\mathfrak{X}^{ord}\rightarrow\mathfrak{X}^{ord}$ which is given by
$E\mapsto E/H_{1}^{can}$ where $H_{1}^{can}\subset E[p]$ is the multiplicative
subgroup.

This map extends to the Igusa tower into a map:
$F:\mathfrak{IG}\^{E}\rightarrow\mathfrak{IG}$, which is given by
$(E,\psi:\mathbb{Z}_{p}\simeq
T_{p}(E)^{et})\mapsto(E/H_{1}^{can},\psi^{\prime}:\mathbb{Z}_{p}\simeq
T_{p}(E/H_{1}^{can})^{et})$ where $\psi^{\prime}$ is defined by
$\mathbb{Z}_{p}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}T_{p}(E)^{et}\tilde{\rightarrow}T_{p}(E/H_{1}^{can})^{et}$.

We deduce that there are two maps, the natural pull-back map on functions:
$F^{\star}\mathscr{O}_{\mathfrak{IG}}\rightarrow\mathscr{O}_{\mathfrak{IG}}$
and the trace map
$\mathrm{tr}_{F}:F_{\star}\mathscr{O}_{\mathfrak{IG}}\rightarrow\mathscr{O}_{\mathfrak{IG}}$.

######  Lemma 4.9.

We have $\mathrm{tr}_{F}(F_{\star}\mathscr{O}_{\mathfrak{IG}})\subset
p\mathscr{O}_{\mathfrak{IG}}$.

###### Proof.

We have $\mathrm{tr}_{F}(F_{\star}\mathscr{O}_{\mathfrak{X}^{ord}})\subset
p\mathscr{O}_{\mathfrak{X}^{ord}}$. Since
$F\times\pi:\mathfrak{IG}\tilde{\rightarrow}\mathfrak{IG}\times_{\pi,\mathfrak{X}^{ord},F}\mathfrak{X}^{ord}$,
we deduce that $\mathrm{tr}_{F}(F_{\star}\mathscr{O}_{\mathfrak{IG}})\subset
p\mathscr{O}_{\mathfrak{IG}}$. ∎

######  Corollary 4.10.

There are two maps
$F:F^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$ and
$U_{p}:F_{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$.

###### Proof.

The maps
$F^{\star}\mathscr{O}_{\mathfrak{IG}}\rightarrow\mathscr{O}_{\mathfrak{IG}}$
and
$\frac{1}{p}\mathrm{tr}_{F}:F_{\star}\mathscr{O}_{\mathfrak{IG}}\rightarrow\mathscr{O}_{\mathfrak{IG}}$
are $\mathbb{Z}_{p}^{\times}$-equivariant. We can tensor with $\Lambda$ and
take the invariants. ∎

####  4.2.4. $U_{p}$, Frobenius and $T_{p}$

We now describe the specialization of these maps at classical weight
$k\in\mathbb{Z}$ in the spirit of section 3.1. Let
$F:F^{\star}\omega^{k}\rightarrow\omega^{k}$ be the specialization of the map
constructed in corollary 4.10. The universal isogeny $\pi:E\rightarrow
E/H_{1}^{can}$ has a differential
$\pi^{\star}:F^{\star}\omega_{E}\rightarrow\omega_{E}$.

######  Lemma 4.11.

$F=p^{-k}(\pi^{\star})^{k}:F^{\star}\omega^{k}\rightarrow\omega^{k}$.

###### Proof.

We have a commutative diagram:

from which it follows easily that $F^{\star}\omega^{k}\rightarrow\omega^{k}$
is given by $((\pi^{D})^{\star})^{-k}=p^{-k}(\pi^{\star})^{k}$. ∎

By duality, there is an étale isogeny $\pi^{D}:E/H_{1}^{can}\rightarrow E$
with differential $(\pi^{D})^{\star}:\omega_{E}\rightarrow
F^{\star}\omega_{E}$, and we can construct a map:

$F_{\star}\omega^{k}\stackrel{{\scriptstyle((\pi^{D})^{\star})^{k}}}{{\rightarrow}}F_{\star}F^{\star}\omega^{k}\stackrel{{\scriptstyle\frac{1}{p}\mathrm{tr}_{F}}}{{\rightarrow}}\omega^{k}.$

######  Lemma 4.12.

The above map coincides with the map
$U_{p}:F_{\star}\omega^{k}\rightarrow\omega^{k}$.

###### Proof.

Similar to lemma 4.11 and left to the reader. ∎

In section 3.1 we constructed a cohomological correspondence $T_{p}$ over
$X_{0}(p)$:

$T_{p}:p_{2}^{\star}\omega^{k}\rightarrow p_{1}^{!}\omega^{k}.$

We can consider the completion $\mathfrak{X}_{0}(p)$ and its ordinary part
$\mathfrak{X}_{0}(p)^{ord}$ which is the disjoint union of two types of
irreducible components:

$\mathfrak{X}_{0}(p)^{ord}=\mathfrak{X}_{0}(p)^{ord,F}\coprod\mathfrak{X}_{0}(p)^{ord,V},$

where on the first components the universal isogeny is not étale, and where it
is étale on the other components.

On $\mathfrak{X}_{0}(p)^{ord,F}$, the map $p_{1}$ is an isomorphism and the
map $p_{2}$ identifies with the Frobenius map $F$. On
$\mathfrak{X}_{0}(p)^{ord,V}$, the map $p_{2}$ is an isomorphism and the map
$p_{1}$ identifies with the Frobenius map $F$. We therefore can think of
$U_{p}$ and $F$ has cohomological correspondences:
$p_{2}^{\star}\omega^{k}\rightarrow p_{1}^{!}\omega^{k}$ supported
respectively on $\mathfrak{X}_{0}(p)^{ord,V}$ and
$\mathfrak{X}_{0}(p)^{ord,F}.$

One can also restrict $T_{p}$ to a cohomological correspondence over
$\mathfrak{X}_{0}(p)^{ord}$ and project it on the components
$\mathfrak{X}_{0}(p)^{ord,F}$ and $\mathfrak{X}_{0}(p)^{ord,V}$. We denote by
$T_{p}^{F}$ and $T_{p}^{V}$ the two projections of the correspondence $T_{p}$.

######  Lemma 4.13.

  1. (1)

We have $T_{p}^{F}=p^{\sup\\{0,k-1\\}}F$.

  2. (2)

We have $T_{p}^{V}=p^{\sup\\{0,1-k\\}}U_{p}$.

  3. (3)

If $k\geq 1$, we have $T_{p}=U_{p}+p^{k-1}F$,

  4. (4)

If $k\leq 1$, we have $T_{p}=F+p^{1-k}U_{p}$.

###### Proof.

This follows from the definitions, compare also with remark 3.4. ∎

We end up this discussion with duality.

######  Lemma 4.14.

We have $D(F)=U_{p}$.

###### Proof.

Compare with proposition 3.6. ∎

####  4.2.5. Higher Hida theory

######  Theorem 4.15.

  1. (1)

$F$ is locally finite on
$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$ and
$e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$ is a finite
projective $\Lambda$-module. Moreover,

$e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\otimes_{\Lambda,k}\mathbb{Z}_{p}=e(T_{p})\mathrm{H}^{1}({X},\omega^{k})$

if $k\leq-1$.

  2. (2)

$U_{p}$ is locally finite on
$\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$ and
$e(U_{p})\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$ is a finite
projective $\Lambda$-module. Moreover,

$e(U_{p})\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\otimes_{\Lambda,k}\mathbb{Z}_{p}=e(T_{p})\mathrm{H}^{0}({X},\omega^{k})$

if $k\geq 3$.

###### Proof.

We first need to justify why $F$ is acting on
$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$. We exhibit a
continuous action on
$\mathrm{H}^{1}_{c}({X}_{n}^{ord},\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n})$,
compatible for all $n$. Let $X_{0}(p)_{n}^{ord}\hookrightarrow X_{0}(p)_{n}$
be the ordinary locus. We have $X_{0}(p)_{n}^{ord}=X_{0}(p)_{n}^{ord,F}\coprod
X_{0}(p)_{n}^{ord,V}$, where $X_{0}(p)_{n}^{ord,F}$ is the component where the
universal isogeny has connected kernel, and $X_{0}(p)_{n}^{ord,V}$ the
component where the universal isogeny has étale kernel. The graph of the map
$F:X^{ord}_{n}\rightarrow X_{n}^{ord}$ is $X_{0}(p)_{n}^{ord,F}$. We can
therefore think of
$F:F^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$ as a
cohomological correspondence on $X_{0}(p)_{n}^{ord}$:

$p_{2}^{\star}\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}\rightarrow
p_{1}^{!}\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}$

which is precisely given by $F$ on the component $X_{0}(p)_{n}^{ord,F}$ and by
$0$ on the component $X_{0}(p)_{n}^{ord,V}$ .

We take a coherent sheaf $\mathscr{F}$ over $X_{n}$ extending
$\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}$. If we denote by
$\mathscr{I}$ the ideal of the complement of $X_{n}^{ord}$ in $X_{n}$ and by
$j:X_{n}^{ord}\rightarrow X_{n}$ the inclusion. We have
$j_{\star}\omega^{\kappa^{un}}/(\mathfrak{m}_{\Lambda})^{n}=\mathrm{colim}~{}\mathscr{I}^{-n}\mathscr{F}$
and we therefore have a cohomological correspondence:
$p_{2}^{\star}(\mathrm{colim}_{l}\mathscr{I}^{-l}\mathscr{F})\rightarrow
p_{1}^{!}(\mathrm{colim}_{l}~{}\mathscr{I}^{-l}\mathscr{F})$ and it follows
that there exists $l$ such that this map induces:
$p_{2}^{\star}(\mathscr{F})\rightarrow p_{1}^{!}(\mathscr{I}^{-l}\mathscr{F})$
and there also exists $m$ such that $p_{2}^{\star}\mathscr{I}^{m}\subset
p_{1}^{\star}\mathscr{I}$. Therefore, we have maps:
$p_{2}^{\star}(\mathscr{I}^{km}\mathscr{F})\rightarrow
p_{1}^{!}(\mathscr{I}^{-l+k}\mathscr{F})$ for all $k\geq 0$. This provides a
map
$F:\mathrm{H}^{i}(X_{n},\mathscr{I}^{km}\mathscr{F})\rightarrow\mathrm{H}^{i}(X_{n},\mathscr{I}^{-l+k}\mathscr{F})$.
Passing to the projective limit over $k$, we finally get a continuous map
$F\in\mathrm{End}(\mathrm{H}^{1}_{c}({X}_{n}^{ord},\omega^{\kappa^{un}}/\mathfrak{m}_{\Lambda}^{n}))$.

We now need to prove that $F$ is locally finite. We first deal with $n=1$. In
that case,
$\mathrm{H}^{1}_{c}({X}_{1}^{ord},\omega^{\kappa^{un}}/\mathfrak{m}_{\Lambda})=\oplus_{k=-p+2}^{0}\mathrm{H}^{1}_{c}({X}_{1}^{ord},\omega^{k})$
(in this last formula, we can let $k$ go through any set of representatives of
$\mathbb{F}_{p}^{\times}$ in $\mathbb{Z}$) and the isomorphism is equivariant
for the action of $F$ on the LHS, and $T_{p}$ on the RHS by lemma 4.13. It
follows from corollary 4.5 that $F$ is locally finite for $n=1$, and also that
$e(F)\mathrm{H}^{1}_{c}({X}_{1}^{ord},\omega^{\kappa^{un}}/\mathfrak{m}_{\Lambda})$
is a finite $\mathbb{F}_{p}$-vector space. We deal with the general case by
induction, using the short exact sequences in cohomology ($\mathfrak{X}^{ord}$
is affine) and proposition 2.14:

$0\rightarrow\mathrm{H}^{1}_{c}({X}_{n+1}^{ord},\omega^{\kappa^{un}}\otimes(\mathfrak{m}_{\Lambda}^{n}/\mathfrak{m}_{\Lambda}^{n+1}))\rightarrow\mathrm{H}^{1}_{c}({X}_{n+1}^{ord},\omega^{\kappa^{un}}/\mathfrak{m}_{\Lambda}^{n+1})\rightarrow\mathrm{H}^{1}_{c}({X}_{1}^{ord},\omega^{\kappa^{un}}/\mathfrak{m}_{\Lambda}^{n})\rightarrow
0$

It follows that $F$ is locally finite and that
$e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})$ is a finite
projective $\Lambda$-module.

Finally, for all $k\leq-1$, we have an isomorphism:
$N\otimes_{\Lambda,k}\mathbb{Z}_{p}=e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{k})$
and there is a canonical map:

$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{k})\rightarrow\mathrm{H}^{1}(X,\omega^{k})$

and by applying projectors on both side we get a map:

$e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{k})\rightarrow
e(T_{p})\mathrm{H}^{1}(X,\omega^{k})$

of finite free $\mathbb{Z}_{p}$-modules, which is an isomorphism modulo $p$ by
corollary 4.5. Therefore this map is an isomorphism. The proof of the second
point of the theorem follows along similar lines. ∎

###  4.3. Serre duality

Recall that we have a residue map
$\mathrm{res}:\mathrm{H}^{1}(X,\Omega^{1}_{X/\mathbb{Z}_{p}})\rightarrow\mathbb{Z}_{p}$.
Therefore, there is a natural map:
$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2}(-D)\otimes\Lambda)\rightarrow\Lambda$
which is obtained as the composite

$\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2}(-D)\otimes\Lambda)\rightarrow\mathrm{H}^{1}(X,\omega^{2}(-D)\otimes\Lambda)\rightarrow\mathrm{H}^{1}(X,\omega^{2}(-D))\otimes\Lambda$
$\stackrel{{\scriptstyle KS\otimes
1}}{{\longrightarrow}}\mathrm{H}^{1}(X,\Omega^{1}_{X/\mathbb{Z}_{p}})\otimes\Lambda\stackrel{{\scriptstyle\mathrm{res}\otimes
1}}{{\longrightarrow}}\Lambda$

Let us denote by
$\omega^{2-\kappa^{un}}(-D)=\omega^{2}\otimes\underline{\mathrm{Hom}}(\omega^{\kappa^{un}},\Lambda\otimes\mathscr{O}_{\mathfrak{X}^{ord}})$.
This is an invertible sheaf of
$\Lambda\otimes\mathscr{O}_{\mathfrak{X}^{ord}}$-modules over
$\mathfrak{X}^{ord}$.

######  Remark 4.16.

The following character $\mathbb{Z}_{p}^{\times}\rightarrow\Lambda^{\times}$,
$t\mapsto t^{2}(\kappa^{un}(t))^{-1}$ induces an automorphism
$d:\Lambda\rightarrow\Lambda$. We have an isomorphism of
$\mathscr{O}_{\mathfrak{X}^{ord}}\hat{\otimes}\Lambda$-modules:
$\omega^{2-\kappa^{un}}(-D)=\omega^{\kappa^{un}}(-D)\otimes_{\Lambda,d}\Lambda$.

We can therefore define a pairing:

$\langle,\rangle:\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\times\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-\kappa^{un}}(-D))\rightarrow\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2}(-D)\otimes_{\mathbb{Z}_{p}}\Lambda)\rightarrow\Lambda$

######  Proposition 4.17.

For any
$(f,g)\in\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\times\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-\kappa^{un}}(-D))$,
we have $\langle U_{p}f,g\rangle=\langle f,Fg\rangle$.

###### Proof.

We have a commutative diagram:

where the vertical maps are injective. It suffices therefore to prove the
identity for the pairing
$\langle,\rangle_{k}:\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{k})\times\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-k}(-D))\rightarrow\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2}(-D))\rightarrow\mathbb{Z}_{p}$

We work modulo $p^{n}$ and let $\mathscr{I}$ be the ideal of the complement of
$X_{n}^{ord}$.

We have
$\mathrm{H}^{1}_{c}({X}_{n}^{ord},\omega^{2-k}(-D))=\lim\mathrm{H}^{1}({X}_{n},\mathscr{I}^{n}\omega^{2-k}(-D))$
and the cohomological correspondence over $X_{0}(p)_{n}$

$F:p_{2}^{\star}\mathscr{I}^{mk}\omega^{2-k}(-D)\rightarrow
p_{1}^{!}\mathscr{I}^{-l+k}\omega^{2-k}(-D)$

Its dual is a map: $D(F)=p_{1}^{\star}\mathscr{I}^{-k+l}\omega^{k}\rightarrow
p_{2}^{!}\mathscr{I}^{-mk}\omega^{k}$ which equals, on the limit over $k$, to
the cohomological correspondence $U_{p}$ by lemma 4.14.

∎

This pairing restricts to a pairing:

$\langle,\rangle:e(U_{p})\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\times
e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-\kappa^{un}}(-D))\rightarrow\Lambda.$

######  Theorem 4.18.

  1. (1)

The pairing $\langle,\rangle$ is a perfect pairing,

  2. (2)

For any $(f,g)\in
e(U_{p})\mathrm{H}^{0}(\mathfrak{X}^{ord},\omega^{\kappa^{un}})\times
e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-\kappa^{un}}(-D))$,

$\langle U_{p}f,g\rangle=\langle f,Fg\rangle,$

  3. (3)

The pairing $\langle,\rangle$ is compatible with the classical pairing in the
sense that for any $k\in\mathbb{Z}$, we have a commutative diagram, where the
bottom pairing is the one deduced from Serre duality on $X$:

###### Proof.

The second point follows from proposition 4.17. The first point will follow
from the third point, since the map $i$ and $j$ are isomorphisms for integers
$k\geq 3$ and the bottom pairing is perfect. Let us prove the last point.
First, we consider the diagram without applying projectors:

which is commutative by construction. For any
$f\in\mathrm{H}^{0}(X,\omega^{k})$ and
$g\in\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-k}(-D))$, we have
$\langle i(f),g\rangle=\langle f,j(g)\rangle$. If we now assume that $f\in
e(T_{p})\mathrm{H}^{0}(X,\omega^{k})$ and $g\in
e(F)\mathrm{H}^{1}_{c}(\mathfrak{X}^{ord},\omega^{2-k}(-D))$, we have that

$\langle e(U_{p})i(f),g\rangle=\langle i(f),e(F)g\rangle=\langle
i(f),g\rangle$

and

$\langle f,e(T_{p})j(g)\rangle=\langle e(T_{p})f,j(g)\rangle=\langle
f,j(g)\rangle$

and the conlusion follows. ∎

##  5\. Higher Coleman theory

###  5.1. Cohomology with support in a closed subspace

We recall the notion of cohomology of an abelian sheaf on a topological space,
with support in a closed subspace. A reference for this material is [Gro05,
I]. Let $X$ be a topological space. Let $i:Z\hookrightarrow X$ be a closed
subspace. We denote by $C_{Z}$ and $C_{X}$ the categories of sheaves of
abelian groups on $Z$ and $X$ respectively.

We have the pushforward functor $i_{\star}:C_{Z}\rightarrow C_{X}$ and the
extension by zero functor $i_{!}:C_{Z}\rightarrow C_{X}$. The functor $i_{!}$
has a left adjoint $i^{!}:C_{X}\rightarrow C_{Z}$. For an abelian sheaf
$\mathscr{F}$ over $X$, we let
$\Gamma_{Z}(X,\mathscr{F})=\mathrm{H}^{0}(X,i_{\star}i^{!}\mathscr{F})$. By
definition this is the subgroup of $\mathrm{H}^{0}(X,\mathscr{F})$ of sections
whose support is included in $Z$. We let $\mathrm{R}\Gamma_{Z}(X,-)$ be the
derived functor of $\Gamma_{Z}(X,-)$.

Let $U=X\setminus Z$ and let $\mathscr{F}$ be an object of $C_{X}$. We have an
exact triangle [Gro05, I, Cor. 2.9]:

$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma(U,\mathscr{F})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

Some properties of the cohomology with support are :

  1. (1)

(Change of support) [Gro05, I, Prop. 1.8] If $Z\subset Z^{\prime}$, there is a
map
$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z^{\prime}}(X,\mathscr{F})$.

  2. (2)

(Pull-back) If we have a cartesian diagram:

and a sheaf $\mathscr{F}$ on $X^{\prime}$, there is a map
$\mathrm{R}\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z}(X,f^{\star}\mathscr{F})$,

  3. (3)

(Change of ambient space) [Gro05, I, Prop. 2.2] If we have $Z\subset U\subset
X$ for some open $U$ of $X$, then the pull back map
$\mathrm{R}\Gamma_{Z}(X,\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z}(U,\mathscr{F})$
is a quasi-isomorphism.

We now discuss the construction of the trace map in the context of adic spaces
and finite flat morphisms.

######  Lemma 5.1.

Consider a commutative diagram of topological spaces:

with $X$ and $X^{\prime}$ adic spaces, $f$ a finite flat morphism of adic
spaces, $Z^{\prime}$ and $Z$ are closed subspaces of $X^{\prime}$ and $X$
respectively. Let $\mathscr{F}$ be a sheaf of
$\mathscr{O}_{X^{\prime}}$-modules. Then there is a trace map
$\mathrm{R}\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\mathrm{R}\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$.

###### Proof.

We first recall that the category of sheaves of $\mathscr{O}_{T}$-modules on a
ringed space $(T,\mathscr{O}_{T})$ has enough injectives ([Sta13, Tag 01DH]).
It follows that it is enough to construct a functorial map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$
for sheaves $\mathscr{F}$ of $\mathscr{O}_{X^{\prime}}$-modules. We have a map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{f^{-1}(Z^{\prime})}(X,f^{\star}\mathscr{F})$.
Therefore, it suffices to consider the case where $Z=f^{-1}(Z^{\prime})$. We
have a trace map $Tr:f_{\star}f^{\star}\mathscr{F}\rightarrow\mathscr{F}$. Let
us complete the above diagram into:

where $U^{\prime}=X^{\prime}\setminus Z^{\prime}$ and $U=X\setminus Z$. We
have a commutative diagram:

Taking global sections and the induced map on the kernel of the two horizontal
morphisms, we deduce that there is a map
$\Gamma_{Z}(X,f^{\star}\mathscr{F})\rightarrow\Gamma_{Z^{\prime}}(X^{\prime},\mathscr{F})$.
∎

###  5.2. The modular curve $X_{0}(p)$

We let $X_{0}(p)$ be the compactified modular curve of level $\Gamma_{0}(p)$
and tame level $\Gamma_{1}(N)$ for some prime to $p$ integer $N\geq 3$, viewed
as an adic space over $\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$. We
let $H_{1}\subset E[p]$ be the universal subgroup of order $p$.

####  5.2.1. Parametrization by the degree

Let $X_{0}(p)^{\mathrm{rk}1}$ be the subset of rank one points of $X_{0}(p)$.
We define a map $\deg:X_{0}(p)^{\mathrm{rk}1}\rightarrow[0,1]$, which sends
$x\in X_{0}(p)^{\mathrm{rk}1}$ to $\deg H_{1}\in[0,1]$ (see [Far10, §4, Def.
3]). For any rational interval $[a,b]\subset[0,1]$, there is a unique quasi-
compact open $X_{0}(p)_{[a,b]}$ of $X$ such that
$\deg^{-1}[a,b]=X_{0}(p)^{\mathrm{rk}1}_{[a,b]}$. We let
$X_{0}(p)_{[0,a[\cup]b,n]}=(X_{0}(p)_{[a,b]})^{c}$.

######  Remark 5.2.

We remark that in this parametrization, the two extremal points $0$ (resp.
$1$) correspond to ordinary semi-abelian schemes equipped with an étale (resp.
multiplicative) subgroup $H_{1}$.

####  5.2.2. The canonical subgroup

We let $X$ be the compactified modular curve of degree prime to $p$. We have
an Hasse invariant $\mathrm{Ha}$ and we can define the Hodge height:

$\mathrm{Hdg}:X^{\mathrm{rk1}}\rightarrow[0,1]$

obtained by sending $x$ to $\inf\\{v_{x}(\tilde{\mathrm{Ha}}),1\\}$ for any
local lift $\tilde{\mathrm{Ha}}$ of the Hasse invariant. For any $v\in[0,1]$,
we let $X_{v}=\\{x\in X,\mathrm{Hdg}(x)\leq v\\}$ (more correctly, $X_{v}$ is
the quasi-compact open whose rank one points are those described as above).

We recall the following theorems:

######  Theorem 5.3 ([Kat73, Thm 3.1]).

If $v<\frac{p}{p+1}$, then over $X_{v}$ we have a canonical subgroup
$H_{1}^{can}\subset E[p]$ which is locally isomorphic to
$\mathbb{Z}/p\mathbb{Z}$ in the étale topology.

######  Theorem 5.4.

  1. (1)

For any rank one point of $X_{0}(p)$, we have the identity $\sum_{H\subset
E[p]}\deg H=1$. Moreover, either all the degrees are equal or there exists a
canonical subgroup $H_{1}^{can}$ and for all $H\neq H_{1}^{can}$, $\deg
H=\frac{1-\deg H_{1}^{can}}{p}$.

  2. (2)

If $a<\frac{1}{p+1}$, $X_{0}(p)_{[0,a]}$ carries a canonical subgroup
$H_{1}^{can}\neq H_{1}$ and $\deg(H_{1})=\frac{\mathrm{Hdg}}{p}$.

  3. (3)

If $a>\frac{1}{p+1}$, $X_{0}(p)_{[a,1]}$ carries a canonical subgroup
$H_{1}^{can}=H_{1}$, and $\deg(H_{1})=1-\mathrm{Hdg}$.

###### Proof.

See [Pil11, A.2] and [Far11, Thm. 6]. ∎

###  5.3. The correspondence $U_{p}$

We let $C$ be the correspondence underlying $U_{p}$. It parametrizes isogenies
of degree $p$ : $(E\rightarrow
E^{\prime},H_{1}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}H_{1}^{\prime})$.
We denote by $H=\mathrm{Ker}(E\rightarrow E^{\prime})$. We have two
projections $p_{1}((E,H_{1},E^{\prime},H^{\prime}_{1}))=(E,H_{1})$,
$p_{2}((E,H_{1},E^{\prime},H^{\prime}_{1}))=(E^{\prime},H^{\prime}_{1})$.

There is actually an isomorphism $X_{0}(p^{2})\rightarrow C$ (where
$X_{0}(p^{2})$ parametrizes $(E,H_{2}\subset E[p^{2}])$, with $H_{2}$ locally
isomorphic to $\mathbb{Z}/p^{2}\mathbb{Z}$), mapping $(E,H_{2})$ to
$(E/H_{1},H_{2}/H_{1},E,H_{1})$ where $H_{1}=H_{2}[p]$, and the isogeny
$E/H_{1}\rightarrow E$ is dual to $E\rightarrow E/H_{1}$.

Let us denote by $C_{[a,b]}=p_{1}^{-1}(X_{0}(p)_{[a,b]})$. If
$a>\frac{1}{p+1}$, then we have a canonical subgroup of order $1$ over
$X_{0}(p)_{[a,1]}$, $H_{1}^{can}=H_{1}$. If $a<\frac{1}{p+1}$, we also have a
canonical subgroup of order $1$ over $X_{0}(p)_{[0,a]}$ and $H_{1}\neq
H_{1}^{can}$. The map $p_{1}:C_{[0,a]}\rightarrow X_{0}(p)_{[0,a]}$ has a
section given by $H_{1}^{can}$. Let $C^{can}_{[0,a]}$ be the image of this
section and let $C^{et}_{[0,a]}$ be its complement, so that
$C_{[0,a]}=C^{can}_{[0,a]}\coprod C^{et}_{[0,a]}$.

######  Proposition 5.5.

  1. (1)

If $a\geq\frac{1}{p+1}$,
$p_{2}(C_{\\{a\\}})=X_{0}(p)_{\\{\frac{p-1}{p}+\frac{a}{p}\\}}$.

  2. (2)

If $a<\frac{1}{p+1}$, we have that
$p_{2}(C^{can}_{\\{a\\}})=X_{0}(p)_{\\{pa\\}}$, and
$p_{2}(C^{et}_{\\{a\\}})=X_{0}(p)_{\\{1-a\\}}$.

  3. (3)

If $a\in]0,1[$, we have $\overline{p_{2}(C_{[a,1]})}\subseteq X_{[a,1]}$.

###### Proof.

We do the case by case argument to check the first two points:

  1. (1)

We have $H_{1}=H_{1}^{can}$. If $H\subset E[p]$ satisfies $H\neq H_{1}^{can}$,
then $\deg H=\frac{1-\deg H_{1}}{p}$ and $\deg E[p]/H=\frac{p-1}{p}+\frac{\deg
H_{1}}{p}$.

  2. (2)

On $C^{can}_{[0,a]}$, the isogeny $E\rightarrow E/H$ is the canonical isogeny
and $\deg E[p]/H=1-\deg H$ where $\deg H={1-p\deg H_{1}}$. On $C^{et}_{[0,a]}$
we have $H\neq H_{1}^{can}$, and $\deg H=\deg H_{1}$. Therefore $\deg
E[p]/H=1-\deg H_{1}$.

We deduce that if $a\in]0,1[$, there exists $b>a$ such that
${p_{2}(C_{[a,1]})}\subseteq X_{[b,1]}$, and the last point follows. ∎

We now give a similar analysis for the transpose of $U_{p}$. It is useful to
use the isomorphism $C\simeq X_{0}(p^{2})$, for which we have
$p_{2}(E,H_{2})=(E,H_{1})$ and $p_{1}(E,H_{2})=(E/H_{1},H_{2}/H_{1})$. We let
$C^{[a,b]}=p_{2}^{-1}(X_{0}(p)_{[a,b]})$.

If $\deg H_{1}<\frac{p}{p+1}$, we deduce that $\deg E[p]/H_{1}>\frac{1}{p+1}$
is the canonical subgroup. If $\deg H_{1}>\frac{p}{p+1}$, we deduce that $\deg
E[p]/H_{1}<\frac{1}{p+1}$ is not the canonical subgroup, but that $E/H_{1}$
admits a canonical subgroup. We denote by $C^{[a,1],can}$ the component where
$H_{2}/H_{1}$ is the canonical subgroup and by $C^{[a,1],et}$ its complement.

######  Proposition 5.6.

  1. (1)

If $a\leq\frac{p}{p+1}$, $p_{1}(C^{\\{a\\}})=X_{0}(p)_{\\{\frac{a}{p}\\}}$.

  2. (2)

If $a>\frac{p}{p+1}$, we have that
$p_{1}(C^{\\{a\\},can})=X_{0}(p)_{\\{1-p(1-a)\\}}$, and
$p_{1}(C^{\\{a\\},et}))=X_{0}(p)_{\\{1-a\\}}$.

  3. (3)

For any $0<a<1$, we have that
$p_{1}(C^{[0,a[})\subseteq\overset{\circ}{X_{0}(p)_{[0,a[}}$.

###### Proof.

We do the case by case argument for the first two points:

  1. (1)

We have $\deg E[p]/H_{1}=1-\deg H_{1}$. This is the canonical subgroup. We
deduce that $\deg H_{2}/H_{1}=\frac{\deg H_{1}}{p}$.

  2. (2)

We have $\deg E[p]/H_{1}=1-\deg H_{1}$, is not the canonical subgroup. In case
$(E,H_{2})\in C^{[a,1],can}$, we deduce that $\deg H_{2}/H_{1}=1-p(1-\deg
H_{1})$. If $(E,H_{2})\in C^{[a,1],et}$, we deduce that $\deg
H_{1}/H_{2}=1-\deg H_{1}$.

We deduce that if $0<a<1$, there exists $b>a$ such that
$p_{1}(C^{[0,a[})\subseteq{X_{0}(p)_{[0,b[}}$. ∎

###  5.4. The $U_{p}$-operator

We work over $X_{0}(p)$. Let $a\in]0,1[\cap\mathbb{Q}$. The cohomologies of
interest are:

  1. (1)

$\mathrm{R}\Gamma(X_{0}(p),\omega^{k})$,

  2. (2)

$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})$,

  3. (3)

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$.

The category of perfect Banach complexes is the homotopy category of the
category of bounded complexes of Banach spaces over $\mathbb{Q}_{p}$.

######  Lemma 5.7.

These cohomologies are objects of the category of perfect Banach complexes.

###### Proof.

In case $(1)$ and $(2)$ we can take a finite affinoid covering to represent
the cohomology. In case $(3)$, the cohomology fits in an exact triangle:

$\mathrm{R}\Gamma_{X_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\rightarrow$

since the two right objects belong to the category of perfect Banach
complexes, so is the last one. ∎

####  5.4.1. Constructing the action

We define a naive cohomological correspondence $U_{p}^{naive}$ has follows.
The two ingredients are the differential map
$p_{2}^{\star}\omega_{E}\rightarrow p_{1}^{\star}\omega_{E}$ and the trace map
$(p_{1})_{\star}\mathscr{O}_{C}\rightarrow\mathscr{O}_{X_{0}(p)}$. Putting all
this together, we obtain

$U_{p}^{naive}:(p_{1})_{\star}p_{2}^{\star}\omega^{k}\rightarrow\omega^{k}.$

######  Proposition 5.8.

We have an action of the $U^{naive}_{p}$-operator on
$\mathrm{R}\Gamma(X_{0}(p),\omega^{k})$,
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})$ and
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$ for any $0<a<1$.
The $U^{naive}_{p}$-operator is compact. Moreover, the $U_{p}^{naive}$
operator acts equivariantly on the triangle:

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

###### Proof.

The operator $U_{p}^{naive}$ is compact on
$\mathrm{R}\Gamma(X_{0}(p),\omega^{k})$ because this later complex is a
perfect complex of finite dimensional $\mathbb{Q}_{p}$-vector spaces. By
proposition 5.5:

$\overline{p_{2}(C_{[a,1]})}\subset X_{0}(p)_{[a,1]}.$

Therefore, we have a map:

$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\rightarrow\mathrm{R}\Gamma(p_{2}(C_{[a,1]}),\omega^{k})\stackrel{{\scriptstyle
p_{2}^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma(C_{[a,1]},p_{2}^{\star}\omega^{k})$
$\rightarrow\mathrm{R}\Gamma(C_{[a,1]},p_{1}^{\star}\omega^{k})\stackrel{{\scriptstyle\mathrm{trace}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k}).$

Therefore, $U^{naive}_{p}$ acts and is compact on
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})$ because the first map
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\rightarrow\mathrm{R}\Gamma(p_{2}(C_{[a,1]}),\omega^{k})$
is. Similarly, by proposition 5.6:

${p_{1}(C^{[0,a[})}\subset\overset{\circ}{X_{0}(p)_{[0,a[}}.$

The operator $U_{p}^{naive}$ acts like the composite of the following maps:

$\mathrm{R}\Gamma_{X_{[0,a[}}(X_{0}(p),\omega^{k})\stackrel{{\scriptstyle
p_{2}^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma_{C^{[0,a[}}(C,p_{2}^{\star}\omega^{k})\rightarrow\mathrm{R}\Gamma_{C^{[0,a[}}(C,p_{1}^{\star}\omega^{k})$
$\rightarrow\mathrm{R}\Gamma_{p_{1}^{-1}p_{1}(C^{[0,a[})}(C,p_{1}^{\star}\omega^{k})\stackrel{{\scriptstyle\mathrm{trace}}}{{\rightarrow}}\mathrm{R}\Gamma_{p_{1}(C^{[0,a[})}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k}).$

We claim that the map
$\mathrm{R}\Gamma_{p_{1}(C{[0,a[})}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$
is compact. This will follow if we prove that the map
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,b[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$
for $0\leq b<a\leq 1$ is compact. To see this, we observe that there is a map
of exact triangles:

and since the two vertical maps on the right are compact, the first vertical
map is also compact.

∎

We also note the following corollary of the proof:

######  Corollary 5.9.

For any $0<a<b<1$, the natural maps

$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[b,1]},\omega^{k})$

and

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,b[}}(X_{0}(p),\omega^{k})$

induce quasi-isomorphism on the finite slope part for $U_{p}^{naive}$.

####  5.4.2.  $U_{p}$ and Frobenius

It is worth spelling out the action of $U_{p}$ on
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$.

If $a<\frac{1}{p+1}$, we have a correspondence

where $p_{1}^{can}$ is actually an isomorphism. We can think of this
correspondence as the graph of the Frobenius map $X_{0}(p)_{[0,pa]}\rightarrow
X_{0}(p)_{[0,a]}$, sending $(E,H_{1})$ to $(E/H_{1}^{can},E[p]/H_{1}^{can})$.
We claim that there is an associated operator:

$U_{p}^{naive,can}:\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k}).$

We first observe that
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})=\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p)_{[0,a]},\omega^{k})$.

We now construct $U_{p}^{naive,can}$ as the composite:

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,pa[}}(X_{0}(p),\omega^{k})\stackrel{{\scriptstyle(p_{2}^{can})^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma_{C^{can}_{[0,{a}[}}(C^{can}_{[0,{a}]},(p_{2}^{can})^{\star}\omega^{k})$
$\rightarrow\mathrm{R}\Gamma_{C^{can}_{[0,{a}[}}(C^{can}_{[0,{a}]},(p_{1}^{can})^{\star}\omega^{k})\tilde{\rightarrow}\mathrm{R}\Gamma_{X_{0}(p)_{[0,{a}[}}(X_{0}(p),\omega^{k}).$

######  Proposition 5.10.

For $a<\frac{1}{p+1}$, we have $U_{p}^{naive,can}=U_{p}^{naive}$ on
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$.

###### Proof.

This amounts to compare the construction of $U_{p}^{naive}$ (given in the
proof of proposition 5.8) and of $U_{p}^{naive,can}$. We see that
$p_{1}(C^{[0,a[})\subset X_{0}(p)_{[0,a]}$ and therefore,

$p_{1}^{-1}p_{1}(C^{[0,a[})=(p_{1}^{-1}p_{1}(C^{[0,a[})\cap
C_{[0,a]}^{can})\coprod(p_{1}^{-1}p_{1}(C^{[0,a[})\cap C_{[0,a]}^{et}).$

Moreover, $p_{2}((p_{1}^{-1}p_{1}(C^{[0,a[})\cap C_{[0,a]}^{et}))\subset
X_{0}(p)_{[1-a,a]}$ and therefore,
$C^{[0,a[}\hookrightarrow(p_{1}^{-1}p_{1}(C^{[0,a[})\cap C_{[0,a]}^{can})$. We
have

$\mathrm{R}\Gamma_{p_{1}^{-1}p_{1}(C^{[0,a[})}(C,p_{1}^{\star}\omega^{k})=$
$\mathrm{R}\Gamma_{(p_{1}^{-1}p_{1}(C^{[0,a[})\cap
C_{[0,a]}^{can})}(C,p_{1}^{\star}\omega^{k})\oplus\mathrm{R}\Gamma_{(p_{1}^{-1}p_{1}(C^{[0,a[})\cap
C_{[0,a]}^{et})}(C,p_{1}^{\star}\omega^{k}).$

Therefore the map
$\mathrm{R}\Gamma_{C^{[0,a[}}(C,p_{1}^{\star}\omega^{k})\rightarrow\mathrm{R}\Gamma_{p_{1}^{-1}p_{1}(C^{[0,a[})}(C,p_{1}^{\star}\omega^{k})$
factors through the direct factor
$\mathrm{R}\Gamma_{(p_{1}^{-1}p_{1}(C^{[0,a[})\cap
C_{[0,a]}^{can})}(C,p_{1}^{\star}\omega^{k})$. ∎

####  5.4.3. Slopes estimates and the control theorem

The following lemma is the key technical input to proving Coleman’s
classicality theorem.

######  Lemma 5.11.

For any $a\in]0,1[\cap\mathbb{Q}$,

  1. (1)

the slopes of $U_{p}^{naive}$ on
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})$ are $\geq 1$.

  2. (2)

the slopes of $U_{p}^{naive}$ on
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$ are $\geq k$.

###### Proof.

We first observe that the finite slope part of the cohomology
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})^{fs}$ is independent of
$a\in]0,1[$ and is therefore supported in degree $0$ by affiness for $a$ close
to 1. It follows that we are left to prove that $U_{p}^{naive}$ has slopes at
least $1$ and this is a $q$-expansion computation. Namely, $U_{p}^{naive}(\sum
a_{n}q^{n})=p\sum a_{np}q^{n}$. For the second cohomology, we again observe
that the cohomology is independent of $a\in]0,1[$. We may therefore suppose
that $a$ is small enough and use that $U^{naive}_{p}=U_{p}^{naive,can}$
(proposition 5.10).

We first claim that for any $s\in\mathbb{Q}$,

$\operatorname{im}(\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})\rightarrow\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})^{=s})$

(where the supscript $=s$ means the slope $s$ part for the action of
$U_{p}^{naive}$) defines a lattice in
$\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})^{=s}$ (an open and
bounded submodule). We can indeed represent the cohomology
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$ by the Cech complex
$C^{\bullet}$ relative to some open covering $\mathcal{U}$, and we can lift
the $U_{p}$-operator to a compact operator $\tilde{U}_{p}$ on the complex.
Note that $C^{\bullet}$ is a complex of Banach modules.

The map from Cech cohomology with respect to $\mathcal{U}$ to cohomology

$\check{\mathrm{H}}^{i}_{\mathcal{U},X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})\rightarrow\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})$

has kernel and cokernel of bounded torsion by [Pil17, Lem. 3.2.2]. It suffices
to prove that

$\operatorname{im}(\check{\mathrm{H}}^{i}_{\mathcal{U},X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})\rightarrow\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})^{=s})$

defines an open and bounded submodule in
$\mathrm{H}^{i}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})^{=s}$. The Cech
cohomology
$\check{\mathrm{H}}^{i}_{\mathcal{U},X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})$
is obtained by taking the cohomology of and open and bounded sub-complex
$C^{+,\bullet}\subset C^{\bullet}$. The image of $C^{+,\bullet}$ in
$C^{\bullet,=s}$ under the continuous projection $C^{\bullet}\rightarrow
C^{\bullet,=s}$ (the target is now a complex of finite dimensional vector
spaces) is again open and bounded and the claim follows.

Moreover, over $C_{[0,a[}^{can}$, we have a universal isogeny which gives an
isomorphism, $p_{2}^{\star}\omega\rightarrow p_{1}^{\star}\omega$, for which
$pp_{1}^{\star}\omega^{+}\subset p_{2}^{\star}\omega^{+}\subset
p^{1-\frac{a}{p}}p_{1}^{\star}\omega^{+}$. We deduce that $U_{p}^{naive}$
induces a map

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),p^{k(1-\frac{a}{p})}\omega^{k,+})$

if $k\geq 0$, and

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k,+})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),p^{k}\omega^{k,+})$

if $k\leq 0$. We deduce that $p^{-k(1-\frac{a}{p})}U_{p}^{naive}$ if $k\geq 0$
and $p^{-k}U_{p}^{naive}$ if $k\leq 0$ stabilize a lattice in the cohomology,
and therefore have only non-negative slope. The lemma follows. ∎

######  Remark 5.12.

The Lemma 3.2.2 in [Pil17] depends on the main result of [Bar78], which in
turn is a key technical ingredient in [Kas06].

We now define $U_{p}=p^{-\inf\\{1,k\\}}U_{p}^{naive}$ and we get:

######  Theorem 5.13.

  1. (1)

$U_{p}$ has slopes $\geq 0$ on $\mathrm{R}\Gamma(X_{0}(p),\omega^{k})$,

  2. (2)

For any $a\in]0,1[\cap\mathbb{Q}$, the map
$\mathrm{R\Gamma}(X_{0}(p),\omega^{k})^{<k-1}\rightarrow\mathrm{R\Gamma}(X_{0}(p)_{[a,1]},\omega^{k})^{<k-1}$
is a quasi-isomorphism,

  3. (3)

For any $a\in]0,1[\cap\mathbb{Q}$, the map
$\mathrm{R\Gamma}_{X_{[0,a[}}(X_{0}(p),\omega^{k})^{<1-k}\rightarrow\mathrm{R\Gamma}(X_{0}(p),\omega^{k})^{<1-k}$
is a quasi-isomorphism.

###### Proof.

We consider the triangle

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p),\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

on which $U_{p}$ acts equivariantly and apply the slope estimates of lemma
5.11. ∎

####  5.4.4.  Going down to spherical level

For $a>\frac{1}{p+1}$, the map $p_{1}:X_{0}(p)_{[a,1]}\rightarrow X_{1-a}$ is
an isomorphism and therefore the pull back map
$\mathrm{R}\Gamma(X_{1-a},\omega^{k})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{k})$
is a quasi-isomorphism. There is a analogue statement for the cohomology with
support that we now explain. Let $a<\frac{1}{p+1}$. We have a Frobenius map
$F:X_{0}(p)_{[0,a]}\rightarrow X_{0}(p)_{[0,pa]}$ given by
$(E,H_{1})\mapsto(E/H_{1}^{can},E[p]/H_{1}^{can})$. There is similarly a
Frobenius map $F:{X}_{\frac{a}{p}}\rightarrow X_{a}$, given by $E\mapsto
E/H_{1}^{can}$.

The Frobenius map fits into the following diagram:

where the diagonal map is given by $E\mapsto(E/H_{1}^{can},E[p]/H_{1}^{can})$.

######  Proposition 5.14.

The pull back map
$\mathrm{R\Gamma}_{X_{\frac{a}{p}}}(X,\omega^{k})\rightarrow\mathrm{R\Gamma}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{k})$
induces a quasi-isomorphism on the finite slope part.

###  5.5. $p$-adic variation

We now consider the problem of interpolation of these cohomologies.

####  5.5.1. Reduction of the torsor $\omega_{E}$

We recall here a construction from [Pil13] and [AIS14]. We let
$\mathcal{T}=\\{w\in\omega_{E},~{}\omega\neq 0\\}$ be the
$\mathbb{G}_{m}$-torsor associated to $\omega_{E}$. We let
$\mathbb{G}_{a}^{+}=\operatorname{Spa}(\mathbb{Q}_{p}\langle
T\rangle,\mathbb{Z}_{p}\langle T\rangle)$ be the unit ball, with its additive
analytic group structure. We fix a positive integer $n$. We have an analytic
subgroup
$\mathbb{Z}_{p}^{\times}(1+p^{n-v\frac{p^{n}}{p-1}}\mathbb{G}_{a}^{+})\hookrightarrow\mathbb{G}_{m}$.

######  Proposition 5.15.

Let $v<\frac{1}{p^{n-1}(p-1)}$. The $\mathbb{G}_{m}$-torsor
$\mathcal{T}\times_{X}X_{v}\rightarrow X_{v}$ has a natural reduction to a
$\mathbb{Z}_{p}^{\times}(1+p^{n-v\frac{p^{n}}{p-1}}\mathbb{G}_{a}^{+})$-torsor
denoted $\mathcal{T}_{v}$.

###### Proof.

We denote by $\omega^{+}_{E}\subset\omega_{E}$ the locally free sheaf of
integral relative differential forms. For $v<\frac{1}{p^{n-1}(p-1)}$, there is
a canonical subgroup of level $n$, $H^{can}_{n}$ over $X_{v}$. The isogeny
$E\rightarrow E/H^{can}_{n}$ yields a map
$\omega_{E/H^{can}_{n}}^{+}\rightarrow\omega_{E}^{+}$, with cokernel
$\omega_{H_{n}}^{+}$. The surjective map
$r:\omega_{E}^{+}\rightarrow\omega_{H^{can}_{n}}^{+}$ induces an isomorphism:

$\omega_{E}^{+}/p^{n-v\frac{p^{n}-1}{p-1}}\tilde{\rightarrow}\omega_{H^{can}_{n}}^{+}/p^{n-v\frac{p^{n}-1}{p-1}}.$

There is a Hodge-Tate map
$\mathrm{HT}:(H^{can}_{n})^{D}\rightarrow\omega_{H^{can}_{n}}^{+}$ (of sheaves
on the étale site), and its linearization $\mathrm{HT}\otimes
1:(H^{can}_{n})^{D}\otimes\mathscr{O}_{X_{v}}^{+}\rightarrow\omega_{H^{can}_{n}}^{+}$
has cokernel killed by $p^{\frac{v}{p-1}}$. We have a diagram:

We now introduce a modification of $\omega_{E}^{+}$: let
$\omega_{E}^{\sharp}=\\{w\in\omega_{E}^{+},~{}r(w)\in\operatorname{im}(\mathrm{HT}\otimes
1)\\}\subset\omega_{E}^{+}$. This is a locally free sheaf of
$\mathscr{O}_{X_{v}}^{+}$-modules on the étale site. The Hodge-Tate map
induces an isomorphism:

$\mathrm{HT}_{v}:(H^{can}_{n})^{D}\otimes\mathscr{O}_{X_{v}}^{+}/p^{n-v\frac{p^{n}}{p-1}}\rightarrow\omega_{E}^{\sharp}/p^{n-v\frac{p^{n}}{p-1}}.$

We let $\mathcal{T}_{v}$ be the torsor under the group
$\mathbb{Z}_{p}^{\times}(1+p^{n-v\frac{p^{n}}{p-1}}\mathbb{G}_{a}^{+})$
defined by

$\mathcal{T}_{v}=\\{\omega\in\omega_{E}^{\sharp},~{}\exists
P\in(H^{can}_{n})^{D},~{}p^{n-1}P\neq 0,\mathrm{HT}_{v}(P)=\omega\mod
p^{n-v\frac{p^{n}}{p-1}}\\}.$

We have a natural map $\mathcal{T}_{v}\hookrightarrow\mathcal{T}$, equivariant
for the analytic group map:
$\mathbb{Z}_{p}^{\times}(1+p^{n-v\frac{p^{n}}{p-1}}\mathbb{G}_{a}^{+})\rightarrow\mathbb{G}_{m}$.
∎

####  5.5.2. Interpolation of the sheaf

We let
$\mathcal{W}=\operatorname{Spa}(\Lambda,\Lambda)\times\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$
be the weight space. We let
$\kappa^{un}:\mathbb{Z}_{p}^{\times}\rightarrow\mathscr{O}_{\mathcal{W}}^{\times}$
be the universal character.

We can write $\mathcal{W}$ as an increasing union of affinoids
$\mathcal{W}=\cup_{0<r<1}\mathcal{W}_{r}$ where $r\in\mathbb{Q}\cap]0,1[$ and
each $\mathcal{W}_{r}$ is a finite union of balls of radius $r$. Over each
$\mathcal{W}_{r}$, there is $t(r)\in\mathbb{Q}_{>0}$ such that the universal
character extends to a character
$\kappa^{un}:\mathbb{Z}_{p}^{\times}(1+p^{t(r)}\mathscr{O}_{W}^{+})\rightarrow\mathscr{O}_{\mathcal{W}}^{\times}$.

We now fix $r$ and we choose $v$ small enough and $n$ large enough such that
$t(r)\leq n-v\frac{p^{n}}{p-1}$ and we define a locally free sheaf
$\omega^{\kappa^{un}}$ over $X_{v}\times\mathcal{W}_{r}$:

$\omega^{\kappa^{un}}=(\mathscr{O}_{\mathcal{T}_{v}}\otimes\mathscr{O}_{\mathcal{W}_{r}})^{\mathbb{Z}_{p}^{\times}(1+p^{n-v\frac{p^{n}}{p-1}}\mathscr{O}_{X_{v}\times\mathcal{W}_{r}}^{+})}.$

Since $\mathcal{T}_{v}\rightarrow X_{v}$ is an étale torsor, the sheaf
$\omega^{\kappa^{un}}$ is a locally free sheaf of
$\mathscr{O}_{X_{v}\times\mathcal{W}_{r}}$-modules in the étale topology. It
is actually a locally free sheaf of
$\mathscr{O}_{X_{v}\times\mathcal{W}_{r}}$-modules in the Zariski topology by
the main result of [BG98].

####  5.5.3. Interpolation of the cohomology

For $a\in]0,\frac{v}{p}]$, we have a map $p_{1}:X_{0}(p)_{[0,a]}\rightarrow
X_{v}$ and we can therefore pull back the sheaf $\omega^{\kappa^{un}}$ to an
invertible sheaf over $X_{0}(p)_{[0,a]}\times\mathcal{W}_{r}$.

If $a\in[1-v,1[$, we have a map $p_{1}:X_{0}(p)_{[a,1]}\rightarrow X_{v}$ and
we can pull back the sheaf $\omega^{\kappa^{un}}$ to an invertible sheaf over
$X_{0}(p)_{[a,1]}\times\mathcal{W}_{r}$.

We consider the cohomologies:

  1. (1)

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}})$

  2. (2)

$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$.

Note that the first cohomology group is well defined because
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}})=\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p)_{[0,a]},\omega^{\kappa^{un}})$.

These cohomologies belong to the category of perfect complexes of Banach
spaces over $\mathscr{O}_{\mathcal{W}_{r}}$ which is the homotopy category of
the category of bounded complexes of projective Banach modules over
$\mathscr{O}_{\mathcal{W}_{r}}$.

####  5.5.4. The $U_{p}$-operator on
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$.

We need to consider the $U_{p}$-correspondence on $X_{0}(p)_{[a,1]}$ where it
reduces to a correspondence

######  Lemma 5.16.

There is a natural isomorphism $p_{2}^{\star}\omega^{\kappa^{un}}\rightarrow
p_{1}^{\star}\omega^{\kappa^{un}}$, and we can define a cohomological
correspondence
$U_{p}:p_{1\star}p_{2}^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$
which specializes in weight $k\geq 1$ to $U_{p}$.

###### Proof.

Over $C_{[a,1]}$, the universal isogeny $\pi:p_{1}^{\star}{E}\rightarrow
p_{2}^{\star}E$ induces an isomorphism on canonical subgroups
$p_{1}^{\star}H^{can}_{n}\simeq p_{2}^{\star}H^{can}_{n}$, and therefore there
is a canonical isomorphism:

This clearly induces an isomorphism $p_{1}^{\star}\mathcal{T}_{v}\rightarrow
p_{2}^{\star}\mathcal{T}_{v}$ and from this we get an isomorphism:

$p_{2}^{\star}\omega^{\kappa^{un}}\rightarrow
p_{1}^{\star}\omega^{\kappa^{un}}$

which specializes to the natural isomorphism
$p_{2}^{\star}\omega^{k}\rightarrow p_{1}^{\star}\omega^{k}$ at weight $k$.

We now define $U_{p}:p_{1\star}p_{2}^{\star}\omega^{\kappa^{un}}\rightarrow
p_{1\star}p_{1}^{\star}\omega^{\kappa^{un}}\stackrel{{\scriptstyle\frac{1}{p}\mathrm{Tr}_{p_{1}}}}{{\longrightarrow}}\omega^{\kappa^{un}}$.

∎

######  Corollary 5.17.

The operator $U_{p}$ is compact on
$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$.

###### Proof.

The operator $U_{p}$ factors as:

$\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[\frac{a}{p},1]},\omega^{\kappa^{un}})\rightarrow\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}}).$

∎

####  5.5.5. The $U_{p}$-operator on
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}})$

We need to consider the $U_{p}$-correspondence on $X_{0}(p)_{[0,a]}$ where
actually only the canonical part of the correspondance is relevant and
therefore, it reduces to a correspondence:

######  Lemma 5.18.

There is a natural isomorphism
$(p^{can}_{2})^{\star}\omega^{\kappa^{un}}\rightarrow(p^{can}_{1})^{\star}\omega^{\kappa^{un}}$,
and a cohomological correspondence
$U_{p}:(p^{can}_{1})_{\star}(p^{can}_{2})^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$
which specializes in weight $k\leq 1$ to $U^{can}_{p}$.

###### Proof.

Over $C^{can}_{[0,a]}$, the dual universal isogeny
$\pi^{D}:(p^{can}_{2})^{\star}{E}\rightarrow(p^{can}_{1})^{\star}E$ induces an
isomorphism on canonical subgroups
$(p^{can}_{2})^{\star}H^{can}_{n}\simeq(p^{can}_{1})^{\star}H^{can}_{n}$, and
therefore there is a canonical isomorphism:

This clearly induces an isomorphism
$(p^{can}_{2})^{\star}\mathcal{T}_{v}\rightarrow(p^{can}_{1})^{\star}\mathcal{T}_{v}$
and from this we get an isomorphism:

$(p^{can}_{2})^{\star}\omega^{\kappa^{un}}\rightarrow(p^{can}_{1})^{\star}\omega^{\kappa^{un}}$

or rather more naturally its inverse:

$(p^{can}_{1})^{\star}\omega^{\kappa^{un}}\rightarrow(p^{can}_{2})^{\star}\omega^{\kappa^{un}}$

which specializes to the natural isomorphism
$(p^{can}_{1})^{\star}\omega^{k}\rightarrow(p^{can}_{2})^{\star}\omega^{k}$
given by $(\pi^{D})^{\star}$, and its inverse
$(p^{can}_{2})^{\star}\omega^{k}\rightarrow(p^{can}_{1})^{\star}\omega^{k}$ is
$p^{-k}(\pi^{\star})$.

Recall that $p_{1}$ is an isomorphism, and we therefore get
$U_{p}:(p^{can}_{1})_{\star}(p^{can}_{2})^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$
which specializes in weight $k\leq 1$ to
$p^{-k}U_{p}^{can,naive}=U^{can}_{p}$.

∎

######  Remark 5.19.

We therefore find that this is really $U_{p}$ and not $U_{p}^{naive}$ that can
be interpolated over the weight space.

######  Corollary 5.20.

The $U_{p}$-operator is compact on
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}})$.

###### Proof.

The operator $U_{p}$ factors as:

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,pa[}}(X_{0}(p),\omega^{\kappa^{un}})\rightarrow\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}}).$

∎

###  5.6. Construction of eigencurves

We use these cohomologies to construct the eigencurve, following the method of
[Col97].

####  5.6.1.  First construction

The cohomology $\mathrm{R}\Gamma(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$ is
concentrated in degree $0$, and is represented by
$\mathrm{H}^{0}(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$ which is a projective
Banach module over $\mathscr{O}_{\mathcal{W}_{r}}$ (see [Pil13, Cor. 5.3]).

The $U_{p}$-operator acts compactly on this space. We let
$\mathcal{P}\in\mathscr{O}_{\mathcal{W}_{r}}[[T]]$ be the characteristic
series of $U_{p}$. This is an entire series. We let
$\mathcal{Z}=V(\mathcal{P})\subset\mathbb{G}_{m}^{an}\times\mathcal{W}_{r}$.
We have a weight map $w:\mathcal{Z}\rightarrow\mathcal{W}_{r}$ which is quasi-
finite, partially proper, and locally on the source and the target a finite
flat map.

Over $\mathcal{Z}$ we have a coherent sheaf $\mathcal{M}$ which is the
universal generalized eigenspace. This is a locally free
$w^{-1}\mathscr{O}_{\mathcal{W}_{r}}$-module and for any
$x=(\kappa,\alpha)\in\mathcal{Z}$,
$x^{\star}\mathcal{M}=\mathrm{H}^{0}(X_{0}(p)_{[a,1]},\omega^{\kappa})^{U_{p}=\alpha^{-1}}$.

We let
$\mathscr{O}_{\mathcal{C}}\subset\mathrm{End}_{\mathcal{Z}}(\mathcal{M})$ be
the subsheaf of $\mathscr{O}_{\mathcal{Z}}$-modules generated by the Hecke
operators of level prime to $Np$, and we let
$\mathcal{C}\rightarrow\mathcal{Z}$ be the associated analytic space.

The construction of $(\mathcal{M},\mathcal{C},\mathcal{Z})$ is compatible when
$r$ changes (and does not depend on auxiliary choices like $a$, $v$, $n$…). We
now let $r$ tend to $1$, glue everything, and with a slight abuse of notation
we have $\mathcal{C}\rightarrow\mathcal{Z}\rightarrow\mathcal{W}$ and a
coherent sheaf $\mathcal{M}$ over $\mathcal{C}$. This is the eigencurve of
[CM98].

####  5.6.2. Second construction

We can perform a similar construction, using instead the cohomology
$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{2-\kappa^{un}}(-D))$
where
$\omega^{2-\kappa^{un}}(-D)=(\omega^{\kappa^{un}})^{\vee}\otimes\omega^{2}(-D)$.
The introduction of this twist is motivated by Serre duality. Recall that the
character $\mathbb{Z}_{p}^{\times}\rightarrow\Lambda^{\times}$, $t\mapsto
t^{2}\kappa^{un}(t)^{-1}$ induces an automorphism
$d:\Lambda\rightarrow\Lambda$, and therefore:

$\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{2-\kappa^{un}}(-D))=\mathrm{R}\Gamma_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{\kappa^{un}}(-D))\otimes^{L}_{\mathscr{O}_{\mathcal{W}_{r}},d}\mathscr{O}_{\mathcal{W}_{r}}.$

This cohomology is a perfect complex of projective Banach modules over
$\mathscr{O}_{\mathcal{W}_{r}}$ and for any morphism
$\operatorname{Spa}(A,A^{+})\rightarrow\mathcal{W}_{r}$, the cohomology
$\mathrm{R}\Gamma_{[0,a[}(X_{0}(p),\omega^{2-\kappa^{un}}(-D))\hat{\otimes}A)$
is supported in degree $1$.

We choose a representative $N^{\bullet}$ for this cohomology, as well as a
compact representative $\tilde{U}_{p}$ representing the action of $U_{p}$. We
let ${\mathcal{Q}}_{i}$ be the characteristic series of $\tilde{U}_{p}$ acting
on $N^{i}$. We let $\tilde{\mathcal{Q}}=\prod\mathcal{Q}_{i}$ and we let
$\mathcal{X}=V(\tilde{\mathcal{Q}})\subset\mathbb{G}_{m}^{an}\times\mathcal{W}_{r}$.
We have a weight map $w:\tilde{\mathcal{X}}\rightarrow\mathcal{W}_{r}$ which
is quasi-finite locally on the source and the target and a bounded complex of
coherent sheaves “generalized eigenspaces” $\tilde{\mathcal{N}}^{\bullet}$
over $\tilde{\mathcal{X}}$. This is a perfect complex of finite projective
$w^{-1}\mathscr{O}_{\mathcal{W}_{r}}$-modules. Moreover,
$\tilde{\mathcal{N}}^{\bullet}$ has cohomology only in degree $1$, and we
deduce that $\mathrm{H}^{1}(\mathcal{N}^{\bullet})=\mathcal{N}$ is a locally
free $w^{-1}\mathscr{O}_{\mathcal{W}_{r}}$-module. We let
$\mathcal{Q}=V(\prod_{i}(-1)^{i}\mathcal{Q}_{i})$ and we set
$\mathcal{X}=V(\mathcal{Q})\subset\tilde{\mathcal{X}}$. The module
$\mathcal{N}$ is supported on $\mathcal{X}$. We let
$\mathscr{O}_{\mathcal{D}}\subset\mathrm{End}_{\mathcal{X}}(\mathcal{N})$ be
the subsheaf generated by the Hecke algebra of prime to $Np$ level, and we let
$\mathcal{D}\rightarrow\mathcal{X}$ be the associated analytic space. We can
now let $r$ tend to $1$, and we have
$\mathcal{D}\rightarrow\mathcal{X}\rightarrow\mathcal{W}$ and the sheaf
$\mathcal{N}$ over $\mathcal{D}$. This is a second eigencurve.

###  5.7. The duality pairing

In this last section, we prove that $\mathcal{Z}$ and $\mathcal{X}$ are
canonically isomorphic, that $\mathcal{M}$ and $\mathcal{N}$ are canonically
dual to each other and that $\mathcal{C}$ and $\mathcal{D}$ are canonically
identified under the pairing between $\mathcal{M}$ and $\mathcal{N}$.

####  5.7.1. Preliminaries

We are going to use the theory of dagger spaces [GK00]. Let $X^{\dagger}$ be a
dagger space over $\operatorname{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$, smooth
of pure relative dimension $d$. Let $\mathscr{F}$ be a coherent sheaf on
$X^{\dagger}$. Then one can define the cohomology groups
$\mathrm{H}^{i}(X^{\dagger},\mathscr{F})$ and
$\mathrm{H}^{i}_{c}(X^{\dagger},\mathscr{F})$. Moreover, these cohomology
groups carry canonical topologies ([vdP92, 1.6]).

By [GK00, Thm. 4.4], there is a residue map:

$\mathrm{res}_{X}:\mathrm{H}^{d}_{c}(X^{\dagger},\Omega^{d}_{X/\mathbb{Q}_{p}})\rightarrow\mathbb{Q}_{p}.$

This residue map has the following two important properties. Let
$f:Y^{\dagger}\rightarrow X^{\dagger}$ be an open immersion. Then the diagram:

is commutative. See [Bey97, Cor. 4.2.12] (although the author is working here
in the “dual” setting of wide open spaces)).

Let $f:Y^{\dagger}\rightarrow X^{\dagger}$ be a finite flat map. Then the
diagram:

(32)

is commutative. See [Bey97, Cor. 4.2.11] (although the author is working here
again in the “dual” setting of wide open spaces).

Let $\mathscr{F}$ be a locally free sheaf of finite rank over $X^{\dagger}$
which is assumed to be affinoid, smooth of pure dimension $d$. We let
$D(\mathscr{F})=\mathscr{F}^{\vee}\otimes\Omega^{d}_{X/\mathbb{Q}_{p}}$.

Then the residue map induces a perfect pairing ([GK00, Thm. 4.4]):

$\mathrm{H}^{0}(X^{\dagger},\mathscr{F})\times\mathrm{H}^{d}_{c}(X^{\dagger},D(\mathscr{F}))\rightarrow\mathbb{Q}_{p}$

for which both spaces are strong duals of each other.

######  Remark 5.21.

Since the topological vector space $\mathrm{H}^{0}(X^{\dagger},\mathscr{F})$
is a compact inductive limit of Banach spaces, we deduce from the theorem that
the topological vector space $\mathrm{H}^{d}_{c}(X^{\dagger},D(\mathscr{F}))$
is a compact projective limit of Banach spaces.

####  5.7.2. The classical pairing

We denote by $w$ the Atkin-Lehner involution over $X_{0}(p)$ and by $\langle
p\rangle$ the diamond operator given by multiplication by $p$. We recall that
$w\circ w=\langle p\rangle$. We recall that $C$ is the Hecke correspondence
underlying $U_{p}$. We can think of it as the moduli space of $(E,H,H_{1})$
where $H,H_{1}$ are distinct subgroups of $E[p]$. We have the projection
$p_{1}(E,H,H_{1})=(E,H_{1})$. We also have a projection
$p_{2}((E,H,H_{1})=(E/H,E[p]/H)$. Exchanging the roles of $H$ and $H_{1}$
yields an automorphism $\iota:C\rightarrow C$ and we let
$q_{i}=p_{i}\circ\iota$. Now one checks easily that $w\circ p_{1}=q_{2}$ and
$w\circ p_{2}=\langle p\rangle\circ q_{1}$. We have a residue map
$\mathrm{H}^{1}(X_{0}(p),\Omega^{1}_{X_{0}(p)/\mathbb{Q}_{p}})\rightarrow\mathbb{Q}_{p}$
and there is a perfect pairing:

$\langle,\rangle_{0}:\mathrm{H}^{0}(X_{0}(p),\omega^{k})\times\mathrm{H}^{1}(X_{0}(p),\omega^{2-k}(-D))\rightarrow\mathbb{Q}_{p}$

where we use the Kodaira-Spencer isomorphism
$\Omega^{1}_{X_{0}(p)/\mathbb{Q}_{p}}\simeq\omega^{2}(-D)$. We modify this
pairing, and set:

$\langle,\rangle=\langle.,w^{\star}.\rangle_{0}.$

######  Lemma 5.22.

For any
$(f,g)\in\mathrm{H}^{0}(X_{0}(p),\omega^{k})\times\mathrm{H}^{1}(X_{0}(p),\omega^{2-k}(-D))$,
we have: $\langle f,U_{p}g\rangle=\langle U_{p}f,g\rangle$.

###### Proof.

For any
$(f,g)\in\mathrm{H}^{0}(X_{0}(p),\omega^{k})\times\mathrm{H}^{1}(X_{0}(p),\omega^{2-k}(-D))$,
we have: $\langle U^{naive}_{p}f,g\rangle_{0}=\langle
f,(U^{naive}_{p})^{t}g\rangle$ where $(U^{naive}_{p})^{t}$ is the operator
associated to the transpose of $C$, and is obtained as follows:

$\mathrm{R}\Gamma(X_{0}(p),D(\omega^{k}))\stackrel{{\scriptstyle
p_{1}^{\star}}}{{\rightarrow}}\mathrm{R}\Gamma(C,p_{1}^{\star}D(\omega^{k}))\rightarrow\mathrm{R}\Gamma(C,p_{2}^{\star}D(\omega^{k}))\stackrel{{\scriptstyle\mathrm{tr}_{p_{2}}}}{{\rightarrow}}\mathrm{R}\Gamma(X_{0}(p),D(\omega^{k})).$

We observe that the determination of the adjoint of $U_{p}^{naive}$ as the
operator associated to the transpose of $C$ uses the compatibility property of
diagram 32. We have a commutative diagram:

We see that $\langle U^{naive}_{p}f,wg\rangle_{0}=\langle
f,w^{\star}U^{naive}_{p}g\rangle_{0}$. We now check that the normalizing
factors are correct so that $\langle U_{p}f,g\rangle=\langle f,U_{p}g\rangle$.
∎

####  5.7.3. The $p$-adic pairing

We now work again over $\mathscr{O}_{\mathcal{W}_{r}}$. We let
$X_{0}(p)^{m,{\dagger}}=\mathrm{colim}_{a\rightarrow 1}X_{0}(p)_{[a,1]}$. We
let $X_{0}(p)^{et,{\dagger}}=\mathrm{colim}_{a\rightarrow 0}X_{0}(p)_{[0,a]}$.
The Atkin-Lehner map is an isomorphism $w:X_{0}(p)^{m,{\dagger}}\rightarrow
X_{0}(p)^{et,{\dagger}}$ and there is an isomorphism
$w:w^{\star}\omega^{\kappa^{un}}\rightarrow\omega^{\kappa^{un}}$ (compare with
sections 5.5.4 and 5.5.5).

######  Lemma 5.23.

We have a canonical perfect pairing
$\langle,\rangle_{0}:\mathrm{H}^{0}(X_{0}(p)^{m,{\dagger}},\omega^{\kappa^{un}})\times\mathrm{H}^{1}_{c}(X_{0}(p)^{m,{\dagger}},\omega^{2-\kappa^{un}}(-D))\rightarrow\mathscr{O}_{\mathcal{W}_{r}}$.
Moreover, $\mathrm{H}^{0}(X_{0}(p)^{m,{\dagger}},\omega^{\kappa^{un}})$ is a
compact inductive limit of projective Banach spaces over
$\mathscr{O}_{\mathcal{W}_{r}}$,
$\mathrm{H}^{1}_{c}(X_{0}(p)^{m,{\dagger}},\omega^{2-\kappa^{un}}(-D))$ is a
compact projective limit of projective Banach spaces over
$\mathscr{O}_{\mathcal{W}_{r}}$, and both spaces are strong duals of each
other.

###### Proof.

The pairing is obtained as follows:

$\mathrm{H}^{0}(X_{0}(p)^{m,{\dagger}},\omega^{\kappa^{un}})\times\mathrm{H}^{1}_{c}(X_{0}(p)^{m,{\dagger}},\omega^{2-\kappa^{un}}(-D))$
$\rightarrow\mathrm{H}^{1}_{c}(X_{0}(p)^{m,{\dagger}},\Omega^{1}_{X_{0}(p)^{m}/\mathbb{Q}_{p}}\hat{\otimes}_{\mathbb{Q}_{p}}\mathscr{O}_{\mathcal{W}_{r}})\stackrel{{\scriptstyle\mathrm{res}_{X_{0}(p)^{m}}}}{{\rightarrow}}\mathscr{O}_{\mathcal{W}_{r}}.$

We prove the remaining claims. Let $\mathcal{W}^{i}$ be the connected
component of the character $i:x\mapsto x^{i}$ for $i=0,\cdots,p-2$ in
$\mathcal{W}$. Then for all $i$,
$\omega^{\kappa^{un}}|_{\mathcal{W}_{r}^{i}}=\omega^{i}\hat{\otimes}_{\mathbb{Q}_{p}}\mathscr{O}_{\mathcal{W}^{i}_{r}}$
is an “isotrivial” sheaf. This follows from the existence of the Eisenstein
family (see [Pil13], the final discussion below Proposition 6.2). Therefore,
all the statements of the lemma are reduced to similar statement for classical
invertible sheaves, and they follow from [GK00, Thm 4.4]. ∎

We deduce form this lemma that there is a canonical pairing:

$\langle,\rangle:\mathrm{H}^{0}(X_{0}(p)^{m,{\dagger}},\omega^{\kappa^{un}})\times\mathrm{H}^{1}_{c}(X_{0}(p)^{et,{\dagger}},\omega^{2-\kappa^{un}}(-D))\rightarrow\mathscr{O}_{\mathcal{W}_{r}}$

by putting $\langle,\rangle=\langle,w^{\star}.\rangle_{0}.$ For this pairing,
$\langle U_{p}.,.\rangle=\langle.,U_{p}.\rangle$. We have by definition that
$\mathrm{H}^{0}(X_{0}(p)^{m,{\dagger}},\omega^{\kappa^{un}})=\mathrm{colim}_{a\rightarrow
1}\mathrm{H}^{0}(X_{0}(p)_{[a,1]},\omega^{\kappa^{un}})$.

######  Lemma 5.24.

We have a canonical isomorphism:
$\mathrm{H}^{1}_{c}(X_{0}(p)^{et,{\dagger}},\omega^{2-\kappa^{un}}(-D))=\lim_{a\rightarrow
0}\mathrm{H}^{1}_{X_{0}(p)_{[0,a[}}(X_{0}(p),\omega^{2-\kappa^{un}}(-D))$.

###### Proof.

We have a short exact sequence for $0<a<b$ small enough:

$0\rightarrow\mathrm{H}^{0}(X_{0}(p)_{[0,b]},\omega^{2-\kappa^{un}}(-D))\rightarrow\mathrm{H}^{0}(X_{0}(p)_{[a,b]},\omega^{2-\kappa^{un}}(-D))$
$\rightarrow\mathrm{H}^{1}_{X_{0}(p)_{[0,a[}}(X_{0}(p)_{[0,b]},\omega^{2-\kappa^{un}}(-D))\rightarrow
0.$

Passing to the limit as $a\rightarrow 0$ proves the lemma. ∎

Therefore the operator $U_{p}$ is compact on both cohomology groups. We deduce
from the pairing that the characteristic series of $U_{p}$ are the same in
degree $0$ and degree $1$, so that $\mathcal{X}=\mathcal{Z}$. We have a
canonical perfect pairing
$\langle,\rangle:\mathcal{M}\times\mathcal{N}\rightarrow
w^{-1}\mathscr{O}_{\mathcal{W}_{r}}$, for which $\langle zf,g\rangle=\langle
f,zg\rangle$ for all
$(z,f,g)\in\mathscr{O}_{\mathcal{Z}}\times\mathcal{M}\times\mathcal{N}$ and
$\langle hf,g\rangle=\langle f,hg\rangle$ for any Hecke operator $h$ of level
prime to $Np$. We deduce that the eigencurves $\mathcal{C}$ and $\mathcal{D}$
are canonically isomorphic.

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