Higher Hida and Coleman theories on the modular curve

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.


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 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 sections.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 [Pil20], [BCGP21] we began to develop them further in order to study the higher coherent cohomology of automorphic vector bundles on the Shimura varieties for the group 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 → Spec Z p be the compactified modular curve of level Γ 1 (N ), where N ≥ 3 is an integer prime to p, and let D be the boundary divisor.Let X 1 → Spec F p be the special fiber and X ord 1 be the ordinary locus.Let ω be the modular line bundle and for a weight k ∈ Z write ω k for ω ⊗k .Theorem 1.1 (Hida's control theorem).There is a Hecke operator T p acting on the cohomology groups RΓ (X 1 , ω k ), RΓ c (X ord 1 , ω k ), and RΓ (X ord 1 , ω k ), and an associated ordinary projector e(T p ).Moreover, we have quasiisomorphisms e(T p )RΓ (X 1 , ω k ) −→ e(T p )RΓ (X ord 1 , ω k ) if k ≥ 3 and e(T p )RΓ c (X ord 1 , ω k ) −→ e(T p )RΓ (X 1 , ω k ) if k ≤ −1.
The coherent cohomology with compact support appearing in the theorem has been defined by Hartshorne [Har72].If SS ⊆ X 1 denotes the (reduced) supersingular divisor, then we have 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 )RΓ (X 1 , ω k ) is concentrated in degree 0 if k ≥ 3, and degree 1 if k ≤ −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, however the first argument is well suited for generalizations to higher dimensional Shimura varieties.
Let Λ = Z p [[Z × p ]] be the Iwasawa algebra.Each integer k ∈ Z defines a character of Z × p , and a morphism k : Λ → Z p .
Theorem 1.2.There are two projective Λ-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 ≥ 3: (1) M ⊗ Λ,k Z p = e(T p )H 0 (X, ω k ), (2) N ⊗ Λ,k Z p = e(T p )H 1 (X, ω 2−k (−D)), Moreover, there is a perfect pairing M × N → Λ 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 Λ-modules.
Let X 0 (p) → Spa(Q p , Z p ) be the adic modular curve of level Γ 1 (N ) ∩ Γ 0 (p).We have two quasi-compact opens X 0 (p) m and X 0 (p) ét 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, † and X 0 (p) ét, † be the corresponding dagger spaces [G-K00].
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 deduce a vanishing theorem for the small slope classical cohomology (without appealing to the Riemann-Roch theorem).
Coleman and Mazur constructed the eigencurve C of tame level Γ 1 (N ).It carries a weight morphism w : C → W where W is the analytic adic space over Spa(Q p , Z p ) associated with the Iwasawa algebra Λ.

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.We thank Ben Heuer and the anonymous referee for their corrections and suggestions.

Finite flat cohomological correspondences
In this section all schemes are Noetherian.We write Coh(X) for the category of coherent sheaves on a scheme X.
Let f : X → Y be a finite flat morphism of schemes.We have a functor f : Coh(X) → Coh(Y ).It induces an equivalence of categories between Coh(X) and the category of coherent sheaves of f O X -modules over Y .
The functor f has a left adjoint f : Coh(Y ) → Coh(X), given by f f F = F ⊗ O Y f O X , as well as a right adjoint f !: Coh(Y ) → Coh(X) given by f f !F = Hom O Y (f O X , F ).For any F ∈ Coh(X), we have an isomorphism A finite flat morphism f :

Cohomological correspondences.
Definition 2.1.A finite flat correspondence over a scheme X is a scheme C equipped with two finite flat morphisms Definition 2.2.Let F be a coherent sheaf on X.A finite flat cohomological correspondence for F is the data of a pair (C, T ) consisting of a finite flat correspondence C and a map T : p 2 F → p ! 1 F .Given a finite flat cohomological correspondence (C, T ) on F , we get a morphism in cohomology that we also denote by T :

Restriction.
Let X be a scheme and Y → X be a closed subscheme defined by a sheaf of ideals I ⊆ O X .For any quasi-coherent sheaf F over X, we let be the subsheaf of sections with scheme theoretic 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 F be a coherent sheaf on Z. Then Proof.Let us denote by I the ideal sheaf of Y in X.The claim follows from the exact sequence:

Local finiteness and ordinary projectors
Let R be a finite Artinian ring.
Lemma 2.4.Suppose M is a finite R-module and T ∈ End R (M).The sequence (T n! ) n∈N is eventually constant and converges to an idempotent e(T ) ∈ 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.T restricts to a bijection on the finite set M 0 , and hence for n large enough T n! is the identity on M 0 .It follows that for all n large enough, T n! is an idempotent projector from M to M 0 .Definition 2.5.Let M be an R-module and let T ∈ End R (M).We say that T is locally finite on M if M is a union of finite R-modules which are stable under T .
Remark 2.6.This is equivalent to: for any finite submodule V ⊆ M there exists an n > 0 such that T n V ⊆ n−1 i=0 T i V .Let M be an R-module and let T ∈ End R (M) be locally finite.Then for each v ∈ M, Lemma 2.4 implies that the sequence T n! v is eventually constant, and we define e(T ) : M → M by e(T )v = T n! v for n sufficiently large.
Lemma 2.4 further implies that e(T ) is an R-linear idempotent which commutes with T , and moreover T is an isomorphism on e(T )M and locally nilpotent on (1−e(T ))M, in the sense that for each v ∈ (1−e(T ))M, there exists n > 0 such that (1) T is locally finite on N if and only if T is locally finite on M and L.
Proof.The only point that requires a proof is that if T is locally finite on M and L, then it is locally finite on N .Let V ⊆ N be a finite submodule.Since T is locally finite on L, there exists n ≥ 1 such that There is furthermore a finite submodule W ⊆ M such that Since T is locally finite on M, there exists m ≥ 1 such that: We finally conclude that T n+m (V +W ) We will also need to work with a different notion of local finiteness, defined on certain topological R-modules.Definition 2.8.A topological R-module M is a profinite R-module if it is homeomorphic to a cofiltered limit lim i∈I M i of finite R-modules, or equivalently, if it is Hausdorff and has a basis of neighborhoods {N i } i∈I of 0 consisting of open cofinite submodules.Definition 2.9.Let M be a profinite R-module, and let T ∈ 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 open, then it is cofinite, and hence must be ∩ n−1 i=0 T −i (V ) for some n > 0.) Let M be a profinite R-module and let T ∈ End R (M) be a locally finite endomorphism.Pick a neighborhood basis {N i } i∈I of 0 with T (N i ) ⊆ N i .Then for all v ∈ M and each i ∈ I, Lemma 2.4 implies that the sequence T n! v is eventually constant in M/N i .It follows that the sequence T n! v converges in M, to a limit that we denote e(T )v.
Lemma 2.4 further implies that e(T ) is a continuous, R-linear idempotent which commutes with T , and moreover T is an isomorphism on e(T )M and topologically nilpotent on (1 − e(T ))M, in the sense that for each v ∈ (1 − e(T ))M, {T n v} n∈N converges to 0.
If f : M → N is a continuous morphism of profinite R-modules, equivariant for a locally finite endomorphism T then we have e(T )f = f e(T ) and so f (e(T )M) ⊆ e(T )N and f Proposition 2.11.Let 0 → M → N → L → 0 be a short exact sequence of profinite R-modules with an equivariant continuous endomorphism T .
(1) T is locally finite on N if and only if T is locally finite on M and L.
Proof.The only point that requires a proof is that if T is locally finite on M and L, then it is locally finite on N .Let V ⊂ N be an open submodule.Since T is locally finite on M, we deduce that there is n ≥ 1 such that It follows that: To shorten notations, let us denote by It follows that: We will also consider the situation that R = lim n R/m n is a complete, semilocal, noetherian ring with Jacobson radical m, and R/m is finite (e.g. is a complete R-module with an endomorphism T , we say that T is locally finite if for all n (or equivalently for a single n by Propositions 2.7 and 2.11) the induced endomorphism T of M/m n M is locally finite in one of the two senses considered above.In this case the idempotent endomorphisms e(T ) of M/m n M are compatible and define an idempotent endomorphism e(T ) of M.

The cohomological correspondence T p
Let N ≥ 3 be an integer and let p be a prime.Let X → Spec Z p be the compactified modular curve of level Γ 1 (N ) ( [DR73]).This is a proper smooth relative curve.Denote by D the boundary divisor, and by E → X the semi-abelian scheme which extends the universal elliptic curve and denote by e the unit section.We let ω E = e Ω 1 E/X .For any k ∈ Z, we denote by ω k = ω ⊗k E .

The cohomological correspondences T p
We denote by p 1 , p 2 : X 0 (p) → X the Hecke correspondence which parametrizes an isogeny π : p 1 E → p 2 E of degree p (compatible with the Γ 1 (N ) level structure).We denote by D 0 (p) the boundary divisor in X 0 (p) (which is reduced, so that D 0 (p) = (p −1 i D) red ).We let π k : p 2 ω k p 1 ω k be the rational map which is deduced from the pull-back map on differentials p 2 ω E → p 1 ω E (this map is regular if k ≥ 0, and is an isomorphism over Q p for all k).We also denote by π −1 k : p 1 ω k p 2 ω k the inverse of π k .We also have a dual isogeny π ∨ : p 2 E → p 1 E and we denote by π ∨ k : p 1 ω k p 2 ω k the rational map which is deduced from the pull-back map on differentials p 1 ω E → p 2 ω E .We also denote by (π ∨ k ) −1 : p 2 ω k p 1 ω k the inverse of π ∨ k .We have the following formula relating π k and π ∨ k : We have natural trace maps tr p 1 : Proof.The boundary divisors in X and X 0 (p) are reduced.The lemma boils down to the statement that the trace of a function vanishing along the boundary on X 0 (p) vanishes on the boundary on X.
We have a "naive" (i.e.unnormalized) cohomological correspondence: which is the rational map defined by taking the tensor product of the map π k : p 2 ω k p 1 ω k and the map tr p 1 : O X 0 (p) → p ! 1 O X , and similarly a map T naive p,k Proof.The map T p,k is a rational map between invertible sheaves over the regular scheme X 0 (p) so we can check that it is defined outside of codimension 2. Since the map is defined over Q p , we can thus localize at a generic point ξ 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 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: obtained by composing the maps: and similarly for cuspidal cohomology.
Remark 3.3.Proposition 3.2 is a particular instance of constructions performed in [FP21], 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 Neben- : it maps a n q n to a np q n + p k−1 χ(p) a n q np if k ≥ 1 and to p 1−k a np q n + χ(p) a n q np if k ≤ 1.
Remark 3.5.Our normalization is consistent with standard conjectures on the existence and properties of Galois representations associated to automorphic forms ( [BG14]).The cohomology groups H i (X, ω k ) ⊗ C can be computed using automorphic forms and for any π = π ∞ ⊗ π f contributing to the cohomology, we find that the infinitesimal character of π ∞ is given by and is indeed C-algebraic.By the Satake isomorphism, we know that T naive 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 → (1, t 1−k ) and the corresponding Hodge polygon has slopes 1−k and 0. We find that T naive ) .The Katz-Mazur inequality predicts that the Newton polygon (which has slopes the p-adic valuations of two eigenvalues of Frob p ) is above the Hodge polygon with same ending and initial point, from which we find that v(T naive p |π p ) ≥ inf{1, k} and that T p is indeed optimally integral.

Duality
We let D Z p = RHom Z p (−, Z p ) be the dualizing functor on the category of bounded complexes of finite type Z p -modules [Har66, Chapter V].We let ω X/Z p and ω X 0 (p)/Z p be the dualizing modules.We recall that ω X/Z p = Ω 1 X/Z p , while ω X 0 (p)/Z p = j Ω 1 U /Z p , where j : U → X 0 (p) is the complement of the supersingular locus in the special fiber, which is also the smooth locus of X 0 (p) → Spec(Z p ). Then we have dualizing functors ) on the derived category of bounded complexes of coherent sheaves on X and X 0 (p).When the context is clear we only write D for any of these dualizing functors.
We now want to understand how this duality isomorphism behaves with respect to Hecke operators.The Hecke T p is defined as a composition and hence dualizes to a composition: We have , according to [Har66, Chapter III, Theorem 11.1 and Chapter V, Proposition 8.5], and it remains to understand D(T p ) : We first recall that we have the Kodaira-Spencer isomorphism over We consider the correspondence If we use the Kodaira-Spencer isomorphism on both sides, we obtain a map: ).We can also consider the transpose correspondence T t p : We can consider the diamond operator p on X and X 0 (p) which multiplies the Γ 1 (N ) level structure by p, as well as the Atkin-Lehner map w : X 0 (p) → X 0 (p), which sends the isogeny π : p 1 E → p 2 E to the dual isogeny π ∨ : p 2 E → p 1 E. We have the formulas w 2 = p , p 1 w = p 2 , and p 2 w = p p 1 , and pulling back by w, we see that T t p acts on cohomology as p −1 T p .The main result of this section is now the following: Proposition 3.6.We have D(T p ) = T t p as maps p 1 (ω −k+2 (−D)) → p ! 2 (ω −k+2 (−D)), and hence we have Lemma 3.7.The following diagram is commutative: Proof.The dual of the map On the other hand, using the Kodaira-Spencer isomorphism we have

The mod p theory
We write X 1 → Spec F p and X 0 (p) 1 → Spec F p for the special fibers of X and X 0 (p).We write X ord 1 ⊆ X 1 for the ordinary locus.
We recall that X 0 (p 1 is the union of the Frobenius and Verschiebung correspondences.We let p F i and p V i be the restrictions of the projections p i to these components.The projection ).We denote by i F and i V the inclusions If k ≤ 0, we have a factorization on X 1 : Proof.By Proposition 2.3, this amounts to checking that the cohomological correspondence T p : p 2 ω k → p ! 1 ω k vanishes at any generic point of multiplicative type in X 0 (p) 1 if k ≥ 2, and at any generic point of étale type in X 0 (p) 1 if k ≤ 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: Proposition 4.3.For all k ≥ 2 and n ∈ Z, the cohomological correspondence T p induces a map: ).For all k ≤ 0 and n ∈ Z, the cohomological correspondence T p induces a map: 1 is totally ramified of degree p and the map p V 2 is an isomorphism.It follows that we have an equality of divisors E identifies with the Hasse invariant and induces an isomorphism: ).We now prove the claim when k ≤ 0. The cohomological correspondence is supported on X 0 (p) F 1 by Lemma 4.1.The map p F 2 is totally ramified of degree p and the map p F 1 is an isomorphism.It follows that we have an equality of divisors (p F 2 ) (SS) = p(p F 1 ) (SS).We deduce that the map ).This proves the claim for k = 0.For k ≤ −1, we remark that the cohomological correspondence ) ω k that we now describe.Informally, this map is deduced from the differential of the isogeny (p F 1 ) E → (p F 2 ) E, after normalizing by a factor p −1 (one can make sense of this over the formal scheme ordinary locus).Equivalently, it is deduced from the differential of the isogeny of the dual map (p F 2 ) E → (p F 1 ) E (the Verschiebung map).We observe that (p F 2 ) ω E ((p F 1 ) ω E ) p and there is a natural isomorphism: (ω E (SS)), and therefore, for all k ≤ 0, an isomorphism: We deduce that there is a map: (1) The T p operator acts on RΓ (X 1 , ω k (nSS)) for all n ≥ 0 and k ≥ 2, and the maps where the diagonal arrow is the map of Proposition 4.3.
For any k, we define as usual . This is a profinite F p -vector space.We also recall that Remark 4.7.Of course, we even have H 1 (X 1 , ω k ) = 0 for k ≥ 2 and H 0 (X 1 , ω k ) = 0 for k < 0 by the Riemann-Roch theorem and the Kodaira-Spencer isomorphism, but the given proof is independent of this, and more suitable for generalizations in higher dimension.

The p-adic theory
Let X be the p-adic completion of X and let X n → Spec Z/p n Z be the scheme obtained by reduction modulo p n .Let X ord and X ord n denote the ordinary loci.

The Igusa tower.
For all n ≥ 1, we have over X ord a canonical multiplicative subgroup is an isomorphism.Passing to the limit over n we obtain the Hodge-Tate map where T p (H can ) D is the pro-étale sheaf lim n (H can n ) D (pro-étale locally Z p ), which induces an isomorphism This defines a Z × p reduction of the principal G m -torsor ω E over X ord (in the pro-étale topology).We can form π : IG = Isom(Z p , T p (H can ) D ) → X ord , the Igusa tower.This is a p-adic formal scheme, and a profinite étale cover of X ord . Let ] be the Iwasawa algebra, and let κ un : Z × p → Λ × be the universal character.For any k ∈ Z, we have an algebraic character Z × p → Z × p , given by x → x k and we denote by k : Λ → Z p the corresponding algebra morphism.We let ω κ un = (O IG ⊗Λ) Z × p where the invariants are taken for the diagonal action.
Lemma 4.8.The sheaf ω κ un is an invertible sheaf of O X ord ⊗Λ-modules and for any k ∈ Z, there is a canonical isomorphism of invertible sheaves over X ord :

Cohomology of the ordinary locus.
We may now consider the following Λ-modules: Let us give the definition of the second module.Let m Λ be the kernel of the reduction map ) as follows: we can take any extension of for I any sheaf of ideals on X n whose vanishing locus is the complement of X ord n .We remark that this is a profinite We remark that for the same reason as above H 1 c (X ord n , ω k ) can also be computed as lim l H 1 (X n , I l F ) for any coherent sheaf F on X n extending ω k on X ord n .We also note that there are natural "corestriction" maps

U p and Frobenius.
There is a lift of Frobenius F : X ord → X ord which is given by E → E/H can 1 , and the Γ 1 (N ) level structure on E/H can 1 induced by the isogeny E → E/H can 1 .This map extends to the Igusa tower as a map: F : IG → IG, which is given by where ψ is defined by We also consider a variant F which is defined in the same way as F except that we give E/H can 1 the Γ 1 (N ) level structure via the dual isogeny E/H can 1 → E, or in other words F = p −1 • F. The inverse of F viewed as a correspondence is a lift over the ordinary locus of the Verschiebung correspondence on the special fiber.
We consider two maps, the natural pull-back map on functions F O IG → O IG and the trace map Lemma 4.9.We have tr

U p , Frobenius, and T p .
We now describe the specializations of these maps at classical weight k ∈ Z and compare them with the cohomological correspondence T p .Over X ord we have the canonical multiplicative isogeny π can : E → E/H can 1 as well as its dual étale isogeny π ∨ can : E/H can 1 → E. We consider the induced maps on differentials π can : For k ∈ Z, specializing the maps constructed in Corollary 4.10 gives maps maps F : Proof.The étale isogeny π ∨ can induces an isomorphism H can /H can 1 → H can and it follows directly from the definition of the Hodge-Tate map that there is a commutative digram We can construct a map: Lemma 4.12.The above map coincides with the map U p : F ω k → ω 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): We can consider the completion X 0 (p) and its ordinary part X 0 (p) ord which is the disjoint union of two types of irreducible components: where on the first components the universal isogeny is not étale, and where it is étale on the other components.On X 0 (p) ord,F , the map p 1 is an isomorphism and the map p 2 identifies with the Frobenius map F. On X 0 (p) ord,V , the map p 2 is an isomorphism and the map p 1 identifies with F .We therefore can think of U p and F as cohomological correspondences p 2 ω k → p ! 1 ω k supported respectively on X 0 (p) ord,V and X 0 (p) ord,F .
One can also restrict T p to a cohomological correspondence over X 0 (p) ord and project it on the components X 0 (p) ord,F and X 0 (p) ord,V .We denote by T F p and T V p the two projections of the correspondence T p .Lemma 4.13.
(1) We have Proof.Parts (3) and (4) follow immediately from parts (1) and (2) (compare also with remark 3.4), while parts (1) and (2) are simply a matter of bookkeeping using Lemmas 4.11 and 4.12 and the definition of T p .
For the benefit of the reader we spell out the details for (1).According to the definition of T p and in particular its normalization (recall the proof of Proposition 3.2), p − sup{0,k−1} T F p exists and is given on X 0 (p) ord,F by the composition of p −k π k : p 2 ω k → p 1 ω k and the trace map for p 1 ω k → p ! 1 ω k .But in fact p 1 is an isomorphism on X 0 (p) ord,F , and making this identification p 2 becomes F, while the universal isogeny over X 0 (p) ord,F becomes π can .Thus we can view p − sup{0,k−1} T F p as a map F ω k → ω k , which is F by Lemma 4.11 We end this discussion with duality.

Higher Hida theory. Theorem 4.15.
(1) There is a locally finite action of F on H 1 c (X ord , ω κ un ) and e(F)H 1 c (X ord , ω κ un ) is a finite projective Λ-module.Moreover, (2) U p is locally finite on H 0 (X ord , ω κ un ) and e(U p )H 0 (X ord , ω κ un ) is a finite projective Λ-module.Moreover, Proof.We will first construct a continuous action of F on H 1 c (X ord n , ω κ un /(m Λ ) n ), compatible for all n.We let I ⊆ O X n be a locally principal sheaf of ideals so that the complement of the corresponding closed subscheme is X ord n (for instance the sheaf of ideals defined by a lift of a power of the Hasse invariant).We take a coherent sheaf F over X n extending ω κ un /(m Λ ) n .We may assume that F is I -torsion free by replacing F with its quotient by the subsheaf of I -power torsion.Then the multiplication map I l ⊗ O X n F → I l F is an isomorphism for all l ≥ 0, and we use this to make sense of I l F for all l ∈ Z. Then if j : X ord n → X n denotes the inclusion, we have j ω κ un /(m Λ ) n = colim l I −l F .
We write X 0 (p) n → Spec Z/p n Z for the reduction mod p n of X 0 (p), and X 0 (p) ord n for its ordinary locus.Then p 1 I and p 2 I can be identified with p −1 1 (I )O X 0 (p) n and p −1 2 (I )O X 0 (p) n , and so we view them as locally principal sheaves of ideals in O X 0 (p) n .They define the same closed subset of X 0 (p) n (the complement of X 0 (p) ord n ) and so in particular we can pick some m with p 2 I m ⊆ p 1 I .We have , where X 0 (p) ord,F n is the component where the universal isogeny has connected kernel, and X 0 (p) ord,V n the component where the universal isogeny has étale kernel.The graph of the map F : We can therefore think of F : F ω κ un → ω κ un as a cohomological correspondence on X 0 (p) ord n : which is given by F on the component X 0 (p) ord,F n and by 0 on the component X 0 (p) ord,V n .Pushing this forward from X ord n to X n we obtain a map It follows that there exists some c for which there is a map p 2 F → p ! 1 (I −c F ). From this and the inclusion p 2 I m → p 1 I we deduce a map p 2 (I lm F ) −→ p ! 1 (I −c+l F ) for all l ≥ 0. Taking cohomology we obtain a map F : H 1 (X n , I lm F ) → H 1 (X n , I −c+l F ). Passing to the limit over all l we obtain a continuous endomorphism F of H 1 c (X ord n , ω κ un /(m Λ ) n ).We now need to prove that F is locally finite.We first deal with n = 1.In that case, (in this last formula, we can let k go through any set of representatives of Z/(p − 1)Z in Z) and this isomorphism is equivariant for the action of F on the left, and T p on the right by Lemma 4.13.It follows from Corollary 4.5 that F is locally finite for n = 1, and also that e(F)H 1 c (X ord 1 , ω κ un /m Λ ) is a finite F p -vector space.We deal with the general case by induction, using the short exact sequences in cohomology (X ord is affine) and Proposition 2.11: It follows that F is locally finite and e(F)H 1 c (X ord , ω κ un ) is a finite projective Λ-module, as it is a complete flat Λ-module whose reduction mod m is finite.
Finally, for all k ∈ Z, we have an isomorphism e(F)H 1 c (X ord , ω κ un ) ⊗ Λ,k Z p = e(F)H 1 c (X ord , ω k ) and we can consider the composition where the maps are inclusion, corestriction, and projection.When k ≤ −1 this is a map of finite free 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.

Serre duality
Recall that we have a residue map res : H 1 (X, Ω 1 X/Z p ) → Z p .Therefore, there is a natural map: where the first map is the corestriction (see Section 4.2.2).Let us denote by ω 2−κ un (−D) = ω 2 (−D) ⊗ Hom(ω κ un , Λ ⊗O X ord ).This is an invertible sheaf of Λ ⊗O X ordmodules over X ord .
Proof.We have a commutative diagram: where the vertical maps are injective.(The injectivity of the first vertical map follows from the fact that for any complete flat 0 by completeness.)It suffices therefore to prove the identity for the pairing As in the proof of Theorem 4.15 we let I be a locally principal sheaf of ideals defining a closed subscheme whose complement is X ord n , and we have a cohomological correspondence over X 0 (p) n F : p 2 I ml ω 2−k (−D) −→ p ! 1 I −c+l ω 2−k (−D) so that passing to cohomology and taking the limit over l gives the action of F on H 1 c (X ord n , ω 2−k (−D)).We can consider the dual D(F) : p 1 I −l+c ω k → p ! 2 I −ml ω k .Passing to cohomology and taking the colimit over l gives the action of p −1 U p on H 0 (X ord n , ω k ) by Lemma 4.14.
This pairing hence restricts to a pairing: Theorem 4.18.
(1) The pairing −, − is a perfect pairing, (2) For any (f , g) ∈ e(U p )H 0 (X ord , ω κ un ) × e(F)H 1 c (X ord , ω 2−κ un (−D)), (3) The pairing −, − is compatible with the classical pairing in the sense that for any k ∈ 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 ≥ 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 ∈ H 0 (X, ω k ) and g ∈ H 1 c (X ord , ω 2−k (−D)), we have i(f ), g = f , j(g) .If we now assume that f ∈ e(T p )H 0 (X, ω k ) and g ∈ e(F)H 1 c (X ord , ω 2−k (−D)), we have that e(U p )i(f ), g = i(f ), e(F)g = i(f ), g and f , e(T p )j(g) = e(T p )f , j(g) = f , j(g) and the conclusion follows.

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, Exposé I].Let X be a topological space.Let i : Z → 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 : C Z → C X .The functor i has a right adjoint i !: C X → C Z .For an abelian sheaf F over X, we let Γ Z (X, F ) = H 0 (X, i i !F ).By definition this is the subgroup of H 0 (X, F ) of sections whose support is included in Z.We let RΓ Z (X, −) be the derived functor of Γ Z (X, −).
Let U = X \ Z and let F be an object of C X .We have an exact triangle [Gro05, I, Corollary 2.9]: Some properties of the cohomology with support are: (1) (Change of support) If Z ⊂ Z , there is a map RΓ Z (X, F ) → RΓ Z (X, F ) (cf. [Gro05, Exposé I, Proposition 1.8] ).
(2) (Pull-back) If we have a cartesian diagram: 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 adic spaces, f a finite flat morphism of adic spaces, Z and Z are closed subspaces of X and X respectively.Let F be a sheaf of O X -modules.Then there is a trace map Proof.We first recall that the category of sheaves of O T -modules on a ringed space (T , O T ) has enough injectives ([Sta22, Tag 01DH]).It follows that it is enough to construct a functorial map Therefore, it suffices to consider the case where Z = f −1 (Z ).We have a trace map Tr : f f F → F .Let us complete the above diagram into: where U = X \ Z and U = X \ 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

The modular curve X 0 (p)
We let X 0 (p) be the compactified modular curve of level Γ 0 (p) and tame level Γ 1 (N ) for some prime to p integer N ≥ 3, viewed as an adic space over Spa(Q p , Z p ).We let H 1 ⊂ E[p] be the universal subgroup of order p.

Parametrization by the degree.
Let X 0 (p) rk1 be the subset of rank one points of X 0 (p).Fargues defines a map deg : X 0 (p) rk1 → [0, 1], which sends x ∈ X 0 (p) rk1 to deg H 1 ∈ [0, 1].We briefly recall the definition (see [Far10, §4, Definition 3] for more details).The group H 1 extends to a finite flat group scheme H1 over Spec k(x) + , by taking the schematic closure in E[p].There is an isomorphism ω H1 = k(x) + /f k(x) + and deg H 1 = v x (f ) for v x the valuation attached to x, normalized by v(p) = 1.For any rational interval 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 .

The canonical subgroup.
We let X be the compactified modular curve of level prime to p.We have an Hasse invariant Ha and we can define the Hodge height: obtained by sending x to inf{v x ( Ha), 1} for any local lift Ha of the Hasse invariant.For any v ∈ [0, 1], we let X v = {x ∈ X | Hdg(x) ≤ v} (more correctly, X v is the quasi-compact open whose rank one points are those described as above).
We recall the following theorems: which is locally isomorphic to Z/pZ in the étale topology.
(1) For any rank one point of X 0 (p), we have the identity

The correspondence U p
We let C be the correspondence over X 0 (p) underlying U p .It parametrizes isogenies of degree p: There is actually an isomorphism X 0 (p 2 ) → C (where ).By Theorem 5.4, if a > 1 p+1 , then we have a canonical subgroup of order p over X 0 (p) [a,1] , H can 1 = H 1 and if a < 1 p+1 , we also have a canonical subgroup of order p over X 0 (p) [0,a] and H 1 H can 1 .The map p 1 : C [0,a] → X 0 (p) [0,a] has a section given by H can 1 .Let C can [0,a] be the image of this section and let C ét [0,a] be its complement, so that Proof.We do the case by case argument to check the first two points, using Theorem 5.4, (1): (1) If a > , 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 X 0 (p 2 ), for which we have p 2 (E, H 2 ) = (E, H 1 ) and Proposition 5.6.
Proof.We do the case by case argument for the first two points, using Theorem 5.4, (1): (1) We have deg , there is no canonical subgroup and the same formula holds.

The U p -operator
We work over X 0 (p).Let a ∈]0, 1[∩Q.The cohomologies of interest are: ( The category of perfect Banach complexes is the homotopy category of the category of bounded complexes of Banach spaces over Q p .A quasi-isomorphism of perfect Banach complexes admits an inverse up to homotopy ([Sch99, Proposition 1.3.22], a quasi-isomorphism is always strict by the open mapping theorem).
Lemma 5.7.The above cohomologies can be canonically represented by objects of the category of perfect Banach complexes by using Čech covers.
Proof.By [Hub94, Lemma 2.6], any finitely generated module over a complete Tate algebra carries a canonical topology and is complete.We deduce that the sections of ω k over an affinoid open subset of X 0 (p) form naturally a Q p -Banach space.In case (1) and (2) we can take a Čech complex for a finite affinoid covering to represent the cohomology.Since any two Čech cover can be refined by a third one, and since quasi-isomorphism admit inverses up to homotopy, we deduce the independence (up to homotopy) of the Čech complex.In case (3), the cohomology fits in an exact triangle: and is quasi-isomorphic to the cone of a continuous map between perfect Banach complexes, hence is a also a perfect Banach complex.

Constructing the action.
A morphism in the category of perfect Banach complexes is called compact if it can be represented by a morphism between complexes which is compact in each degree.
We define a naive cohomological correspondence U naive p has follows.The two ingredients are the differential map p 2 ω E → p 1 ω E and the trace map (p 1 ) O C → O X 0 (p) .Putting all this together, we obtain Proposition 5.8.We have an action of the U naive p -operator on RΓ (X 0 (p), ω k ), RΓ (X 0 (p) [a,1] , ω k ) and -operator is compact.Moreover, the U naive p -operator acts equivariantly on the triangle: Proof.The operator U naive p is compact on RΓ (X 0 (p), ω k ) because this later complex is a perfect complex of finite dimensional Q p -vector spaces.By Proposition 5.5: It follows that we have a compact morphism RΓ (X 0 (p) [a,1] , ω k ) → RΓ (p 2 (C [a,1] ), ω k ).Therefore, we have a map: We deduce that U naive p acts and is compact on RΓ (X 0 (p) [a,1] , ω k ) because the first map is.Similarly, by Proposition 5.6: The operator U naive p acts like the composite of the following maps: We claim that the map 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.
For any h ∈ Q we can consider a direct factor RΓ (X 0 (p), ω k ) ≤h , RΓ (X 0 (p) [a,1] , ω k ) ≤h and respectively RΓ X 0 (p) [0,a[ (X 0 (p), ω k ) ≤h of RΓ (X 0 (p), ω k ), RΓ (X 0 (p) [a,1] , ω k ) and respectively RΓ X 0 (p) [0,a[ (X 0 (p), ω k ) called the slope less than h part.It is obtained by representing U naive p by a compact map of complexes and by applying the slope less than h projector in each degree ( [Ser62]).The slope ≤ h complexes are perfect complexes of Q p -vector spaces (i.e. they are bounded complexes with finite dimensional cohomology).The finite slope part (denote by a supscript f s) is the inverse limit of the ≤ h-part for h → ∞.
We also note the following corollary of the proof: Corollary 5.9.For any 0 < a < b < 1, the natural maps induce quasi-isomorphism on the finite slope part for U naive p .

U p and Frobenius.
It is worth spelling out the action of where p can 1 is actually an isomorphism.We can think of this correspondence as the graph of the Frobenius map X 0 (p) [0,pa] → X 0 (p) [0,a] , sending (E, H 1 ) to (E/H can 1 , E[p]/H can 1 ).We claim that there is an associated operator: We first observe that RΓ X 0 (p) [0,a[ (X 0 (p), ω k ) = RΓ X 0 (p) [0,a[ (X 0 (p) [0,a] , ω k ).We now construct U naive, can p as the composite: Proof.This amounts to comparing the construction of U naive p (given in the proof of Proposition 5.8) and of U naive, can p .We see that p 1 (C [0,a[ ) ⊂ X 0 (p) [0,a] and therefore,

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 ∈]0, 1[∩Q, (1) the slopes of Proof.We first observe that the finite slope part of the cohomology RΓ (X 0 (p) [a,1] , ω k ) f s is independent of a ∈]0, 1[ and is therefore supported in degree 0 by affineness for a close to 1.It follows that we are left to prove that U naive p has slopes at least 1 and this is a q-expansion computation.Namely, U naive p ( a n q n ) = p a np q n .For the second cohomology, we again observe that the cohomology is independent of a ∈]0, 1[.We may therefore suppose that a is small enough and use that U naive p = U naive, can p (Proposition 5.10).We define an invertible sheaf of O + X 0 (p) -modules, ω + E ⊆ ω E .A section f ∈ ω E (U ) belongs to ω + E (U ) if for any x ∈ U , f x defines a section of the conormal sheaf on the extension Ẽ → Spec k(x) + of the semi-abelian scheme E at x.Alternatively, we have a specialization morphism sp : X 0 (p) → X where X is the formal scheme equal to the completion of the prime-to-p level modular curve X viewed as a scheme over Spec Z p .Then where ω E is the modular sheaf over X.We let ω k,+ = (ω + E ) ⊗k .We first claim that for any s ∈ Q, Moreover, over C can [0,a[ , we have a universal isogeny which gives an isomorphism, p 2 ω → p 1 ω, for which pp 1 ω + ⊂ p 2 ω + ⊂ p 1− a p p 1 ω + .We deduce that U naive p induces a map if k ≤ 0 stabilize a lattice in the cohomology, and therefore have only non-negative slope.The lemma follows.
Remark 5.12.Lemma 3.2.2 in [Pil20] 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 naive p and we get: Theorem 5.13.
Proof.We consider the triangle on which U p acts equivariantly and apply the slope estimates of Lemma 5.11.
There is an analogous statement for the cohomology with support that we now explain.We first introduce a cohomology with support at spherical level.For a < 1 we define X <a ⊆ X as the complement in X of the quasi-compact open whose rank one points are {x ∈ X rk1 | Hdg(x) ≥ a} (see Section 5.2.2).We may then form the cohomology with support RΓ X <a (X, ω k ).
Let a < 1 p+1 .We have a Frobenius map F : X 0 (p) [0,a] → X 0 (p) [0,pa] given by There is similarly a Frobenius map F : X a p → X a , given by E → E/H can 1 .The Frobenius map fits into the following diagram: where the diagonal map is given by E → (E/H can 1 , E[p]/H can 1 ).We can define an endomorphism F of RΓ X < a p (X, ω k ) as the composite where the first map is the change of support for the inclusion X a p ⊆ X a , which is compact as in the proof of Proposition 5.8, the third map is pullback by F, and the fourth map is induced by the map F ω k → ω k which comes from the isogeny E → E/H can 1 .The same construction applied to X 0 (p) [0,a] yields the operator U naive, can which is deduced from the definitions and the commutative diagram above.

p-adic variation
We now consider the problem of interpolation of these cohomologies.

Reduction of the torsor ω E .
We recall here a construction from [Pil13] and [AIS14].We let T = {w ∈ ω E , ω 0} be the G m -torsor associated to ω E .We let G + a = Spa(Q p T , Z p T ) be the unit ball, with its additive analytic group structure.We have subgroups Z × p (1 + p r G + a ) → G m for any positive rational number r.
Proof.We denote by ω + E ⊂ ω E the locally free sheaf of integral relative differential forms.For v < There is a Hodge- We now introduce a modification of ω + E : let ω E = {w ∈ ω + E , r(w) ∈ Im(HT ⊗ 1)} ⊂ ω + E .This is a locally free sheaf of O + X v -modules on the étale site.The Hodge-Tate map induces an isomorphism: We let T v be the torsor under the group We have a natural map T v → T , equivariant for the analytic group map:

Interpolation of the sheaf.
We let W = Spa(Λ, Λ) × Spa(Q p , Z p ) be the weight space.We let κ un : Z × p → O × W be the universal character.We can write W as an increasing union of affinoids W = ∪ 0<r<1 W r where r ∈ Q∩]0, 1[ and each W r is a finite union of balls of radius r.Over each W r , there is t(r) ∈ Q >0 such that the universal character extends to a character κ un : We now fix r and we choose v small enough and n large enough such that t(r) ≤ n − v p n p−1 and we define a locally free sheaf ω κ un over X v × W r : Since T v → X v is an étale torsor, the sheaf ω κ un is a locally free sheaf of O X v ×W r -modules in the étale topology.It is actually a locally free sheaf of O X v ×W r -modules in the Zariski topology by the main result of [BG98].

Interpolation of the cohomology.
For a ∈]0, v p ], we have a map p 1 : X 0 (p) [0,a] → X v and we can therefore pull back the sheaf ω κ un to an invertible sheaf over X 0 (p) [0,a] × W r .
If a ∈ [1 − v, 1[, we have a map p 1 : X 0 (p) [a,1] → X v and we can pull back the sheaf ω κ un to an invertible sheaf over X 0 (p) [a,1] × W r .
We consider the cohomologies: (1) Note that the first cohomology group is well defined because These cohomologies belong to the category of perfect complexes of Banach spaces over O W r which is the homotopy category of the category of bounded complexes of projective Banach modules over O W r .

The U p -operator on RΓ (X 0 (p) [a,1] , ω κ un ).
We need to consider the U p -correspondence on X 0 (p) [a,1] for a ≥ 1 − v > 1 p+1 , where by Proposition 5.5, it induces to a correspondence This clearly induces an isomorphism p 1 T v → p 2 T v and from this we get an isomorphism: p 2 ω κ un −→ p 1 ω κ un which specializes to the natural isomorphism p 2 ω k → p 1 ω k at weight k.We now define Corollary 5.17.The operator U p is compact on RΓ (X 0 (p) [a,1] , ω κ 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 by Proposition 5.10, and therefore it reduces to a correspondence: Remark 5.19.We therefore find that this is really U p and not U naive p that can be interpolated over the weight space.
Proof.The operator U p factors as:

Construction of eigencurves
We use these cohomologies to construct the eigencurve, following the method of [Col97].
The U p -operator acts compactly on this space.We let P ∈ O W r [[T ]] be the characteristic series of U p .This is an entire series.We let Z = V (P ) ⊂ G an m × W r .We have a weight map π : Z → W r which is quasi-finite, partially proper, and locally on the source and the target a finite flat map.
Over Z we have a coherent sheaf M which is the universal generalized eigenspace.This is a locally free as a π −1 O W r -module and for any We let O C ⊆ End Z (M) be the subsheaf of O Z -modules generated by the Hecke operators of level prime to N p, and we let C → Z be the associated analytic space.
The construction of (M, C, 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 C → Z → W and a coherent sheaf M over C.This is the eigencurve of [CM98].
We choose a representative N • for this cohomology, as well as a compact representative p −1 U p representing the action of p −1 U p .We let Q i be the characteristic series of p −1 U p acting on N i .We let Q = Q i and we let X = V ( Q) ⊆ G an m × W r .We have a weight map π : X → W r which is quasi-finite locally on the source and the target and a bounded complex of coherent sheaves "generalized eigenspaces" Ñ • over X .This is a perfect complex of finite projective π −1 O W r -modules.Moreover, Ñ • has cohomology only in degree 1, and we deduce that H 1 (N • ) = N is a locally free π −1 O W r -module.We let Q = V ( i Q (−1) i i ) and we set X = V (Q) ⊆ X .The module N is supported on X .We let O D ⊆ End X (N ) be the subsheaf generated by the Hecke algebra of prime to N p level, and we let D → X be the associated analytic space.We can now let r tend to 1, and we have D → X → W and the sheaf N over D. This is a second eigencurve.

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

Preliminaries.
We are going to use the theory of dagger spaces [G-K00].Let X † be a dagger space over Spa(Q p , Z p ), smooth of pure relative dimension d.Let F be a coherent sheaf on X † .Then one can define the cohomology groups H i (X † , F ) and H i c (X † , F ).Moreover, these cohomology groups carry canonical topologies ([vdP92, 1.6]).
By [G-K00, Theorem 4.4], there is a residue map: This residue map has the following two important properties.Let f : Y † → X † be an open immersion.Then the diagram: Q p is commutative.See [Bey97, Corollary 4.2.12](although the author is working here in the "dual" setting of wide open spaces)).
Let f : Y † → X † be a finite flat map.Then the diagram: (5.1) Remark 5.21.Since the topological vector space H 0 (X † , F ) is a compact inductive limit of Banach spaces, we deduce from the theorem that the topological vector space H d c (X † , D(F )) is a compact projective limit of Banach spaces.

The classical pairing.
We denote by w the Atkin-Lehner involution over X 0 (p) and by p the diamond operator given by multiplication by p.We recall that w • w = p .We recall that C is the Hecke correspondence underlying U p (see Section 5.3).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 ι : C → C and we let q i = p i • ι.Now one checks easily that w • p 1 = q 2 and w • p 2 = p • q 1 .We have a residue map H 1 (X 0 (p), Ω 1 X 0 (p)/Q p ) → Q p and there is a perfect pairing: −, − 0 : H 0 (X 0 (p), ω k ) × H 1 (X 0 (p), ω 2−k (−D)) → Q p where we use the Kodaira-Spencer isomorphism Ω 1 X 0 (p)/Q p ω 2 (−D).We modify this pairing, and set: −, − = −, w (−) 0 .Lemma 5.22.For any (f , g) ∈ H 0 (X 0 (p), ω k ) × H 1 (X 0 (p), ω 2−k (−D)), we have: U p f , g = f , p −1 U p g .Proof.For any (f , g) ∈ H 0 (X 0 (p), ω k ) × H 1 (X 0 (p), ω 2−k (−D)), we have: U naive p f , g 0 = f , (U naive p ) t g 0 where (U naive p ) t is the operator associated to the transpose of C, and is obtained as follows: RΓ (X 0 (p), D(ω k )) We see that U naive p f , w g 0 = f , w p −1 U naive p g 0 .We now check that the normalizing factors are correct so that U p f , g = f , p −1 U p g .

Moreover
• H 0 (X 0 (p) m, † , ω κ un ) is a compact inductive limit of projective Banach spaces over O W r , • H 1 c (X 0 (p) m, † , ω 2−κ un (−D)) is a compact projective limit of projective Banach spaces over O W r , and both spaces are strong duals of each other.
Proof.The pairing is obtained as follows: We prove the remaining claims.Let W i be the connected component of the character i : x → x i for i = 0, . . ., p − 2 in W . Then for all i, ω κ un | W i r = ω i ⊗Q p O 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, and similarly for cuspidal cohomology.All the statements of the lemma are reduced to the similar statements for the classical invertible sheaves ω i , and they follow from the usual duality for coherent cohomology of affinoid dagger spaces recalled in Section 5.7.1.

A
finite flat morphism f : X → Y has a trace map tr f : f O X → O Y .This trace is by definition a global section of f !O Y or equivalently a morphism f O Y → f !O Y .It follows that the trace map provides a natural transformation f ⇒ f ! .
equivariant for a locally finite endomorphism T (i.e.we have locally finite endomorphisms T ∈ End R (M) and T ∈ End R (N ) with f T = T f ) then we have e(T )f = f e(T ) and so f (e(T )M) ⊆ e(T )N and f ((1 − e(T ))M) ⊆ (1 − e(T ))N .Proposition 2.7.Let 0 → M → N → L → 0 be a short exact sequence of R-modules with an equivariant endomorphism T .

Corollary 4. 10 .
There are two maps F : F ω κ un → ω κ un and U p : F ω κ un → ω κ un .Proof.The maps F O IG → O IG and 1 p tr F : F O IG → O IG are Z × p -equivariant.We can tensor with Λ and take the invariants.

(
where the supscript = s means the slope s part for the action of U naive p ) defines a lattice inH i X 0 (p) [0,a[ (X 0 (p), ω k ) =s (an open and bounded submodule).We can indeed represent the cohomology RΓ X 0 (p) [0,a[ (X 0 (p), ω k ) by the Čech complex C • relative to some open covering U , and we can lift the U p -operator to a compact operator Ũp on the complex.Note that C • is a complex of Banach modules.The map from Čech cohomology with respect to U to cohomologyȞi U ,X 0 (p) [0,a[ X 0 (p), ω k,+ −→ H i X 0 (p) [0,a[ X 0 (p), ω k,+has kernel and cokernel of bounded torsion by [Pil20, Lemma 3.2.2].It suffices to prove thatIm Ȟi U ,X 0 (p) [0,a[ X 0 (p), ω k,+ −→ H i X 0 (p) [0,a[ X 0 (p), ω k =sdefines an open and bounded submodule inH i X 0 (p) [0,a[ (X 0 (p), ω k ) =s .The Čech cohomology Ȟi U ,X 0 (p) [0,a[ X 0 (p), ω k,+is obtained by taking the cohomology of an open and bounded sub-complexC +,• ⊂ C • .The image of C +,• in C •,=s under the continuous projection C • → C •,=s(the target is now a complex of finite dimensional vector spaces) is again open and bounded and the claim follows.
p = U naive p of Section 5.4.2 as the correspondence C can [0,a] is the graph of F. Proposition 5.14.The pull back map p 1 : RΓ X < a p (X, ω k ) → RΓ X 0 (p) [0,a[ (X 0 (p), ω k ) induces a quasi-isomorphism on the finite slope parts for F and U naive p Proof.This follows from the existence of a commutative diagram commutative.See [Bey97, Corollary 4.2.11](although the author is working here again in the "dual" setting of wide open spaces).Let F be a locally free sheaf of finite rank over X † which is assumed to be affinoid, smooth of pure dimension d.We letD(F ) = F ∨ ⊗ Ω d X/Q p .Then the residue map induces a perfect pairing ([G-K00, Theorem 4.4]):H 0 (X † , F ) × H d c (X † , D(F )) −→ Q p forwhich both spaces are strong duals of each other.
Note that since p 1 and p 2 are local complete intersection morphisms, p ! 1 O X and p ! 2 O X are invertible sheaves.The restrictions of tr p 1 and tr p 2 to O X 0 (p) (−D 0 (p)) factor through maps tr p 1 : which is étale locally isomorphic to µ p n .We can consider the étale Cartier dual (H can n ) D = Hom(H can n , µ p n ) (away from the boundary, the canonical principal polarization on E defines an isomorphism between (H can n ) D and the étale quotient E[p n ] ét = E[p n ]/H can n ).We can consider the Hodge-Tate map: HT : (H can n ) D −→ ω E /p n which sends a local section s : H can n → µ p n of (H can n ) D to s dx x ∈ ω H can n = ω E /p n , where dx x is the canonical generator of the co-lie algebra of µ p n = Spec Z[x]/(x p n − 1).If s is a local generator of (H can n ) D , then HT(s) is a local generator of ω E /p n , and hence the linearized Hodge-Tate map H⊂E[p]deg H = 1.Moreover, either all the degrees 1 p+1 , we have H 1 = H can 1 .If H ⊂ E[p] satisfies H H can 1 , then deg H = , the isogeny E → E/H is the canonical isogeny and deg E[p]/H = 1−deg H where deg H = 1 − p deg H 1 .On C ét [0,a] we have H H can 1 , and deg and hence E[p]/H 1 is the canonical subgroup.If deg H 1 > p p+1 , we deduce that deg E[p]/H 1 < 1 p+1 , and hence E[p]/H 1 is not the canonical subgroup, but 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],ét its complement.
1) , there is a canonical subgroup of level n, H can n over X v .The isogeny E → E/H can n yields a map ω + There is a natural isomorphism p 2 ω κ un → p 1 ω κ un , and we can define a cohomological correspondence U p : p 1 p 2 ω κ un → ω κ un which specializes in weight k ≥ 1 to U p .Proof.Over C [a,1] , the universal isogeny π : p 1 E → p 2 E induces an isomorphism on canonical subgroups p 1 H can a,1] Lemma 5.16.
Recall that p 1 is an isomorphism, and we therefore get U p : (p can 1 ) (p can 2 ) ω κ un → ω κ un which specializes in weight k ≤ 1 to p −k U naive, can un→ ω κ un which specializes in weight k ≤ 1 to U can p .k → (p can 2 ) ω k given by (π D ) , and its inverse(p can 2 ) ω k → (p can 1 ) ω k is p −k (π ).p = U can p .