An integral model of the perfectoid modular curve

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight $2$ and trace maps.


Introduction
Throughout this document we fix a prime number p, C p the p-adic completion of an algebraic closure of Q p , and {ζ m } m∈N ⊂ C p a compatible system of primitive roots of unity.Given a non-archimedean field K we let O K denote its valuation ring.We let F p be the residue field of O Cp and Zp = W (F p ) ⊂ C p the ring of Witt vectors.Let Z cyc p and Zcyc p denote the p-adic completions of the p-adic cyclotomic extensions of Z p and Zcyc p in C p respectively.Let M ≥ 1 be an integer and Γ(M ) ⊂ GL 2 (Z) the principal congruence subgroup of level M .We fix N ≥ 3 an integer prime to p.For n ≥ 0 we denote by Y (N p n )/ Spec Z p the integral modular curve of level Γ(N p n ) and X(N p n ) its compactification, cf.[KM85].We denote by X(N p n ) the completion of X(N p n ) along its special fiber, and by X (N p n ) its analytic generic fiber seen as an adic space over Spa(Q p , Z p ), cf.[Hub96].
In [Sch15], Scholze constructed the perfectoid modular curve of tame level Γ(N ).He proved that there exists a perfectoid space X (N p ∞ ), unique up to a unique isomorphism, satisfying the tilde limit property see definition 2.4.1 of [SW13] and definition 2.4.2 of [Hub96].
The first result of this paper is the existence of a Katz-Mazur integral model of the perfectoid modular curve.More precisely, we prove the following theorem, see Section 2 for the notion of a perfectoid formal scheme Theorem 0.1.The inverse limit X(N p ∞ ) = lim ← −n X(N p n ) is a formal perfectoid scheme over Spf Z cyc p whose analytic generic fiber is naturally isomorphic to the perfectoid modular curve X (N p ∞ ).
The integral perfectoid modular curve X(N p ∞ ) was previously constructed by Lurie in [Lur20], his method reduces the proof of perfectoidness to the ordinary locus via a mixed characteristic version of Kunz Theorem.The strategy in this paper is more elementary: we use faithfully flat descent to deduce perfectoidness of X(N p ∞ ) from the description of the stalks at the F p -points.Then, we deal with three different kind of points: • The ordinary points where we use the Serre-Tate parameter to explicitly compute the deformation rings, cf.[Kat81,§2].
• The cusps where we have explicit descriptions provided by the Tate curve, cf.
• The supersingular points where even though we do not compute explicitly the stalk, one can proves that the Frobenius map is surjective modulo p.
It worth to mention that the study of the ordinary locus in Lurie's approach and the one presented in this document are very related, see Proposition 2.2 of [Lur20] and Proposition 1.2 down below.
As an application of the integral model we can prove vanishing results for the coherent cohomology of the perfectoid modular curve.Let E sm /X(N ) be the semiabelian scheme extending the universal elliptic curve over Y (N ), cf.[DR73].Let e : X(N ) → E sm be the unit section and ω E = e * Ω 1 E sm /X(N ) the sheaf of invariant differentials.For n ≥ 0 we denote by ω E,n the pullback of ω E to X(N p n ), and D n ⊂ X(N p n ) the reduced cusp divisor.Let k ∈ Z, we denote ω k E,n = ω ⊗k E,n and ω k E,n,cusp = ω k E,n (−D n ).Let ω k E,∞ be the pullback of ω k E to X(N p ∞ ), and ω k E,∞,cusp the p-adic completion of the direct limit of the cuspidal modular sheaves ω k E,n,cusp .In the following we consider almost mathematics with respect to the maximal ideal of Z cyc p .
2. For k ∈ Z and i, s ≥ 0, we have H i (X(N p ∞ ), F )/p s = H i (X(N p ∞ ), F /p s ) and H i (X(N p ∞ ), F ) = lim ← −s H i (X(N p ∞ ), F /p s ).
3. The cohomology groups H i (X(N p ∞ ), F ) are torsion free.
Next, we use Serre duality and Pontryagin duality to construct a local duality theorem for the modular curves at finite level.In the limit one obtains the following theorem Theorem 0.3.Let X n be the connected component of X(N p n ) Zp defined as the locus where the Weil pairing of the universal basis of where the transition maps in the RHS are given by normalized traces, and F ∨ n is the dual sheaf of F n .
Finally, we specialize to the case F = O X∞ where the completed cohomology appears.Let X n be a connected component of X(N p n ) Zp as in the previous theorem.
Let i ≥ 0 and let H i = lim ← −s lim − →n H i et (X n,Cp , Z/p s Z) be the completed i-th cohomology group, where X n,Cp = X n × Spec Zp[ζp n ] Spec C p .Note that this is a slightly different version of Emerton's completed cohomology [Eme06], where one considers the étale cohomology with compact supports of Y n,Cp ⊂ X n,Cp .Nevertheless, both cohomologies are related via the open and closed immersions Y n ⊂ X n ⊃ D n .Following the same ideas of [Sch15, §4.2] one can show that H i ⊗ Zp O Cp is almost equal to H i an (X ∞,Cp , O + X∞ ), in particular it vanishes for i ≥ 2 and H 0 = Z p .Using the theorem above we obtain the following result Theorem 0.4.There is a GL The outline of the paper is the following.In Section 1 we recall the construction of the integral modular curves at finite level; they are defined as the moduli space parametrizing elliptic curves endowed with a Drinfeld basis of the torsion subgroups, we will follow [KM85].Then, we study the deformation rings of the modular curves at F p -points.For ordinary points we use the Serre-Tate parameter to describe the deformation ring at level Γ(N p n ).We show that it represents the moduli problem parametrizing deformations of the p-divisible group E[p ∞ ], and a split of the connected-étale short exact sequence For cusps we refer to the explicit computations of [KM85, §8 and 10].Finally, in the case of a supersingular point we prove that any element of the local deformation ring at level Γ(N p n ) admits a p-th root modulo p at level Γ(N p n+1 ).
In Section 2 we introduce the notion of a perfectoid formal scheme.We prove Theorem 0.1 reducing to the formal deformation rings at F p -points via faithfully flat descent.We will say some words regarding Lurie's construction of X(N p ∞ ).It is worth to mention that the tame level Γ(N ) is taken only for a more clean exposition, by a result of Kedlaya-Liu about quotients of perfectoid spaces by finite group actions (Theorem 3.3.26 of [KL19]), there are integral models of any tame level.
In Section 3, we use Serre and Pontryagin duality to define a local duality pairing for the coherent cohomology of vector bundles over an lci projective curve over a finite extension of Z p .
In Section 4, we compute the dualizing complexes of the modular curves at finite level.We prove the cohomological vanishing of Theorem 0.2 and its comparison with the cohomology of the perfectoid modular curve.We prove the duality theorem at infinite level, Theorem 0.3, and specialize to F = O X∞ to obtain Theorem 0.4.
Acknowledgments.The construction of the integral perfectoid modular curve initiated as a problem of a mémoire of M2 in 2019.I should like to convey my special gratitude to Vincent Pilloni for encouraging me to continue with the study of the coherent cohomology at perfectoid level, and its application to the completed cohomology.I would like to thank George Boxer and Joaquin Rodrigues Jacinto for all the fruitful discussions of the subject.I wish to express special thanks to the anonymous referee for the careful proofreading of this paper, and for the numerous suggestions and corrections that have notably improved the presentation of this text.Particularly, for the remark that Theorem 0.3 holds for all the sheaves involved and not only for the structural sheaf.This work has been done while the author was a PhD student at the ENS de Lyon.

A brief introduction to the Katz-Mazur integral modular curves
Let N ≥ 3 be an integer prime to p and n ∈ N. Let Γ(N p n ) ⊂ GL 2 (Z) be the principal congruence subgroup of level N p n .

Drinfeld bases
We recall the definition of a Drinfeld basis for the M -torsion of an elliptic curve Definition 1.1.Let M be a positive integer, S a scheme and E an elliptic curve over such that the following equality of effective divisors holds We also write (P, Q) = (ψ(1, 0), ψ(0, 1)) for the Drinfeld basis ψ.
Remark 1.1.The left-hand-side of (1.1) is an effective divisor of E/S being a finite flat group scheme over S. The right-hand-side is a sum of effective divisors given by the sections Let M ≥ 3. From Theorem 5.1.1 and Scholie 4.7.0 of [KM85], the moduli problem parametrizing elliptic curves E/S and Drinfeld bases (P, Q) of E[M ] is representable by an affine and regular curve over Z.We denote this curve by Y (M ) and call it the (affine) integral modular curve of level Γ(M ).By an abuse of notation, we will write Y (M ) for its scalar extension to Z p .
The j-invariant is a finite flat morphism of Z p -schemes j : Zp in Y (M ) via the j-invariant.The cusps or the boundary divisor D is the closed reduced subscheme of X(M ) defined by 1 j = 0.The curve X(M ) is projective over Z p and a regular scheme.We refer to X(M ) and Y (M ) simply as the modular curves of level Γ(M ).
Let E univ /Y (M ) be the universal elliptic curve and (P univ,M , Q univ,M ) the universal Drinfeld basis of E univ [M ].Let Φ M (X) be the M -th cyclotomic polynomial, and The map e M extends uniquely to a map e M : X(M ) → Spec Z p [µ × M ] by normalization.In addition, e M is geometrically reduced, and has geometrically connected fibers.
Taking N as in the beginning of the section, and n ∈ N varying, we construct the commutative diagram the upper horizontal arrows being induced by the map (P univ,N p n+1 , Q univ,N p n+1 ) → (pP univ,N p n+1 , pQ univ,N p n+1 ) = (P univ,N p n , Q univ,N p n ), and the lower horizontal arrows by the natural inclusions.In fact, the commutativity of the diagram is a consequence of the compatibility of the Weil pairing with multiplication by p e N p n+1 (pP univ,N p n+1 , pQ univ,N p n+1 ) = e N p n (P univ,N p n , Q univ,N p n ) p cf. Theorems 5.5.7 and 5.6.3 of [KM85].

Deformation rings at F p -points
Let k = F p be an algebraic closure of F p .Let {ζ N p n } n∈N be a fixed sequence of compatible primitive N p n -th roots of unity, set ζ p n = ζ N N p n .Let Zp = W (k) denote the ring of integers of the p-adic completion of the maximal unramified extension of Q p .In the next paragraphs we will study the deformation rings of the modular curve at the closed points X(N p n )(k).We let X(N p n ) Zp denote the compactified modular curve over Zp of level Γ(N p n ).Proposition 8.6.7 of [KM85] implies that X(N p n ) Zp = X(N p n ) × Spec Zp Spec Zp .

There is an isomorphism Zp
given by fixing a primitive N -th root of unity in Zp .Let X(N p n ) • Zp be the connected component of the modular curve which corresponds to the root ζ N .In other words, X(N p n ) • Zp is the locus of X(N p n ) Zp where e N p n (P univ,N p n , Q univ,N p n ) = ζ N p n .We denote Finally, given an elliptic curve E/S, we denote by E the completion of E along the identity section.

The ordinary points
Let Art k be the category of local artinian rings with residue field k, whose morphisms are the local ring homomorphisms compatible with the reduction to k.Any object in Art k admits an unique algebra structure over Zp .Let Zp [ζ p n ]-Art k denote the subcategory of Art k of objects endowed with an algebra structure of Zp [ζ p n ] compatible with the reduction to k.Following [Kat81], we use the Serre-Tate parameter to describe the deformation rings at ordinary k-points of X(N p n ) Zp .
Let E 0 be an ordinary elliptic curve over k and R an object in Art k .A deformation of E 0 to R is a pair (E, ι) consisting of an elliptic curve E/R and an isomorphism ι : E ⊗ R k → E 0 .We define the deformation functor Ell E0 : Art k → Sets by the rule Then Ell E0 sends an artinian ring R to the set of deformations of E 0 to R modulo isomorphism.
Let Q be a generator of the physical Tate module . Let G m be the multiplicative group over Zp and G m its formal completion along the identity.We have the following pro-representability theorem Theorem 1.1.[Kat81, Theo.2.1] 1.The Functor Ell E0 is pro-representable by the formal scheme The isomorphism is given by the Serre-Tate parameter q, which sends a deformation E/R of E 0 to a bilinear form By evaluating at the fixed generator Q of T p E 0 (k), we obtain the more explicit description where X = q(E univ / Ell E0 ; Q, Q) − 1.
2. Let E 0 and E ′ 0 be ordinary elliptic curves over k, let π 0 : E 0 −→ E ′ 0 be a homomorphism and π t 0 : E ′ 0 −→ E 0 its dual.Let E and E ′ be liftings of E 0 and E ′ 0 to R respectively.A necessary and sufficient condition for π 0 to lift to a homomorphism π : We deduce the following proposition describing the ordinary deformation rings of finite level:

Zp
(k) be an ordinary point, say given by a triple (E 0 , P 0 , Q 0 ), and write (P Then there is an isomorphism such that: ii. the variable 1 + X is equal to the Serre-Tate parameter q(E univ /A x ; Q, Q); iii. the variable 1+T is equal to the Serre-Tate parameter q(E ′ univ /A x , (π t ) −1 (Q), (π t ) −1 (Q)) of the universal deformation π : E univ → E ′ univ of the étale isogeny π 0 : Proof.The group scheme E 0 [N ] is finite étale over k, which implies that a deformation of (E 0 , P 0 , Q 0 ) is equivalent to a deformation of (E 0 , P modulo the maximal ideal.By Cartier duality, there is a unique We also have a natural equivalence We obtain the isomorphism where ] is a union of Igusa curves with intersections at the supersingular points [KM85, Theo.13.10.3].The Igusa curves are smooth over F p [KM85, Theo.12.6.1],which implies that the deformation ring of X(N p n ) at x is isomorphic to a power series ring Zp [[T n ]] (cf.discussion after Remark 3.4.4 of [Wei13]).The content of the previous proposition is the explicit relation between the variables T n in the modular tower, see also Proposition 2.2 of [Lur20].
We consider the ring Z p [[q]] as the completed stalk of P 1 Zp at infinity.The Tate curve provides a description of the modular curve locally around the cusps, for that reason one can actually compute the formal deformation rings by means of this object, see [KM85] and [DR73].In fact, let Cusps[Γ(N p n )] be the completion of the modular curve X(N p n ) Zp along the cusps.From the theory developed in [KM85, Ch. 8 and 10], more precisely Theorems 8.11.10 and 10.9.1, we deduce the following proposition: on each respective connected component.

The supersingular points
Let (x n ∈ X(N p n ) Zp ) n∈N be a sequence of compatible supersingular points and E 0 the elliptic curve defined over x n .We denote by A xn the deformation ring of X(N p n ) Zp at x n .Let E univ /A xn be the universal elliptic curve and (P We fix a formal parameter T of E univ .Since x n is supersingular, any p-power torsion point belongs to E univ .We will use the following lemma as departure point: By the Serre-Tate Theorem [Kat81, Theo.1.2.1], and the general moduli theory of 1-dimensional formal groups over k [LT66], the deformation ring of X(N ) Zp at a supersingular point is isomorphic to Zp [[X]].Moreover, the p-multiplication modulo p can be written as 0 → E 0 .Without loss of generality we assume that V has the form Using the Weierstrass Preparation Theorem we factorize V (T ) as Proposition 1.4.The parameter X is a p-power in A x1 /p.Moreover, the generators T (P univ ) of the maximal ideal of A xn are p-powers in A xn+1 /p.Proof.The second claim follows from the first and the equality [p](T ) ≡ V (T p ) mod p.Consider n = 1 and write P = P (1) 0 → E 0 denote the Frobenius and Verschiebung homomorphisms respectively.Using the action of GL 2 (Z/pZ), we can assume that P and F (Q) are generators of ker F and ker V respectively (cf.Theorem 5.5.2 of [KM85]).We have the equality of divisors on (1.4) The choice of the formal parameter T gives a formal parameter of ] we are done.
Corollary 1.1.The Frobenius ϕ : lim − →n Proof.By induction on the graded pieces of the filtration defined by the ideal (T (P n ), T (Q n )), one shows that A xn /p is in the image of the Frobenius restricted to A xn+1 /p.
Remark 1.3.The completed local ring at a geometric supersingular point x of X(N p n ) is difficult to describe.For example, its reduction modulo p is the quotient of the power series ring k[[X, Y ]] by some explicit principal ideal which is written in terms of the formal group law of E at x [KM85, Theo.13.8.4].Weinstein gives in [Wei16] an explicit description of the deformation ring at a supersingular point of the modular curve at level Γ(N p ∞ ).In fact, Weinstein finds an explicit description of the deformation ring at infinite level of the Lubin-Tate space parametrizing 1-dimensional formal O K -modules of arbitrary height.In particular, he proves that the m x0 -adic completion of the direct limit lim − →n A xn is a perfectoid ring.The Corollary 1.1 says that the p-adic completion of lim − →n A xn is perfectoid, which is a slightly stronger result.

Construction of the perfectoid integral model Perfectoid Formal spaces
In this section we introduce a notion of perfectoid formal scheme which is already considered in [BMS18, Lemma 3.10], though not explicitly defined.We start with the affine pieces Definition 2.1.An integral perfectoid ring is a topological ring R containing a non zero divisor π such that p ∈ π p R, satisfying the following conditions: i. the ring R is endowed with the π-adic topology.Moreover, it is separated and complete.
We call π satisfying the previous conditions a pseudo-uniformizer of R.
Remark 2.1.The previous definition of integral perfectoid rings is well suited for padic completions of formal schemes.We do not consider the case where the underlying topology is not generated by a non-zero divisor, for example, the ring As is pointed out in Remark 3.8 of [BMS18], the notion of being integral perfectoid does not depend on the underlying topology, however to construct a formal scheme it is necessary to fix one.
Let R be an integral perfectoid ring with pseudo-uniformizer π, we attach to R the formal scheme Spf R defined as the π-adic completion of Spec R. We say that Spf R is a perfectoid formal affine scheme.The following lemma says that the standard open subschemes of Spf R are perfectoid Proof.Let n, k ≥ 0, as π is not a zero divisor we have a short exact sequence Localizing at f and taking inverse limits on n we obtain Then R f −1 is π-adically complete and π is not a zero-divisor.On the other hand, localizing at f the Frobenius map ϕ : R/π which proves that R f −1 is an integral perfectoid ring.
Definition 2.2.A perfectoid formal scheme X is a formal scheme which admits an affine cover X = i U i by perfectoid formal affine schemes.
Let F be equal to Q p or F p ((t)), O F denote the ring of integers of F and ̟ be a uniformizer of O F .Let Int-Perf OF be the category of perfectoid formal schemes over O F whose structural morphism is adic, i.e. the category of formal perfectoid schemes X/ Spf O F such that ̟O X is an ideal of definition of O X .Let Perf F be the category of perfectoid spaces over Spa(F, O F ).
Proposition 2.1.Let R be an integral perfectoid ring and π a pseudo-uniformizer.The ring R[ 1 π ] is a perfectoid ring in the sense of Fontaine [Fon13].Furthermore, there is a unique "generic fiber" functor Moreover, given X a perfectoid formal scheme over O F , its generic fiber is universal for morphisms from perfectoid spaces to X. Namely, if Y is a perfectoid space and ) is a morphism of locally and topologically ringed spaces, then there is a unique map Y → X µ making the following diagram commutative Remark 2.2.The universal property of the functor (−) µ is Huber's characterization of the generic fiber of formal schemes in the case of perfectoid spaces, see [Hub94, Prop.4.1].
Proof.The first statement is Lemma 3.21 of [BMS18].For the construction of the functor, let X be a perfectoid formal scheme over O F .One can define X η to be the glueing of the affinoid spaces Spa(R[ 1 ̟ ], , R + ) for Spf R ⊂ X an open perfectoid formal afine subscheme, this is well defined after Lemma 2.1.
We prove the universal property of the generic fiber functor.Let Y ∈ Perf K and let f : (Y, O + Y ) → (X, O X ) be a morphism of locally and topologically ringed spaces.First, if Y = Spa(S, S + ) is affinoid perfectoid and X = Spf R is perfectoid formal affine, f is determined by the global sections map f * : R → S + .Then, there exists a unique map of affinoid perfectoid rings f * η : (R[ 1 ̟ ], R + ) → (S, S + ) extending f * .By glueing morphisms from affinoid open subsets for a general Y, one gets that ) satisfies the universal property.For an arbitrary X, one can glue the generic fibers of the open perfectoid formal affine subschemes of X.
We end this subsection with a theorem which reduces the proof of the perfectoidness of the integral modular curve at any tame level to the level Γ(N p ∞ ).

The main construction
Let X (N p ∞ ) denote Scholze's perfectoid modular curve [Sch15].Let Z cyc p be the padic completion of the p-adic cyclotomic integers lim − →n Z p [µ p n ].Let X(N p n ) be the completion of X(N p n ) along its special fiber.We have the following theorem Theorem 2.2.The inverse limit X(N p ∞ ) := lim ← −n X(N p n ) is a p-adic perfectoid formal scheme, it admits a structural map to Spf Z cyc p [µ N ], and its generic fiber is naturally isomorphic to the perfectoid modular curve X (N p ∞ ).Furthermore, let n ≥ 0, let Spec R ⊂ X(N p n ) be an affine open subscheme, Spf R its p-adic completion and Remark 2.3.The previous result gives a different proof of Scholze's theorem that the generic fiber X (N p ∞ ) is a perfectoid space by more elementary means.
Proof.The maps between the (formal) modular curves are finite and flat.Then • .Suppose that the claim holds, it is left to show that X(N p ∞ ) η is the perfectoid modular curve X (N p ∞ ).There are natural maps of locally and topologically ringed spaces (X (N where we use the notion of tilde limit [SW13, Def.2.4.1].Then, by p-adically completing the inverse limit of the tower, we obtain a map of locally and topologically ringed spaces This provides a map f : and the tilde limit is unique in the category of perfectoid spaces [SW13, Prop.2.4.5], the map f is actually an isomorphism. Proof of the Claim.First, by Lemma A.2.2.3 of [Heu19] the ring ( R n [ 1 p ]) • is the integral closure of R n in its generic fiber.By Lemma 5.1.2 of [Bha17] and the fact that R n is a regular ring one gets that Moreover, R ∞ is integrally closed in its generic fiber, and by Lemma 5.1.2 of loc.cit.
In particular, there exists π ∈ R ∞ such that π p = pa with a ∈ Z cyc,× p .To prove that R ∞ is integral perfectoid we need to show that the absolute Frobenius map ϕ : R ∞ /π −→ R ∞ /p is an isomorphism.The strategy is to prove this fact for the completed local rings of the stalks of Spec R ∞ /p and use faithfully flat descent.
Injectivity is easy, it follows from the fact that R ∞ is integrally closed in R ∞ [1/p].To show that ϕ is surjective, it is enough to prove that the absolute Frobenius is surjective after a profinite étale base change.Indeed, the relative Frobenius is an isomorphism for profinite étale base changes.Let S = R ⊗ Zp Zp and let S = R ⊗ Zp Zp be the p-adic completion of S. We use similar notation for S n = R n ⊗ Zp Zp , S n , S ∞ and S ∞ .We have to show that the absolute Frobenius which is an inverse limit of F p -points of Spf S n .Write x n0 simply by x 0 .Then, it is enough to show that ϕ is surjective after taking the stalk at x. Let S n,xn be the localization of S n at the prime x n and S ∞,x = lim − →n S n,xn .Let S n,xn be the completion of S n,xn along its maximal ideal.Recall that the ring S n is finite flat over S, this implies that The scheme X(N p n ) is of finite type over Z p , in particular every point has a closed point as specialization.Thus, by faithfully flat descent, we are reduced to prove that for every is surjective (even an isomorphism).We have the following commutative diagram The ring R n is of finite type over Z p , so that the absolute Frobenius ϕ : R n /π → R n /p is finite.This implies that S n,xn /p is a finite S x0 -module via the module structure induced by the Frobenius.Then, the following composition is an isomorphism where m S is the maximal ideal of S x0 .Thus, we are reduced to prove that the absolute Frobenius ϕ : lim − →n S n,xn /π → lim − →n S n,xn /p is surjective.Finally, we deal with the cusps, the supersingular and the ordinary points separately; we use the descriptions of Section 1: From the proof of Proposition 1.2, one checks that the inclusion S n,xn → S n+1,xn+1 is given by X n = (1 + X n+1 ) p − 1.Then, one obtains the surjectivity of Frobenius when reducing modulo p.
• Finally, if we are dealing with a cusp x, the ring S n,xn is isomorphic to Zp [ζ p n ][[q1/N p n ]] and S n,xn → S n+1,xn+1 is the natural inclusion by Proposition 1.3.The surjectivity of ϕ is clear.

Relation with Lurie's stack
In this subsection we make more explicitly the relation between Lurie's construction of X(N p ∞ ) and the one presented in this document.The key result is the following theorem Theorem 2.3 ( [Lur20, Theo.1.9]).Let π ∈ Z p [µ p 2 ] be a pseudo-uniformizer such that π p = ap where a is a unit.For n ≥ 3 there exists a unique morphism θ : where ϕ is the absolute Frobenius.
This theorem can be deduced from the local computations made in Section 1. Indeed, let x n ∈ X(N p n )(F p ) be a F p -point and x n−1 ∈ X(N p n−1 )(F p ) its image.We have proven that there exists a unique map of the deformation rings at the points x n−1 and x n θ * : O X(N p n−1 ),xn−1 /p → O X(N p n ),xn /π making the following diagram commutative This corresponds to Propositions 1.2, 1.3 and 1.4 for x n ordinary, a cusp and a supersingular point respectively.Then, one constructs θ using faithfully flat descent from the completed local rings to the localized local rings at x n , and glueing using the uniqueness of θ * .

Cohomology and local duality for curves over O K
Let K be a finite extension of Q p and O K its valuation ring.In this section we recall the Grothendieck-Serre duality theorem for local complete intersection (lci) projective curves over O K , we will follow [Har66].Then, we use Pontryagin duality to define a local duality paring of coherent cohomologies.
Let X be a locally noetherian scheme and D(X) the derived category of O Xmodules.We use subscripts c, qc on D(X) for the derived category of O X -modules with coherent and quasi-coherent cohomology, the subscript f T d refers to the subcategory of complexes with finite Tor dimension.We use superscripts +, −, b for the derived category of bounded below, bounded above and bounded complexes respectively.For instance, D b c (X) f T d is the derived category of bounded complexes of O X -modules of finite Tor dimension and coherent cohomology.If X = Spec A is affine, we set D(A) := D qc (X), the derived category of A-modules.
for F ∈ D − qc (X) and G ∈ D + qc (Y ).Moreover, the formation of the exceptional inverse image is functorial.More precisely, given a composition X Y Z f g with f, g and gf projectively embeddable, there is a natural isomorphism (gf ) ! ∼ = f !g ! .This functor commutes with flat base change.Namely, let u : Y ′ → Y be a flat morphism, f ′ : X ′ → Y ′ the base change of X to Y ′ and v : X ′ → X the projection.Then there is a natural isomorphism of functors v * f != f ′ !u * .
Proof.We refer to [Har66, Theo.III.8.7] for the existence of f !, its functoriality and compatibility with flat base change.See Theorems III.10.5 and III 11.1 of loc.cit.for the existence of Tr and the adjunction θ respectively.
Example 3.1.Let f : X → Y be a morphism of finite type of noetherian schemes of finite Krull dimension.
1. We can define the functor f ! for finite morphisms as The duality theorem in this case is equivalent to the (derived) ⊗-Hom adjunction, see [Har66, §III.6].

Let f be smooth of relative dimension n, then one has f
Lemma 3.1.Let f : X → Y be an lci morphism of relative dimension n between locally noetherian schemes of finite Krull dimension.Then Proof.Working locally on Y and X, we may assume that f factors as X S Y ι g , where g is a smooth morphism of relative dimension m, and ι is a regular closed imersion of codimension m − n defined by an ideal Let K(f ) be the Koszul complex of the regular sequence which is an invertible sheaf of O X -modules as required.
Remark 3.1.Let f : X → Y be a regular closed immersion of codimension n defined by the ideal I From the proof of Lemma 3.1 one can deduce that The compatibility of f ! with tensor products allows us to compute f !F in terms of f * F and f !O Y : Proposition 3.1 ([Har66, Prop.III.8.8]).Let f : X → Y be an embeddable morphism of locally noetherian schemes of finite Krull dimension.Then there are functorial isomorphisms dualizing sheaf of f .We now prove the local duality theorem for vector bundles over lci projective curves: Proposition 3.2.Let f : X → Spec O K be an lci projective curve, and let ω • X/OK be the dualizing sheaf of f , i.e. the invertible sheaf such that ω Let F be a locally free O X -module of finite rank, then: 1. R f * F is representable by a perfect complex of lenght [0, 1];

we have a perfect pairing
given by the composition of the cup product and the trace Tr : As F is a vector bundle and f is projective of relative dimension 1, the cohomology groups R i f * F are finitely generated over O K and concentrated in degrees 0 and 1.Then, R f * F is quasi-isomorphic to a complex 0 M 2 finite free O K -modules.Moreover, the complex 0 which translates in the desired statement.
Remark 3.2.The previous proposition relates two notions of duality.Namely, Serre and Pontryagin duality.We can deduce the following facts: ) is free of rank r.In that case, the module H 0 (X, F ) is free and H 0 (X, F )/p n → H 0 (X, F /p n ) is an isomorphism for all n ∈ N. Furthermore, Serre duality provides a perfect pairing ) is free (resp.co-free) for any finite locally free O X -module.
3. In the notation of the previous proof, Pontryagin duality implies which is equivalent to a perfect pairing

Cohomology of modular sheaves
Let N ≥ 3 be an integer prime to p. Let X(N p n ) be the modular curve over Z p of level Γ(N p n ).Let Zp = W (F p ) and let X(N p n ) Zp be the extension of scalars of X(N p n ) to Zp .We denote by Let E sm /X be the semi-abelian scheme over X extending the universal elliptic curve to the cusps, cf [DR73].Let e : X → E sm be the unit section and ω E := e * Ω 1 E sm /X the modular sheaf, i.e., the sheaf of invariant differentials of E sm over X.For k ∈ Z we define ω k E = ω ⊗k E the sheaf of modular forms of weight k, we denote by ω k E,n the pullback of ω k E to X n .Let D n ⊂ X n be the (reduced) cusp divisor and ω k E,n,cusp := ω k E,n (−D n ) the sheaf of cusp forms of weight k over X n .By an abuse of notation we will also write D n for the pullback p * n+1 D n to X n+1 , by Proposition 1.3 we have that D n = pD n+1 .
Finally, we let X n be the completion of X n along its special fiber and X ∞ = lim ← −n X n the integral perfectoid modular curve, see Theorem 2.2.Let X n be the analytic generic fiber of X n and X ∞ ∼ lim − →n X n the Scholze's perfectoid modular curve.

Dualizing sheaves of modular curves
Consider the tower of modular curves On the other hand, the Kodaira-Spencer map provides an isomorphism KS : and by an abuse of notation p n : X n → X ′ n−1 the induced map.Let π ′ n−1 : X ′ n−1 → O n be the structural map and pr 1 : X ′ n−1 → X n−1 the first projection.We also write ω k E,n−1 for the pullback of ω k E,n−1 to X ′ n−1 .Note that the compatibility of the exceptional inverse image functor with flat base change (Theorem 3.1) implies that π Proposition 4.1.There exists a natural isomorphism ξ n : Proof.By Proposition 3.1 we have an isomorphism The map p n is finite flat, then It suffices to consider the ordinary points and the cusps, indeed, the supersingular points are of codimension 2 in X n .Let x ∈ X ′ n−1 (F p ) be an ordinary point.We have a cartesian square By Proposition 1.2 we have isomorphisms On the other hand, let x ∈ X ′ n−1 (F p ) be a cusp.We have a cartesian square (4.2) and by Proposition 1.3 isomorphisms Taking the different ideal we obtain the equality The previous computations show that the trace of Then, from (4.1) we have an isomorphism be an an ordinary point or a cusp.Let Tr n : Proof.Localizing at x we find where q 1/p n−1 is invertible if x is ordinary, or a generator of D n−1 if it is a cusp.The explicit descriptions found in the previous proposition show that Tr n is surjective on each direct summand.Finally, looking at an ordinary point x, it is clear that there are p different points x n in the fiber of x, this implies Tr n (1) = p.
Proof.Part (2) is Proposition 3.2.Part (1) follows from Remark 3.2 i) and the previous proposition.Indeed, if k < 0, the vanishing of H 0 (X n , F ⊗ K n /O n ) implies that H 1 (X n , F ) is torsion free.As the cohomology group is of finite type over O n , it is a finite free O n -module.The other cases are proved in a similar way.
Next, we will prove some cohomological vanishing results for the modular sheaves ω k E and ω k E,cusp at infinite level.Particularly, we will show that the cohomology of ω k E over X ∞ is concentrated in degree 0 if k > 0. The case k > 2 will follow from Proposition 4.2, one can also argue directly for k = 2. What is remarkable is the vanishing for k = 1, in which case we use the perfectoid nature of X ∞ . Let , and the pullback of ω k E,n,cusp injects into ω k E,m,cusp .We define ω k E,∞,cusp as the p-adic completion of the direct limit lim − →n is no longer a coherent sheaf over X ∞ ; its reduction modulo p is a direct limit of line bundles which is not stationary at the cusps.One way to think about an element in ω k E,∞,cusp is via q-expansions: Then, an element f ∈ ω k E,∞,x can be written as a power series satisfying certain convergence conditions.The element f belongs to the localization at x of ω k E,∞,cusp if and only if a 0 = 0.For a detailed treatment of the cusps at perfectoid level we refer to [Heu20], particularly Theorem 3.17.
2. For all m, i ≥ 0 and k ∈ Z, we have 3. The sheaves ω k E,∞ and ω k E,∞,cusp have cohomology concentrated in degree 0 for k > 0. Similarly, the sheaves ω k E,∞ and ω k E,∞,cusp have cohomology concentrated in degree 1 for k < 0.
or ω k E,n,cusp respectively.We show (1) assuming part (2).By evaluating F at formal affine perfectoids of X ∞ arising from finite level, one can use Lemma 3.18 of [Sch13] to deduce that F = R lim ← −s F /p s : the case F = ω k E,∞ is clear as it is a line bundle.Otherwise, we know that F /p s = lim − →n F n /p s = lim − →n (F n /p s ⊗ Xn O X∞ ) is a direct limit of O X∞ /p s -line bundles, so that it is a quasi-coherent sheaf over X ∞ , and the system {F /p s } s∈N satisfies the Mittag-Leffler condition on formal affine perfectoids.One obtains the quasi-isomorphism whose cohomology translates into short exact sequences But part (2) implies that H i (X ∞ , F /p s ) = lim − →n H i (X n , F n /p s ) for all s ∈ N. As X n is a curve over O n and F n /p s is supported in its special fiber, we know that H i (X n , F n /p s ) = 0 for i ≥ 2 and that the inverse system {H 1 (X n , F n /p s )} s∈N satisfies the ML condition.This implies that H i (X ∞ , F /p s ) = 0 for i ≥ 2 and that the ML condition holds for {H 1 (X ∞ , F /p s )} s∈N .From (4.3) one obtains that H i (X ∞ , F ) = 0 for i ≥ 2.
We prove part (2).Let U = {U i } i∈I be a finite affine cover of X, let U n (resp.U ∞ ) be its pullback to X n (resp.X ∞ ).As F /p s = lim − →n F n /p s is a quasi-coherent O X∞ /p s -module, and the (formal) schemes X ∞ and X n are separated, we can use the Čech complex of U n (resp.U ∞ ) to compute the cohomology groups.By definition we have follows as filtered direct limits are exact.The vanishing results of Proposition 4.2 imply (3) for k < 0 and k > 2. Let k = 1, 2 and p 1/p ∈ O cyc be such that |p 1/p | = |p| 1/p .As X ∞ is integral perfectoid, the Frobenius F : X ∞ /p → X ∞ /p 1/p is an isomorphism.Moreover, ).This proves (3) for k = 1, 2. Finally, part (4) follows from part (2), Proposition 4.2 (3), and the fact that ) for all i, s ≥ 0. In particular, the cohomology groups H i (X ∞ , F ) are p-adically complete and separated.Moreover, they are all torsion free.Proof.The case k = 0 follows since the cohomology complexes R Γ(X ∞ , F /p s ) are concentrated in only one degree, and R Γ(X ∞ , F ) = R lim ← −s R Γ(X ∞ , F /p s ).The case k = 0 follows by part (4) of the previous theorem.Namely, H 0 (X ∞ , O X∞ (−D ∞ )/p s ) = 0 and H 0 (X ∞ , O X∞ /p s ) = O cyc /p s for all s ≥ 0. Hence, the inverse system of H 0cohomology groups satisfy the Mittag-Leffler condition, and the R 1 lim ← − appearing in the derived inverse limit disappears for the H 1 -cohomology.
As an application of the previous vanishing theorem, we obtain vanishing results for the coherent cohomology of the perfectoid modular curve.Let (X ∞ , O + X∞ ) → (X ∞ , O OX ∞ ) be the natural map of locally and topologically ringed spaces provided by the generic fiber functor, see Proposition 2.1 and Theorem 2.2.We define , where the completed tensor product is with respect to the p-adic topology.As usual, we denote O + X∞ (−D ∞ ) = ω 0,+ E,cusp .In the following we consider almost mathematics with respect to the maximal ideal of O cyc .

H
Proof.We first prove the corollary for F = ω k E,∞ .Let F + η denote the pullback of F to (X ∞ , O + X∞ ).Let U = {U i } i∈I be an open cover of X ∞ given by formal affine perfectoids arising from finite level such that ω E,∞ | Ui is trivial.By Theorem 2.2, the generic fiber U i,η of U i is an open affinoid perfectoid subspace of X ∞ .Let U η := {U i,η } i∈I , note that U η is a covering of X ∞ and that the restriction of F + η to U i,η is trivial.By Scholze's Almost Acyclicity Theorem for affinoid perfectoids, F + η | Ui,η is almost acyclic for all i ∈ I.The Čech-to-derived functor spectral sequence gives us an almost quasi-isomorphism On the other hand, by the proof of Theorem 4.1 there is a quasi-isomorphism But by definition of F + η , and the fact that we argue as follows: note that we can write To apply the same argument as before we only need to show that O + X∞ (−D ∞ ) is almost acyclic over affinoid perfectoids of X ∞ .Let V (D ∞ ) ⊂ X ∞ be the perfectoid closed subspace defined by the cusps.Note that O X∞ (−D ∞ ) is the ideal sheaf of V (D ∞ ), see the proof of [Sch15, Theo.IV.2.1] or the explicit description of the completed stalks at the cusps of the integral perfectoid modular curve.Then, we have an almost short exact sequence for all s ∈ N 0 As the intersection of an affinoid perfectoid of X ∞ with V (D ∞ ) is affinoid perfectoid, and the second map of (4.5) is surjective when evaluating at affinoid perfectoids of X ∞ , Scholze's almost acyclicity implies that O + X∞ (−D ∞ )/p s is almost acyclic in affinoid perfectoids.Taking inverse limits and noticing that {O + X∞ (−D ∞ )/p s } s∈N satisfies the ML condition in affinoid perfectoids, we get that O + X∞ (−D ∞ ) is almost acyclic in affinoid perfectoids of X ∞ .The corollary follows from the vanishing results at the level of formal schemes.
Remark 4.2.As it was mentioned to me by Vincent Pilloni, the cohomological vanishing of the modular sheaves at infinite level provides many different exact sequences involving modular forms and the completed cohomology of the modular tower (to be defined in the next subsection).Namely, the primitive comparison theorem permits to compute the C p -scalar extension of the completed cohomology as H 1 an (X ∞,Cp , O X∞ ).On the other hand, the Hodge-Tate exact sequence via the universal trivialization of T p E. Then, taking the cohomology of (4.6) one obtains an exact sequence Another is example is given by tensoring (4.6) with ω E and taking cohomology.One finds It may be interesting a more careful study of these exact sequences.

Duality at infinite level
or ω k E,ncusp respectively.Let C be a non archimedean field extension of K cyc and O C its valuation ring.Let X ∞,C be the extension of scalars of the integral modular curve to O C .Corollary 4.2 says that the cohomology groups H i (X ∞ , F ) are torsion free, p-adically complete and separated.In particular, we can endow H i (X ∞,C , F )[ 1 p ] with an structure of C-Banach space with unit ball H i (X ∞,C , F ).The local duality theorem extends to infinite level in the following way Theorem 4.2.Let F and F n be as above, and let Next, we recall the GL 2 (Q p )-action in both sides of (4.7).Without loss of generality we take C for the conjugation x → gxg −1 .We denote by X(N p n ) c(g) be the modular curve of level Γ(N ) ∩ Γ(p n ) ∩ gΓ(p n )g −1 , let X n,c(g) be the locus where the Weil pairing of the universal basis of E[N ] is equal to ζ N ∈ Zp .We let ω • be the dualizing sheaf of X n,c(g) , i.e. the exceptional inverse image of O n,c(g) := H 0 (X n,c(g) , O X n,c(g) ) over X n,cg .
The maps with g an isomorphism.Notice that the modular sheaves ω k E are preserved by the pullbacks of q 1 , q 2 and g.Let F and F n be as in Theorem 4.2, we have induced maps of cohomology R Γ(X n , F n /p s ) Taking direct limits we obtain a map R Γ(X ∞ , F /p s ) Finally, taking derived inverse limits one gets the action of g ∈ GL 2 (Q p ) on the cohomology R Γ(X ∞ , F ).The action of GL 2 (Q p ) on cohomology is not O cyc -linear.In fact, it is ψ-semilinear; this can be shown by considering the Cartan decomposition GL 2 (Q p ) = n1≥n2 GL 2 (Z p ) p n1 0 0 p n2 GL 2 (Z p ) and using the compatibility of the Weil pairing with the determinant.
The action of GL 2 (Q p ) on lim ← −n, Trn H 1−i (X n,O cyc , F ∨ n ⊗ω 2 E,n,cusp ) is defined in such a way that the isomorphism (4.7) is equivariant.Namely, there is a commutative diagram of local duality pairings provided by the functoriality of Serre duality The maps Tr q1 and Tr q2 are induced by the Serre duality traces of q 1 and q 2 respectively, cf.Remark 4.3 (1).Thus, the right action of g ∈ GL 2 (Q p ) on a tuple f = (f n ) ∈ lim ← −n, Trn H 1−i (X n,O cyc , F ∨ n ⊗ ω 2 E,n,cusp ) is given by f | g = ((f | g ) n ) n∈N , where for m big enough, and q 1 , q 2 as in (4.8).
Proof of Theorem 4.2.Without loss of generality we take C = K cyc .Let F = ω k E,∞ or ω k E,∞,cusp .By Corollary 4.2 we have Therefore On the other hand, we have where the transition maps are given by pullbacks.By local duality, Proposition 3.2, we have a natural isomorphism The isomorphism is GL 2 (Q p )-equivariant by the diagram (4.8).
We end this section with an application of the local duality theorem at infinite level to the completed cohomology.We let X n,proet be the pro-étale site of the finite level modular curve as in §3 of [Sch13], and X ∞,proet the pro-étale site of the perfectoid modular curve as in Lecture 8 of [SW20].Definition 4.1.Let i ≥ 0. The i-th completed cohomology group of the modular tower {X n } n≥0 is defined as Remark 4.4.The previous definition of completed cohomology is slightly different from the one of [Eme06].Indeed, Emerton consider the étale cohomology with compact support of the affine modular curve Y n .Let j : Y n → X n be the inclusion and ι : D n → X n be the cusp divisor, both constructions are related by taking the cohomology of the short exact sequence 0 → j !(Z/p s Z) → Z/p s Z → ι * ι * Z/p s Z → 0.
Moreover, the cohomology at the cusps can be explicitely computed, and many interesting cohomology classes already appear in H 1 .
Namely, we have O + X∞ /p s = lim − →n O + Xn /p s as sheaves in the étale site of X ∞ .In fact, let U ∞ be an affinoid perfectoid in the étale site of X ∞ which factors as a composition of rational localizations and finite étale maps.By Lemma 7.5 of [Sch12] there exists n 0 ≥ 0 and an affinoid space U n0 ∈ X n,et such that U ∞ = X ∞ × Xn 0 U n0 .For n ≥ n 0 denote the pullback of U n0 to X n,et by U n , then We also obtain that H i /p s = H i et (X et,Cp , Z/p s Z) for all i ∈ N. Taking inverse limits in (4.10), and using (4.11) and (4.12) one obtains the corollary.
We obtain a description of the dual of the completed cohomology in terms of cuspidal modular forms of weight 2: Theorem 4.3.There is a GL 2 (Q p )-equivariant isomorphism of almost O Cp -modules Proof.This is a consequence of Proposition 4.4 and the particular case of Theorem 4.2 when F = O X∞ and C = C p .
a Drinfeld basis if and only if it is an isomorphism of group schemes, cf.[KM85, Lem.1.5.3].Proposition 1.1.Let E/S be an elliptic curve.Let (P, Q) be a Drinfeld basis of E[M ] and e M : E[M ] × E[M ] → µ M the Weil pairing.Then e M (P, Q) ∈ µ × M (S) is a primitive root of unity , i.e. a root of the M -th cyclotomic polynomial.Proof.[KM85, Theo.5.6.3].
Theorem 2.1 (Kedlaya-Liu).Let A be a perfectoid ring on which a finite group G acts by continuous ring homomorphisms.Then the invariant subring A G is a perfectoid ring.Moreover, if R ⊂ A is an open integral perfectoid subring of A then R G is an open integral perfectoid subgring of A G .Proof.The first statement is Theorem 3.3.26 of [KL19].The second statement follows from the description of open perfectoid subrings of A as p-power closed subrings of A • , i.e open subrings of A • such that x p ∈ R implies x ∈ R, see Corollary 2.2 of [Mor17].

Definition 3. 1 .
Let f : X → Y be a morphism of schemes.1.The map f is embeddable if it factors as X ι − → S → Y where ι is a finite morphism and S is smooth over Y .2. The map f is projectively embeddable if it factors as composition X ι − → P n Y → Y for some n ≥ 0, where ι a finite morphism.3. The map f is a local complete intersection if locally on Y and X it factors as X ι − → S → Y , where S is a smooth Y -scheme, and ι is a closed immersion defined by a regular sequence of S. The length of the regular sequence is called the codimension of X in S. Theorem 3.1 (Hartshorne).Let f : X → Y be a projectively embeddable morphism of noetherian schemes of finite Krull dimension.Then there exist an exceptional inverse image functor f !: D(Y ) → D(X), a trace map Tr : Rf * f !→ 1 in D + qc (Y ), and an adjunction

Zp
the connected component of X(N p n ) Zp given by fixing the Weil pairing e N (P N , Q N ) = ζ N , where (P N , Q N ) is the universal basis of E[N ] and ζ N ∈ Zp a primitive N -th root of unity.We also write X = X 0 .Let O n = Zp [µ p n ] be the n-th cyclotomic extension of Zp , O cyc the p-adic completion of lim − →n O n , K n and K cyc the field of fractions of O n and O cyc respectively.We set O = Zp and K = O[ 1 p ]. Let π n : X n → Spec O n denote the structural map defined by the Weyl pairing of the universal basis of E[p n ].We also denote p n : X n → X n−1 the natural morphism induced by p-multiplication of Drinfeld bases.
The group SL 2 (Z/p n Z) acts transitively on the set of Drinfeld bases of E 0 [p n ] with Weil pairing ζ p n .Without loss of generality, we can assume that P Let E univ denote the universal elliptic curve over A x and C ⊂ E univ [p n ] the subgroup generated by Q Conversely, let R be an object in Zp [ζ p n ]-Art k and E/R a deformation of E 0 .Let C be an étale subgroup of E[p n ] of rank p n .Then there exists a unique U ∞ ∼ lim ← −n≥n0 U n and O + (U ∞ )/p s = lim − →n≥n0 O + (U n )/p s .The sheaf O + X∞ /p s is almost acyclic on affinoid perfectoids, this implies that the RHS of (4.10) is equal to H i an (X ∞,Cp , O + X∞ /p s ).Then, the proof of Corollary 4.3 allows us to compute the above complex using the formal model X ∞H i an (X ∞,Cp , O + X∞ /p s ) = ae H i (X ∞ , O X∞ /p s ) ⊗ O cyc O Cp .(4.11)The Corollary 4.2 shows that the inverse system {H i (X ∞ , O X∞ /p s )} s satisfy the Mittag-Leffler condition.As O Cp /p s is a faithfully flat Z/p s Z-algebra, the inverse system {H i et (X ∞,Cp , Z/p s Z)} s also satisfies the Mittag-Leffler condition.One deduces from Proposition 4.3 that H i proet (X ∞,Cp , Z p ) = H i .(4.12)