Overconvergent Modular Forms

and Perfectoid Shimura Curves

Przemysław Chojecki, D. Hansen, and C. Johansson

Received: January 18, 2016 Revised: July 28, 2016

Communicated by Otmar Venjakob

Abstract. We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze’s infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate ‘z’ in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over ℚ, proving stronger analogues of results of Andreatta-Iovita-Stevens.

2010 Mathematics Subject Classification: 11F33 11F33, 11F70, 11G18

Contents

1. Introduction 192 1.1. Perfectoid modular curves and w-ordinarity 194 1.2. Modular curves vs. Shimura curves 196 1.3. The overconvergent Eichler-Shimura map 197 1.4. Notation, conventions and an outline of the paper 200 Acknowledgements 201 2. Overconvergent modular forms 201 2.1. Weights and characters 201 2.2. Shimura curves 204 2.3. w-ordinary false elliptic curves 206 2.4. The fundamental period and a non-vanishing section 211 2.5. A perfectoid definition of overconvergent modular forms 212 2.6. Locally projective of rank one 214 2.7. Comparison with other definitions of overconvergent modular

forms 219

3. Overconvergent modular symbols 221 3.1. Basic definitions and the filtrations 222 3.2. Slope decompositions 223 4. Sheaves on the pro-étale site 225 4.1. A handy lemma 225 4.2. Sheaves of overconvergent distributions 226 4.3. Completed sheaves of overconvergent modular forms 228 4.4. The overconvergent Eichler-Shimura map 232 4.5. Factorization for weights k ≥ 2 234 5. The overconvergent Eichler-Shimura map over the eigencurve 236 5.1. Sheaves on the eigencurve 236 5.2. Faltings’s Eichler-Shimura map 241 5.3. Results 243 6. Appendix 247 6.1. Mixed completed tensor products 247 6.2. Mixed completed tensor products on rigid-analytic varieties 249 6.3. Analogues of the theorems of Tate and Kiehl 252 6.4. Quotients of rigid spaces by finite groups. 257 References 259

1. Introduction

Let N ≥ 5 be an integer, and let Y_{1}(N)(ℂ) be the usual analytic modular curve. A holomorphic modular form f of weight k and level N admits two rather distinct interpretations, which one might call the algebraic and analytic points of view:

Algebraic: f is a global section ω(f) of the line bundle ω^{⊗k} = ω_{E/Y_{1}(N)}^{⊗k} on Y_{1}(N)(ℂ) (extending to X_{1}(N)(ℂ)). Equivalently, f is a rule which assigns to each test object - an isomorphism class of triples (S, E, η) consisting of a complex analytic space S, a (generalized) elliptic curve π : E → S with level N structure, and η an O_{S}-generator of ω_{E/S} := (LieE/S)^{∗} - an element f(S, E, η) ∈ O(S) such that f commutes with base change in S and satisfies f(S, E, aη) = a^{−k}f(S, E, η) for any a ∈ O(S)^{×}. These two interpretations are easily seen to be equivalent (for simplicity we are ignoring cusps): for any test object, there is a canonical map s : S → Y_{1}(N) realizing E/S as the pullback of the universal curve over Y_{1}(N), and one has s^{∗}ω(f) = f(S, E, η)η^{⊗k}.

Analytic: f is a holomorphic function on the upper half-plane 𝔥 of moderate growth, satisfying the transformation rule f(\frac{az+b}{cz+d}) = (cz + d)^{k}f(z) for all γ ∈ Γ_{1}(N).

How do we pass between these points of view? The key is that 𝔥 may be identified with the universal cover { \atop Y_{1}(N)} of Y_{1}(N), in the category of complex analytic spaces, and the pullback of the line bundle ω to { \atop Y_{1}(N)} is canonically trivialized. Precisely, { \atop Y_{1}(N)} consists of pairs (E, β) where E/ℂ is an elliptic curve and β = {β_{1}, β_{2}} ∈ H_{1}(E(ℂ), ℤ) is an oriented basis^{1}. This space admits a left action of Γ_{1}(N) by {β_{1}, β_{2}} ↦ {aβ_{1} + bβ_{2}, cβ_{1} + dβ_{2}}. Let p : { \atop Y_{1}(N)} → Y_{1}(N) be the natural projection. Defining the period z(E, β) ∈ ℂ by ∫_{β_{1}} η = z(E, β) ∫_{β_{2}} η, where η ≠ 0 ∈ H^{0}(E(ℂ), Ω_{E/ℂ}^{1}) is any nonzero holomorphic one-form on E, the map

{ \atop Y_{1}(N)} →^{∼} 𝔥
(E, β) ↦ z(E, β)

is a Γ_{1}(N)-equivariant isomorphism of Riemann surfaces. The pullback p^{∗}ω is then trivialized by the differential η_{can} characterized by ∫_{β_{2}} η_{can} = 1. Defining f(z) by p^{∗}ω(f) = f(z)η_{can}^{⊗k}, a calculation shows that η_{can}(γz) = (cz + d)^{−1}η_{can}(z), from which the transformation law of f(z) follows.

In the p-adic setting a priori one only has the algebraic definition valid for integral weights. Given the importance of non-integral weights, it seems natural to hope for a direct algebraic definition of modular forms with non-integral weights. More precisely, given a continuous character κ : ℤ_{p}^{×} → L^{×} for some L/ℚ_{p} finite, one would like to define a “p-adic modular form of weight κ and level N” as a rule which assigns to each test object an isomorphism class of triples (R, E, η) consisting of a p-adically separated and complete ring R, a (generalized) elliptic curve E/SpecR with level N structure, and η a generator of ω_{E/R} - an element g(R, E, η) ∈ R ⊗_{ℤ_{p}} L such that g commutes with base change in R and satisfies g(R, E, rη) = κ(r)^{−1}g(R, E, η) for any r ∈ R^{×}. The problem with this naive definition is that the expression κ(r) does not make sense in general. In fact, the only characters for which this is unambiguously defined and functorial in R are the power characters κ : r → r^{k}, k ∈ ℤ. However, Andreatta-Iovita-Stevens and Pilloni ([AIS1, Pil]) discovered (independently) a remarkable fix to this problem, whereby for a given character κ one only allows “certain elliptic curves” and, more importantly, “certain differentials” in the definition of test objects. The admissible elliptic curves are those whose Hasse invariant has suitably small valuation. The admissible η’s are defined using torsion p-adic Hodge theory and the theory of the canonical subgroup; they are not permuted by all of R^{×}, but only by a subgroup of elements p-adically close enough to ℤ_{p}^{×} so that κ(r) is defined.

Our first goal in this paper is to develop a p-adic analogue of the analytic picture above. Of course, the most pressing question here is: what are the correct analogues of { \atop Y_{1}(N)}, 𝔥 ⊆ ℙ_{ℂ}^{1} and z? The following theorem gives a partial answer.

1One also requires that β_{2} generate the Γ_{1}(N)-structure under H_{1}(E(ℂ), ℤ/Nℤ) ≅ E(ℂ)[N]

Theorem 1.1. Given N and p ∤ N as above, let

X_{∞} ∼ \underleftarrow{lim}_{n} X_{K_{1}(N)K(p^{n})}^{ad}

be the infinite-level perfectoid (compactified) modular curve of tame level N ([Sch4]), with its natural right action of GL_{2}(ℚ_{p}). There is a natural family of K_{0}(p)-stable open affinoid perfectoid subsets X_{∞,w} ⊆ X_{∞} parametrized by rationals w ∈ ℚ_{>0}, with X_{∞,w′} ⊆ X_{∞,w} for w ≤ w′, and there is a canonical global section 𝔷 ∈ O(X_{∞,w}) (compatible under changing w) such that γ^{∗}𝔷 = \frac{a𝔷+c}{b𝔷+d} for all γ ∈ K_{0}(p). For any κ as above and any w ≫_{κ} 0, the space

M_{κ,w}(N) := {f ∈ O(X_{∞,w}) ⊗_{ℚ_{p}} L | κ(b𝔷 + d) · γ^{∗}f = f ∀γ ∈ K_{0}(p)}

is well-defined, and the module M_{κ}^{†}(N) = lim_{w→∞} M_{κ,w}(N) is canonically isomorphic with the modules of overconvergent modular forms of weight κ and tame level N defined by Andreatta-Iovita-Stevens and Pilloni.

Thus, in our approach, X_{∞,w} plays the role of { \atop Y_{1}(N)} and the “fundamental period” 𝔷 defined below plays the role of z. Key to this theorem is that ω is trivialized over any X_{∞,w} (by one of the “fake” Hasse invariants constructed in [Sch4]), which is similar to the fact that ω is trivialized over 𝔥. One reason why the latter is true is because the complex period map is not surjective onto ℙ_{ℂ}^{1}. By contrast, the p-adic period morphism π_{HT} : X_{∞} → ℙ^{1} constructed in [Sch4] is surjective, but becomes non-surjective when restricted to any X_{∞,w}. This is the reason that our description works for overconvergent modular forms but not for classical modular forms.

The second goal of the paper is to apply our “explicit” point of view to redefine and analyze the overconvergent Eichler-Shimura map of [AIS2], which compares overconvergent modular symbols to overconvergent modular forms. Our perspective gives a short and transparent definition of these maps, and (we believe) clarifies the ideas involved. We also make use of certain new filtrations on overconvergent distribution modules to obtain a more “global” point of view on the Eichler-Shimura maps: after first defining them in the setting of Coleman families, we glue them into a morphism of coherent sheaves over the whole eigencurve.

We now describe these results in more detail.

1.1. Perfectoid modular curves and w-ordinarity. Let Y_{n} denote the modular curve over ℚ_{p} with Y_{n}(S) parametrizing elliptic curves E/S with a point P ∈ E(S)[N] of exact order N together with an isomorphism

α_{n} : (ℤ/p^{n}ℤ)^{2} →^{∼} E(S)[p^{n}].

Let X_{n} denote the usual compactification of Y_{n}, and let Y_{n} ⊆ X_{n} be the associated adic spaces over Spa(ℚ_{p}, ℤ_{p}). These form compatible inverse systems (X_{n})_{n≥1}, (Y_{n})_{n≥1} with compatible actions of GL_{2}(ℚ_{p}). Fundamental to all our considerations is Scholze’s construction of the infinite level modular curves Y_{∞} ∼ \underleftarrow{lim}_{n} Y_{n}, X_{∞} ∼ \underleftarrow{lim}_{n} X_{n} and the GL_{2}(ℚ_{p})-equivariant Hodge-Tate period map π_{HT} : X_{∞} → ℙ^{1} (a morphism of adic spaces over ℚ_{p}).

Let us say a few more words about π_{HT}. Let C/ℚ_{p} be a complete algebraically closed field extension with ring of integers O_{C}, and let A be an abelian variety over C of dimension g, with dual A^{∨}. Set ω_{A} = H^{0}(A, Ω_{A/C}^{1}) ≅ Hom_{C}(Lie A, C), a C-vector space of rank g. Then we have a natural linear map HT_{A} : T_{p}A ⊗_{ℤ_{p}} C ↠ ω_{A^{∨}}, the Hodge-Tate map of A, which fits into a short exact sequence

0 ⟶ (Lie A)(1) ⟶^{HT_{A^{∨}}^{∨} (1)} Hom(T_{p}A^{∨}, C(1)) = T_{p}A ⊗_{ℤ_{p}} C ⟶^{HT_{A}} ω_{A^{∨}} ⟶ 0

where −(1) denotes a Tate twist. If E/C is an elliptic curve and α : ℤ_{p}^{2} →^{∼} T_{p}E is a trivialization, then (E, α) defines a point in Y_{∞}(C, O_{C}) and π_{HT} sends (E, α) to the line (α ⊗ 1)^{−1}(Lie E) ⊆ C^{2}.

Our first key definition is a new gauge for the ordinarity of an abelian variety, defined in terms of the Hodge-Tate map HT_{A}. Let F_{A} = HT_{A}(T_{p}A ⊗_{ℤ_{p}} O_{C}). This is an O_{C}-lattice inside ω_{A^{∨}}.

Definition 1.2. Let w ∈ ℚ_{>0}.

(1) An abelian variety A/C is w-ordinary if there is a basis b_{1}, …, b_{2g} of T_{p}A such that HT_{A}(b_{i}) ∈ p^{w}F_{A} for all 1 ≤ i ≤ g.

(2) For A/C w-ordinary and 0 < n < w + 1, the pseudocanonical subgroup H_{n} of level n is defined to be the kernel of the natural map

A[p^{n}](C) → F_{A} ⊗_{O_{C}} O_{C}/p^{min(n,w)}O_{C}

induced by HT_{A}.

(3) For A/C w-ordinary, a trivialization α : ℤ_{p}^{2g} →^{∼} T_{p}A is strict if α(e_{1}), …, α(e_{g}) mod p ∈ A[p](C) form a basis for the pseudocanonical subgroup of level one.

Let u and v be the homogeneous coordinate with respect to the standard basis e_{1} = ( \begin{matrix}1 \\ 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 \\ 1 \end{matrix} ) of ℚ_{p}^{2}. We define z = −v/u, a coordinate function on ℙ^{1}. For w ∈ ℚ_{>0} we let ℙ_{w}^{1} ⊆ ℙ^{1} be the locus

ℙ_{w}^{1} = {z | inf_{a∈pℤ_{p}}|z − a| ≤ |p|^{w}}.

For γ ∈ K_{0}(p) one sees that γ^{∗}z = \frac{az+c}{bz+d} and that ℙ_{w}^{1} is a K_{0}(p)-stable affinoid, where K_{0}(p) is the usual Iwahori subgroup of GL_{2}(ℤ_{p}). Define X_{∞,w} = π_{HT}^{−1}(ℙ_{w}^{1}) and Y_{∞,w} = X_{∞,w} ∩ Y_{∞}. These loci are K_{0}(p)-stable. Finally, define the fundamental period 𝔷 = π_{HT}^{∗}z ∈ O_{X_{∞}}^{+}(X_{∞,w}).

Theorem 1.3. A point (E, α) ∈ Y_{∞}(C, O_{C}) is contained in X_{∞,w} if and only if E is w-ordinary and α is strict. Furthermore, X_{∞,w} is the preimage of a canonical affinoid X_{w} ⊆ X_{K_{0}(p)}, and X_{∞,w} is affinoid perfectoid. The (C, O_{C})- points of Y_{w} = X_{w} ∩ Y_{K_{0}(p)} are the pairs (E, H) where E is w-ordinary and H is the pseudocanonical subgroup of level one, and (X_{w})_{w} is a cofinal set of strict neighbourhoods of the ordinary multiplicative locus in X_{K_{0}(p)}.

Taking these infinite-level objects as our basic ingredients, we are able to give a short definition (Definition 2.18) of sheaves of overconvergent modular forms ω_{κ,w}^{†} on X_{w} whose global sections yield the module M_{κ,w}(N) of Theorem 1.1. This construction also works in families of weights. As a guide to our constructions so far, we offer the following table of analogies:

ℂ  ℚ_{p}
Y = Y_{1}(N)(ℂ), a complex analytic space  Y_{w} ⊂ Y_{K_{0}(p)}, an adic space
E/ℂ an elliptic curve  E/C a w-ordinary elliptic curve
(E, β) , β = {β_{1}, β_{2}} ∈ H_{1}(E(ℂ), ℤ)

an oriented basis

(E, α), α : ℤ_{p}^{2} →^{∼} T_{p}E

a strict trivialization

(E, β) ∈ Ỹ, the universal cover of Y  x = (E, α) ∈ Y_{∞,w}(C) ⊂ X_{∞,w}(C)
Γ_{1}(N) ⟳ Ỹ →^{∼} 𝔥 ⊂ ℙ_{/ℂ}^{1}  π_{HT} : X_{∞,w} ↠ ℙ_{w}^{1} ⊂ ℙ_{/ℚ_{p}}^{1}⟲ K_{0}(p)
Ỹ = 𝔥 and \overline{𝔥} are contractible  X_{∞,w} is affinoid perfectoid
z, the coordinate on 𝔥  𝔷 ∈ O^{+}(X_{∞,w}), the fundamental period
z = z(E, β) characterized by  𝔷(x) ∈ C characterized by
∫_{β_{1}} η = z ∫_{β_{2}} η  HT_{E}(α(e_{1})) = 𝔷(x)HT_{E}(α(e_{2}))
∫_{β_{2}} η_{can} = 1  𝔰 = HT_{E}(α(e_{2}))
η_{can}(γz) = \frac{1}{cz+d} η_{can}(z)  γ^{∗}𝔰 = (b𝔷 + d)𝔰
(cz + d)^{k} ∈ O(𝔥)  κ(b𝔷 + d) ∈ O(X_{∞,w}) ⊗_{ℚ_{p}} L (w ≥ w_{λ})
M_{k}(N) = \{f ∈ O(𝔥) | f(gz) = (cz + d)^{k}f(z)\} M_{κ,w}(N) = \{ \begin{matrix}f ∈ O(X_{∞,w}) ⊗_{ℚ_{p}} L, \\ κ(b𝔷 + d)γ^{∗}f = f \end{matrix} \}

M_{κ}^{†} = lim_{w→∞} M_{κ,w}

Table 1.1: Analogies.

As we have already mentioned, sheaves of overconvergent modular forms have been constructed previously by [AIS1] and [Pil], and their constructions (which appear slightly different on the surface) are known to give equivalent notions of overconvergent modular forms. We prove (Theorem 2.33) that our definition is also equivalent to these previous constructions. As emphasized to us by a referee, we remark that our definition of the sheaf ω_{κ,w}^{†} is essentially as a line bundle on X_{∞,w} with a descent datum to X_{w}. However pro-étale descent of vector bundles is not effective in general (this follows for example from [KL2, Example 8.1.6]). In our case we show with relative ease that the resulting sheaf is indeed a line bundle.

1.2. Modular curves vs. Shimura curves. While we have written this introduction so far in the setting of modular curves, we have chosen to work with compact Shimura curves associated with an indefinite quaternion division algebra B/ℚ split at (our fixed) p in the body of the paper. There are two reasons for this. The first reason is that the local p-adic geometry of these Shimura curves is entirely analogous to the local p-adic geometry of modular curves, but the global geometry is simpler without the presence of boundary divisors, compactifications, and their attendant complications. We believe that working in a boundaryless setting helps to clarify the point of view adopted in this paper.

Having said this, many of our ideas extend to the case of modular curves. In particular, all the definitions and results in §2 have exact analogues for classical modular curves, and Theorems 1.1 and 1.3 are true as stated. The techniques we use for Shimura curves work over the open modular curve, and one may extend over the boundary using “soft” techniques (one does not need the advanced results of [Sch4, §2]).

The second reason is that our point of view on the overconvergent Eichler-Shimura map does not immediately generalize to modular curves. While we believe that the underlying philosophy should adapt, there are certain technical aspects of the construction (in particular Proposition 4.4) which seem difficult to adapt. As far as we can tell, this is a purely technical issue with the proétale site as defined in [Sch3]. In work in progress ([DT]), Diao and Tan are developing a logarithmic version of the pro-étale site, and we believe that our constructions would work well in that setting.

1.3. The overconvergent Eichler-Shimura map. From now on we work with Shimura curves attached to an indefinite quaternion division algebra B/ℚ_{p} split at p and use the same notation that we previously used for modular curves.

After Coleman’s construction of Coleman families and the globalization of these to the Coleman-Mazur eigencurve ([Clm, CM]), different constructions of families of finite slope eigenforms and eigencurves for GL_{2/ℚ} were given by Stevens ([Ste], completed by Bellaïche [Bel]) and Emerton ([Eme]). These constructions give the same eigencurve. Roughly speaking, each approach first constructs a p-adic Banach/Fréchet space (or many such spaces) of ”overconvergent” objects interpolating modular forms/cohomology classes, and then creates a geometric object out of this (these) space(s). All spaces have actions of certain Hecke algebras and the fact that these give the same eigencurves amounts to saying that they contain the same finite slope systems of Hecke eigenvalues. However, not much is known about the direct relation between these spaces. One can rephrase the problem in the following way: each construction of the eigencurve remembers the spaces it came from in the form of a coherent sheaf on it, and one may ask if there are relations between these sheaves.

In [AIS2], Andreatta, Iovita and Stevens study the relationship between overconvergent modular forms and overconvergent modular symbols. While Coleman’s overconvergent modular forms p-adically interpolate modular forms, Stevens’s overconvergent modular symbols interpolate classical modular symbols, i.e. classes in the singular cohomology groups H^{1}(Y_{1}(N), Sym^{k−2} H^{1}), where H^{1} is the first relative singular cohomogy of the universal elliptic curve. Classical Eichler-Shimura theory, which one may view as an elaboration of Hodge Theory for these particular varieties and coefficient systems, gives a Hecke-equivariant isomorphism

H^{1}(Y_{1}(N), Sym^{k−2} H^{1}) ⊗_{ℤ} ℂ ≅ M_{k} ⊕ S_{k}

where M_{k} is the space of weight k and level N modular forms and S_{k} is its subspace of cusp forms. Faltings ([Fal]) constructed a p-adic Hodge-theoretic analogue of this isomorphism, replacing singular cohomology with étale cohomology. This construction was then adapted to the overconvergent context in [AIS2].

Let us describe these ideas and our work in more detail. We refer to the main body of the paper for exact definitions. Recall ([AS, Han1]) the overconvergent distribution modules D_{κ}^{s} where κ is a character of ℤ_{p}^{×} as above. It may be interpreted as a local system on X_{K_{0}(p)}(ℂ). The singular cohomology H^{1}(X_{K_{0}(p)}(ℂ), D_{κ}^{s}) is the space of overconvergent modular symbols. Following [Han2], we construct a filtration on the integral distribution module D_{κ}^{s,∘} for which the corresponding topology is profinite. From this one gets a sheaf on the pro-étale site of X_{K_{0}(p)} whose cohomology is isomorphic to H^{1}(X_{K_{0}(p)}(ℂ), D_{κ}^{s}), but also carries a Galois action. To compare this to overconvergent modular forms of weight κ, one introduces a “fattened” version OD_{κ}^{s} of D_{κ}^{s} which has the explicit description

V ↦ (D_{κ} ˆ⊗Ô_{X_{K_{0}(p)}}(V_{∞}))^{K_{0}(p)}

for V ∈ X_{K_{0}(p),proét} quasi-compact and quasi-separated (qcqs), where V_{∞} := V ×_{X_{K_{0}(p)}} X_{∞} and Ô_{X_{K_{0}(p)}} is the completed structure sheaf on X_{K_{0}(p),proét}. After restricting to X_{w} it turns out to be easy to give a morphism to the completed version ω_{κ,w}^{†}: for V ∈ X_{w,proét} qcqs there is a K_{0}(p)-equivariant morphism

D_{κ} ˆ⊗Ô_{X_{w}}(V_{∞}) → Ô_{X_{w}}(V_{∞}) ⊗_{ℚ_{p}} L

given on elementary tensors by

μ ⊗ f ↦ μ(κ(1 + 𝔷x))f.

The formula is heavily inspired by a formula of Stevens for the comparison map between overconvergent distributions and polynomial distributions (whose cohomology computes classical modular symbols) which does not seem to be used a lot in the literature (see the paragraph before Definition 3.2). One then passes to K_{0}(p)-invariants to obtain the desired morphism of sheaves. This is our analogue of the maps denoted by δ_{κ}^{∨}(w) in [AIS2]. It induces a map on cohomology groups over ℂ_{p} which gives the desired map of spaces using that H_{proét}^{1}(X_{w,ℂ_{p}}, ω_{κ,w}^{†}) ≅ H^{0}(X_{w,ℂ_{p}}, ω_{κ,w}^{†} ⊗_{O_{Xw}} Ω_{X_{w}}^{1}). The strategy is the same as in [AIS2], except that they work with the so-called Faltings site instead of the pro-étale site. It is the presence of infinite level Shimura curves in the pro-étale sites of finite level Shimura curves that accounts for the clean explicit formulas we obtain: they provide the correct “local coordinates” for the problem at hand.

We analyze these maps by carrying out the above constructions in families of weights (as in [AIS2]). To define Galois actions one needs to work with families parametrized by certain affine formal schemes instead of the more commonly used affinoid rigid spaces. Whereas the filtrations defined in [AIS2] only work when the formal scheme is an open unit disc near the center of weight space, our filtrations are defined over arbitrary Spf R where R is finite over ℤ_{p}[[X_{1}, .., X_{d}]] for some d. This enables us to glue the morphisms for different families of weights into a morphism of sheaves over the whole eigencurve.

Denote by C the eigencurve and let C_{ℂ_{p}} be its base change to ℂ_{p}. It carries coherent sheaves V, resp. M^{†}, coming from overconvergent modular symbols resp. forms. We denote by V_{ℂ_{p}}, resp. M_{ℂ_{p}}^{†}, their base changes to ℂ_{p}, which may also be viewed as sheaves of O_{C} ˆ⊗_{ℚ_{p}}ℂ_{p}-modules on C. In the latter point of view, one may think of ℂ_{p} as a (rather primitive) period ring. After gluing, the overconvergent Eichler-Shimura map is a morphism ES : V_{ℂ_{p}} → M_{ℂ_{p}}^{†}(−1) of sheaves.

Theorem 1.4 (Theorem 5.11, Theorem 5.14). Let C^{sm} be the smooth locus of C.

(1) V and M^{†} are locally free over C^{sm}, and the kernel K and image I of ES are locally projective sheaves of O_{C^{sm}} ˆ⊗_{ℚ_{p}}ℂ_{p}-modules (or equivalently locally free sheaves on C_{ℂ_{p}}^{sm}). The support of M_{ℂ_{p}}^{†}(−1)/I on C_{ℂ_{p}} is Zariski closed of dimension 0.

(2) Let ϵ_{C^{sm}} be the character of G_{ℚ_{p}} defined by the composition

G_{ℚ_{p}} ⟶^{ϵ} ℤ_{p}^{×} ⟶^{χ_{W}} O_{W}^{×} ⟶ (O_{C^{sm}}^{×} ˆ⊗_{ℚ_{p}}ℂ_{p})^{×}

where ϵ is the p-adic cyclotomic character of G_{ℚ_{p}} and χ_{W} is the universal character of ℤ_{p}^{×}. Then the semilinear action of G_{ℚ_{p}} on the module K(ϵ_{C^{sm}}^{−1}) is trivial.

(3) The exact sequence

0 → K → V_{ℂ_{p}} → I → 0

is locally split. Zariski generically, the splitting may be taken to be equivariant with respect to both the Hecke- and G_{ℚ_{p}}-actions, and such a splitting is unique.

These are stronger analogues of results in [AIS2], where the authors prove the analogous results for modular curves in some small (unspecified) open neighbourhood of the set of non-critical classical points.

We believe that our perspective on overconvergent modular forms and the overconvergent Eichler-Shimura map should generalize to higher-dimensional Shimura varieties. In particular, it should be reasonably straightforward to adapt the methods of this paper to (the compact versions of) Hilbert modular varieties.

1.4. Notation, conventions and an outline of the paper. Throughout this text, we let p denote a fixed prime.

For the purposes of this paper, a small ℤ_{p}-algebra is a ring R which is reduced, p-torsion-free, and finite as a ℤ_{p}[[X_{1}, ..., X_{d}]]-algebra for some (unspecified) d ≥ 0. Any such R carries a canonical adic profinite topology, and is also complete for its p-adic topology. For convenience we will fix a choice of ideal 𝔞 = 𝔞_{R} defining the profinite topology (a “canonical” example of such a choice is the Jacobson radical). All constructions made using this choice will be easily verified to not depend upon it.

We will need various completed tensor product constructions, some of which are non-standard. We will need to take a form of completed tensor product between small ℤ_{p}-algebras and various Banach spaces or other ℤ_{p}-modules. These will always be denoted by an undecorated completed tensor product ˆ⊗. We explain our conventions for the unadorned ˆ⊗ in Convention 2.2 and Definitions 6.3 and 6.6 below. Any adorned ˆ⊗_{A} is a standard one, with respect to the natural topology coming from A.

We will use Huber’s adic spaces as our language for non-archimedean analytic geometry in this paper. In particular, a “rigid analytic variety” will refer to the associated adic space, and all open subsets and open covers are open subsets resp. open covers of the adic space (i.e. we drop the adjective “admissible” used in rigid analytic geometry). The pro-étale site of [Sch3] is key to our constructions; we will freely use notation and terminology from that paper. For perfectoid spaces we use the language of [KL1] for simplicity (e.g. we speak of perfectoid spaces over ℚ_{p}), but any perfectoid space appearing is a perfectoid space is the sense of [Sch1] (i.e. it lives over a perfectoid field).

Let us finish the introduction by briefly outlining the contents of the paper. In §2 we give our new definitions of sheaves of overconvergent modular forms in families and prove their basic properties, including a comparison with the definitions of [AIS1, Pil]. In §3 we recall the basic definitions from the theory of overconvergent modular symbols and define the filtrations mentioned above. We make some technical adjustments when defining slope decompositions. In particular, we do not need the concept of a weak orthonormal basis used in [AIS2]; all slope decompositions can be defined using standard orthonormal bases and formal operations. In §4 we define our overconvergent Eichler-Shimura maps, and §5 glues them over the eigencurve and proves the properties stated above.

The paper concludes with an appendix, collecting various technical results and definitions that are needed in the main text; some of these results may be of independent interest. In §6.1 we define our non-standard completed tensor products and prove some basic properties. While we only need this ˆ⊗ for small ℤ_{p}-algebras R, it turns out that the ring structure only serves to obfuscate the situation. Accordingly, we define ˆ⊗ for a class of ℤ_{p}-module that we call profinite flat. Throughout the text we will need to consider sheaves of rings like O_{X} ˆ⊗R and Ô_{X} ˆ⊗R on X where X is a rigid space and R is a small ℤ_{p}- algebra. We prove some technical facts about these sheaves of rings and their modules in §6.2-6.3. Finally §6.4 discusses quotients of rigid spaces by finite group actions, proving a standard existence result that we were not able to locate in the literature.

Acknowledgements. This collaboration grew out of discussions at the conference in Lyon in June 2013 between P.C. and C.J, and then conversations in Jussieu in October 2013 between P.C. and D.H. We thank the organizers of the Lyon conference for the wonderful gathering. All of the authors benefited from shorter or longer stays at Columbia University and the University of Oxford. D.H. would also like to thank Johan de Jong, Michael Harris and Shrenik Shah for some helpful conversations, and C.J. would like to thank Hansheng Diao for conversations relating to this work and [DT]. The authors would like to thank Judith Ludwig for pointing out a gap in our original proof of Proposition 6.24, as well as the anonymous referees for their comments and corrections.

P.C. was partially funded by EPSRC grant EP/L005190/1. D.H. received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 290766 (AAMOT). C. J. was supported by EPSRC Grant EP/J009458/1 and NSF Grants 093207800 and DMS-1128155 during the work on this paper.

2. Overconvergent modular forms

2.1. Weights and characters. In section we recall some basic notions about weights. We may define weight space as the functor from complete affinoid (ℤ_{p}, ℤ_{p})-algebras (A, A^{+}) to abelian groups given by

(A, A^{+}) ↦ Hom_{cts}(ℤ_{p}^{×}, A^{×}).

This functor is representable by (ℤ_{p}[[ℤ_{p}^{×}]], ℤ_{p}[[ℤ_{p}^{×}]]). The proof is well known. The key fact that makes the arguments work in this generality is that A^{∘}/A^{∘∘} is a reduced ring of characteristic p, where A^{∘∘} is the set of topologically nilpotent elements in A. We define 𝔚 = Spf(ℤ_{p}[[ℤ_{p}^{×}]]) and let W = Spf(ℤ_{p}[[ℤ_{p}^{×}]])^{rig} be the associated rigid analytic weight space, with its universal character χ_{W} : ℤ_{p}^{×} → O(W)^{×}.

We embed ℤ into W by sending k to the character χ_{k}(z) = z^{k−2}.

Definition 2.1. (1) A small weight is a pair U = (R_{U}, χ_{U}) where R_{U} is a small ℤ_{p}-algebra and χ_{U} : ℤ_{p}^{×} → R_{U}^{×} is a continuous character such that χ_{U}(1 + p) − 1 is topologically nilpotent in R_{U} with respect to the p-adic topology.

(2) An affinoid weight is a pair U = (S_{U}, χ_{U}) where S_{U} is a reduced Tate algebra over ℚ_{p} topologically of finite type and χ_{U} : ℤ_{p}^{×} → S_{U}^{×} is a continuous character.

(3) A weight is a pair U = (A_{U}, χ_{U}) which is either a small weight or an affinoid weight.

We shall sometimes abbreviate A_{U} by A when U is clear from context. In either case, we can make A_{U}[\frac{1}{p}] into a uniform ℚ_{p}-Banach algebra by letting A_{U}^{∘} be the unit ball and equipping it with the corresponding spectral norm. We will denote this norm by | · |_{U}. Note then that there exists a smallest integer s ≥ 0 such that |χ_{U}(1 + p) − 1|_{U} < p^{− \frac{1}{p^{s}(p−1)}} . We denote this s by s_{U}. When U is small, the existence uses that χ_{U}(1+p)−1 is p-adically topologically nilpotent. We will also make the following convention:

Convention 2.2. Let U be a weight and let V be a Banach space over ℚ_{p}. We define V ˆ⊗A_{U} as follows:

(1) If U is small, then V ˆ⊗R_{U} is a mixed completed tensor product in the sense of the appendix.

(2) If U is affinoid, then V ˆ⊗S_{U} := V ˆ⊗_{ℚ_{p}}S_{U}.

Remark 2.3. Many (though not all) of the results in this paper involving a choice of some weight U make equally good sense whether U is small or affinoid. In our proofs of these results, we typically give either a proof which works uniformly in both cases, or a proof in the case where U is small, which is usually more technically demanding.

When U = (R_{U}, χ_{U}) is a small weight the universal property of weight space gives us a canonical morphism

Spf(R_{U}) → 𝔚,

which induces a morphism

Spf(R_{U})^{rig} → W.

When U = (S_{U}, χ_{U}) is an affinoid weight we get an induced morphism

Spa(S_{U}, S_{U}^{∘}) → W.

We make the following definition:

Definition 2.4. (1) A small weight U = (R_{U}, χ_{U}) is said to be open if R_{U} is normal and the induced morphism Spf(R_{U})^{rig} → W is an open immersion.

(2) An affinoid weight U = (S_{U}, χ_{U}) is said to be open if the induced morphism Spa(S_{U}, S_{U}^{∘}) → W is an open immersion.

(3) A weight U = (A_{U}, χ_{U}) is said to be open if it is either a small open weight or an affinoid open weight.

Note that if U is a small weight such that U^{rig} → W is an open immersion, then the normalization of U is a small open weight.

Let B be any uniform ℚ_{p}-Banach algebra. Let us say that a function f : ℤ_{p} → B is s-analytic for some nonnegative integer s if, for any fixed a ∈ ℤ_{p}, there is some φ_{f,a} ∈ B ⟨T ⟩ such that φ_{f,a}(x) = f(p^{s}x + a) for all x ∈ ℤ_{p}. In other words, f can be expanded in a convergent power series on any ball of radius p^{−s}. This is naturally a Banach space which we denote by C^{s−an}(ℤ_{p}, B).

Theorem 2.5 (Amice). The polynomials e_{j}^{s}(x) = ⌊p^{−s}j⌋! ( \begin{matrix}x \\ j \end{matrix} ) form an or-

thonormal basis of C^{s−an}(ℤ_{p}, B) for any uniform ℚ_{p}-Banach algebra B. Furthermore, we have e_{j}^{s}(ℤ_{p} + p^{s}B^{∘}) ⊆ B^{∘}.

Proof. This is well known, see e.g. [Col, Theorem 1.7.8] and its proof. □

We can now prove that characters extend over a bigger domain.

Proposition 2.6. Let U = (A_{U}, χ_{U}) be a weight and let B be any uniform ℚ_{p}-Banach algebra. Then for any s ∈ ℚ_{>0} such that s ≥ s_{U}, χ_{U} extends canonically to a character

χ_{U} : B_{s}^{×} → (A_{U}^{∘} ˆ⊗_{ℤ_{p}}B^{∘})^{×} ⊂ (A_{U} ˆ⊗B)^{×},

where B_{s}^{×} := ℤ_{p}^{×} ·(1+p^{s+1}B^{∘}) ⊆ (B^{∘})^{×} and p^{s+1}B^{∘} is shorthand for {b ∈ B^{∘} | |b| ≤ p^{−s−1}} (where | − | is the spectral norm on B).

Proof. Without loss of generality we may assume that s is an integer (e.g by replacing s with ⌊s⌋). We may decompose any b ∈ B_{s}^{×} uniquely as b = ω(b) ⟨b⟩, with ω(b) ∈ μ_{p−1} and ⟨b⟩ ∈ 1+pℤ_{p} +p^{s+1}B^{∘}. We will show that for any s ≥ s_{U} and b ∈ B_{s}^{×}, the individual terms of the series

f(b) = χ_{U}(ω(b)) ∑_{j=0}^{∞} (χ_{U}(1 + p) − 1)^{j} ( j^{\frac{log⟨b⟩}{log(1+p)}} )
“ = ” χ_{U}(ω(b)) · χ(1 + p)^{\frac{log⟨b⟩}{log 1+p}}

lie in A_{U}^{∘} ⊗_{ℤ_{p}} B^{∘} and tend to zero p-adically, so this series converges to an element of A_{U}^{∘} ˆ⊗_{ℤ_{p}}B^{∘} and a fortiori to an element of A_{U} ˆ⊗B. We claim this series defines a canonical extension of χ_{U}.

Using the well known formula for the p-adic valuation of factorials we see that v_{p}(⌊p^{−s}j⌋!) ≤ \frac{j}{p^{s}(p−1)}. In particular, writing

χ_{U}(ω(b)) ∑_{j=0}^{∞} (χ_{U}(1 + p) − 1)^{j} ( \begin{matrix}x \\ j \end{matrix} ) = χ_{U}(ω(b)) ∑_{j=0}^{∞} \frac{(χ_{U}(1 + p) − 1)^{j}}{⌊p^{−s}j⌋!} e_{j}^{s}(x),

our assumption on s implies that |χ_{U}(1 + p) − 1|_{U} < p^{− \frac{1}{p^{s}(p−1)}} , so we see that this series converges to an element of A_{U}^{∘} ˆ⊗_{ℤ_{p}}B^{∘} for any x ∈ B such that e_{j}^{s}(x) ∈ B^{∘} for all j. By Theorem 2.5 it then suffices to verify that if b ∈ B_{s}^{×} then x = \frac{log⟨b⟩}{log 1+p} ∈ ℤ_{p} + p^{s}B^{∘}. But we know that the function

η(b) = \frac{log⟨b⟩}{log 1+p} defines a homomorphism from 1 + pℤ_{p} + p^{s+1}B^{∘} to ℤ_{p} + p^{s}B^{∘}, so we are done. The character property follows by calculating directly from the definitions. Finally, to show that this character extends χ_{U}, note that for b ∈ μ_{p−1} × (1 + p)^{ℤ_{≥0}}, f(b) becomes a finite sum which equals χ_{U}(b) by the binomial theorem, so f|_{ℤ_{p}^{×}} = χ_{U} by continuity. □

2.2. Shimura curves. We write C for an algebraically closed field containing ℚ_{p}, complete with respect to a valuation v : C → ℝ ∪ {+∞} with v(p) = 1 (so v is nontrivial), and we write O_{C} for the valuation subring. Fix an embedding ℂ_{p} ⊆ C. We fix a compatible set of p^{n}-th roots of unity in ℂ_{p} and use this choice throughout to ignore Tate twists (over any C). We let B denote an indefinite non-split quaternion algebra over ℚ, with discriminant d which we assume is prime to p. We fix a maximal order O_{B} of B as well as an isomorphism O_{B} ⊗_{ℤ} ℤ_{p} ≅ M_{2}(ℤ_{p}), and write G for the algebraic group over ℤ whose functor of points is

R ↦ (O_{B} ⊗_{ℤ} R)^{×},

where R is any ring. We fix once and for all a neat compact open subgroup K^{p} ⊆ G(ℤ̂^{p}) such that K^{p} = ∏_{ℓ≠p} K_{ℓ} for compact open subgroups K_{ℓ} ⊆ GL_{2}(ℤ_{ℓ}) and (for simplicity) det(K^{p}) = (ℤ̂^{p})^{×}.

Recall (e.g. from [Buz1, §1]) that a false elliptic curve over a ℤ[\frac{1}{d}]-scheme S is a pair (A/S, i) where A is an abelian surface over S and i : O_{B} ↪ End_{S}(A) is an injective ring homomorphism. We refer to [Buz1] for more information and definitions regarding false elliptic curves, in particular the definition of level structures. Let X_{GL_{2}(ℤ_{p})} be the moduli space of false elliptic curves with K^{p}-level structure as a scheme over ℤ_{p}. We denote by X_{GL_{2}(ℤ_{p})} the Tate analytification of its generic fibre, viewed as an adic space over Spa(ℚ_{p}, ℤ_{p}). For any compact open subgroup K_{p} ⊆ GL_{2}(ℤ_{p}) we use a subscript −_{K_{p}} to denote the same objects with a K_{p}-level structure added. We will mostly use the standard compact open subgroups K_{0}(p^{n}) or K(p^{n}), for n ≥ 1. Since we will mostly work with the Shimura curves with K_{0}(p)-level structure, we make the following convention:

Convention 2.7. We define X := X_{K_{0}(p)}, X = X_{K_{0}(p)}, et cetera. A Shimura curve with no level specified has K_{0}(p)-level at p.

The following striking theorem of Scholze is key to all constructions in this paper.

Theorem 2.8 (Scholze). There exist a perfectoid space X_{∞} over Spa(ℚ_{p}, ℤ_{p}) such that

X_{∞} ∼ \underleftarrow{lim}_{n} X_{K(p^{n})}.

It carries an action of GL_{2}(ℚ_{p}) and there exists a GL_{2}(ℚ_{p})-equivariant morphism

π_{HT} : X_{∞} → ℙ^{1}

of adic spaces over Spa(ℚ_{p}, ℤ_{p}). Let ℙ^{1} = V_{1} ∪ V_{2} denote the standard affinoid cover. Then V_{1} = π_{HT}^{−1}(V_{1}) and V_{2} = π_{HT}^{−1}(V_{2}) are both affinoid perfectoid, and there exists an N and affinoid opens S_{1}, S_{2} ⊆ X_{K(p^{N})} such that V_{i} is the preimage of S_{i}. Moreover we have ω = π_{HT}^{∗}O(1) on X_{∞}, where ω is obtained by pulling back the usual ω (defined below) from any finite level X_{K(p^{n})}.

A few remarks are in order. For the definition of ∼ we refer to [SW, Definition 2.4.1]. This theorem is essentially a special case of [Sch4, Theorem IV.1.1] except for the difference in base field and the target of π_{HT}; there one obtains a perfectoid space over some algebraically closed complete C/ℚ_{p} and π_{HT} takes values in a larger (partial) flag variety. The version here is easily deduced in the same way; we now sketch the argument. The tower (X_{K(p^{n})})_{n} embeds into the tower of Siegel threefolds (over ℚ_{p}), and the same argument as in the proof of [Sch4, Theorem IV.1.1] gives the existence of X_{∞} and a map π_{HT} which takes values in the partial flag variety Fl of GSp_{4} with respect to the Siegel parabolic. Using the M_{2}(ℤ_{p})-action (see below) one sees that it takes values in ℙ^{1} ⊆ Fl. Since the first version of this paper, such results have appeared in the case of general Hodge type Shimura varieties; see [CS, Theorems 2.1.2 and 2.1.3]. Finally, one easily sees that the standard affinoid opens of ℙ^{1} come by pullback from standard affinoid opens of ℙ^{5} via the embeddings ℙ^{1} ⊆ Fl ⊆ ℙ^{5}, where Fl ⊆ ℙ^{5} is the Plücker embedding.

Let us now discuss some standard constructions and define the sheaf ω mentioned in Theorem 2.8. For any false elliptic curve A over some ℤ_{p}-scheme S the p-divisible group A[p^{∞}] carries an action O_{B} ⊗_{ℤ} ℤ_{p} ≅ M_{2}(ℤ_{p}). Put G_{A} = eA[p^{∞}], where e ∈ M_{2}(ℤ_{p}) is an idempotent that we will fix throughout the text (take e.g. ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )). This is a p-divisible group over S of height 2 and we have A[p^{∞}] ≅ G_{A}^{⊕2} functorially; we will fix this isomorphism. For all purposes G_{A} behaves exactly like the p-divisible group of an elliptic curve and we may use it to define ordinarity, supersingularity, level structures et cetera. We will often just write G instead of G_{A} if the false elliptic curve A is clear from the context. The line bundle ω is the dual of e(Lie(A^{univ}/X)), where A^{univ} is the universal false elliptic curve. We will also write G^{univ} = G_{A^{univ}}. The same definitions and conventions apply to the adic versions, and to other level structures.

Next, we specify the right action of GL_{2}(ℚ_{p}) on (C, O_{C})-points on both sides of the Hodge-Tate period map X_{∞} → ℙ^{1}. First we consider ℙ^{1}: g ∈ GL_{2}(ℚ_{p}) acts from the left on C^{2} (viewed as column vectors) and a line L ⊆ C^{2} is sent by g to g^{∨}(L), where g ↦ g^{∨} is the involution

g = (\begin{matrix}a & b \\ c & d \end{matrix}) ↦ g^{∨} = det(g)g^{−1} = (\begin{matrix}d & −b \\ −c & a \end{matrix}) .

This defines a right action. A (C, O_{C})-point of X_{∞} consists of a false elliptic curve A/C and an isomorphism α : ℤ_{p}^{2} → T_{p}G (and the K^{p}-level structure which we ignore). Let g ∈ GL_{2}(ℚ_{p}) and fix n ∈ ℤ such that p^{n}g ∈ M_{2}(ℤ_{p}) but p^{n−1}g ∉ M_{2}(ℤ_{p}). For m ∈ ℤ_{≥0} sufficiently large the kernel of p^{n}g^{∨} modulo p^{m} stabilizes and we denote the corresponding subgroup of G[p^{m}] under α by H. We define (A, α).g to be (A/H^{⊕2}, β), where β is defined as the composition

ℤ_{p}^{2} ⟶^{p^{n}g} ℚ_{p}^{2} ⟶^{α} V_{p}G ⟶^{(f^{∨})_{∗}^{−1}} V_{p}(G/H).

Here H^{⊕2} is viewed as a subgroup scheme of A[p^{∞}] via the functorial isomorphism A[p^{∞}] ≅ G^{⊕2}, V_{p}(−) denotes the rational Tate module and (f^{∨})_{∗} : V_{p}(G/H) → V_{p}G is the map induced from the dual of the natural isogeny f : G → G/H (note that β is isomorphism onto T_{p}(G/H)). In particular, if g ∈ GL_{2}(ℤ_{p}), then (A, α).g = (A, α · g) where (α · g)(e_{1}) = aα(e_{1}) + cα(e_{2}), (α · g)(e_{2}) = bα(e_{1}) + dα(e_{2}). Here and everywhere else in the text e_{1} and e_{2} are the standard basis vectors ( \begin{matrix}1 \\ 0 \end{matrix} ) and ( \begin{matrix}0 \\ 1 \end{matrix} ) of ℤ_{p}^{2}.

2.3. w-ordinary false elliptic curves. Let H be a finite flat group scheme over O_{C} killed by p^{n}. We let ω_{H} denote the dual of Lie(H). It is a torsion O_{C}- module and hence isomorphic to ⨁_{i} O_{C}/a_{i}O_{C} for some finite set of a_{i} ∈ O_{C}. The degree deg(H) of H is defined to be ∑_{i} v(a_{i}). The Hodge-Tate map HT_{H} is the morphism of fppf abelian sheaves H → ω_{H^{∨}} over O_{C} defined on points by

f ∈ H = (H^{∨})^{∨} ↦ f^{∗}(dt/t) ∈ ω_{H^{∨}}

where we view f as a morphism f : H^{∨} → μ_{p^{n}}, dt/t ∈ ω_{μ_{p^{n}}} is the invariant differential and −^{∨} denotes the Cartier dual. We will often abuse notation and use HT_{H} for the map on O_{C}-points, and there one may identify the O_{C}-points of H with the C-points of its generic fibre.

Now let G be a p-divisible group over O_{C}. Taking the inverse limit over the Hodge-Tate maps for the G[p^{n}] we obtain a morphism HT_{G} : T_{p}G → ω_{G^{∨}}, which we will often linearize by tensoring the source with O_{C}. Taking this morphism for G^{∨} and dualizing it we obtain a morphism Lie(G) → T_{p}G⊗_{ℤ_{p}} O_{C}. Putting these morphisms together we get a sequence

0 → Lie(G) → T_{p}G ⊗_{ℤ_{p}} O_{C} → ω_{G^{∨}} → 0

which is in fact a complex with cohomology groups killed by p^{1/(p−1)} ([FGL, Théorème II.1.1]).

Let A/C be a false elliptic curve. Then A has good reduction and we will denote its unique model over O_{C} by A. We have the Hodge-Tate sequence of G_{A}[p^{∞}]:

0 → Lie(G_{A}) → T_{p}G ⊗_{ℤ_{p}} O_{C} → ω_{G_{A}^{∨}} → 0.

Here we have dropped the subscript −_{A} in the notation of the Tate module for simplicity; this should not cause any confusion. We will write HT_{A} for HT_{G_{A}[p^{∞}]}. The image and kernel of HT_{A} are free O_{C}-modules of rank 1 that we will denote by F_{A} and F_{A}^{1} respectively. Note that p^{1/(p−1)}ω_{G_{A}^{∨}} ⊆ F_{A} ⊆ ω_{G_{A}^{∨}}.

Recall that e_{1} and e_{2} are the standard basis vectors of ℤ_{p}^{2} and let w be a positive rational number.

Definition 2.9. Let A/C be a false elliptic curve with model A/O_{C}. Let w ∈ ℚ_{>0}.

(1) Let α be a trivialization of T_{p}G. We say that α is w-ordinary if HT_{A}(α(e_{1})) ∈ p^{w}F_{A}.

(2) A is called w-ordinary if there is a w-ordinary trivialization of T_{p}G.

Note that if A is w-ordinary, then it also w′-ordinary for all w′ < w. Note also that A is ordinary (in the classical sense) if and only if it is ∞-ordinary (i.e. A-ordinary for all w > 0).

Definition 2.10. Let A/C be a w-ordinary false elliptic curve and assume that n ∈ ℤ_{≥1} is such that n < w + 1. Then the kernel of the morphism G[p^{n}](C) → F_{A}/p^{min(n,w)}F_{A} induced by HT_{A} is an étale subgroup scheme H_{n} of G_{A}[p^{n}] isomorphic to ℤ/p^{n}ℤ which we call the pseudocanonical subgroup of level n.

We will use the notation H_{n} to denote the pseudocanonical subgroup of level n (when it exists) whenever the false elliptic curve A is clear from the context. When there are multiple false elliptic curves in action we will use the notation H_{n,A}. Since H_{n} is naturally equipped with an inclusion into G_{A}[p^{n}] we may take its schematic closure inside G_{A}[p^{n}]. This is a finite flat group scheme of rank p^{n} over O_{C} with generic fibre H_{n} and we will abuse notation and denote it by H_{n} as well.

When n = 1 we will refer to H_{1} simply as the pseudocanonical subgroup and drop ”of level 1”. Note that if α : T_{p}G → ℤ_{p}^{2} is a w-ordinary trivialization with n − 1 < w ≤ n then (α^{−1} mod p^{n})|_{ℤ/p^{n}ℤ⊕0} trivializes the pseudocanonical subgroup. We record a simple lemma:

Lemma 2.11. Let A/C be a false elliptic curve and let α be a w-ordinary trivialization of T_{p}G. Assume that w > n ∈ ℤ_{≥1} and let m ≤ n be a positive integer. Then A/H_{m,A}^{⊕2} is (w − m)-ordinary, and for any m′ ∈ ℤ with m < m′ ≤ n, H_{m′−m,A/H_{m}^{⊕2}} = H_{m′,A}/H_{m,A}.

Proof. Let g ∈ GL_{2}(ℚ_{p}) denote the matrix ( \begin{matrix}1 & 0 \\ 0 & p^{m} \end{matrix} ). Then (A, α).g = (A/H_{m,A}^{⊕2}, β) where β is defined by this equality. Let f denote the natural isogeny G → G/H_{m}. From the definitions we get a commutative diagram

\xymatrix{ℤ_{p}^{2} \ar[d]^{g^{∨}} \ar[r]^{α} & T_{p}G \ar[d]^{f_{∗}} \ar[r]^{^{HT^{A}}} & F_{A} \ar[d]^{(f^{∨})^{∗}} \\ ℤ_{p}^{2} \ar[r]^{β} & T_{p}(G/H_{m}) \ar[r]^{_{HT_{A/H{⊕2 \atop m}}}} & F_{A/H_{m}^{⊕2}}}

and direct computation gives that p^{m} HT_{A/H_{m}^{⊕2}}(β(e_{1})) = (f^{∨})^{∗} HT_{A}(α(e_{1})). Since HT_{A}(α(e_{1})) ∈ p^{w}F_{A} we see that HT_{A/H_{m}^{⊕2}}(β(e_{1})) ∈ p^{w−m}F_{A/H_{m}^{⊕2}} which proves the first assertion. For the second assertion, observe that by definition H_{m′−m,A/H_{m}^{⊕2}} and H_{m′,A}/H_{m,A} are generated by β(e_{1}) mod p^{m′−m} and f(α(e_{1})) mod p^{m} respectively, and that these are equal. □

Remark 2.12. The commutativity of the diagram in the proof above is also what essentially proves the GL_{2}(ℚ_{p})-equivariance of the Hodge-Tate period map π_{HT}, and the first assertion may be viewed more transparently as a direct consequence of this equivariance for the element g. Note also that the second assertion of the Lemma mirrors properties of the usual canonical subgroups of higher level.

Next we recall some calculations from Oort-Tate theory which are recorded in [Far, §6.5 Lemme 9] (we thank an anonymous referee for pointing out this reference). For the last statement, see Proposition 1.2.8 of [Kas] (for further reference see Remark 1.2.7 of loc.cit and §3 of [Buz2]; note that these references treat elliptic curves but the results carry over verbatim).

Lemma 2.13. Let H be a finite flat group scheme over O_{C} of degree p. Then H is isomorphic to Spec(O_{C}[Y]/(Y^{p} − aY)) for some a ∈ O_{C} and determined up to isomorphism by v(a), and the following holds:

(1) ω_{H} = (O_{C}/aO_{C}).dY and hence deg(H) = v(a).

(2) The image of the (linearized) Hodge-Tate map HT_{H^{∨}} : H^{∨}(C)⊗O_{C} →

ω_{H} is equal to (cO_{C}/aO_{C}).dY, where v(c) = (1 − v(a))/(p − 1).

Moreover, if A/C is a false elliptic curve such that H ⊆ G_{A}[p] and deg(H) > 1/(p + 1), then H is the canonical subgroup of G_{A}.

We will use these properties freely in this section. Using this we can now show that the pseudocanonical subgroup coincides with the canonical subgroup for sufficiently large w (as a qualitative statement this is implicit in [Sch4], cf. Lemma III.3.8).

Lemma 2.14. Let A/C be a w-ordinary false elliptic curve and assume that p/(p^{2} − 1) < w ≤ 1. Then H_{1} is the canonical subgroup of G_{A}.

Proof. Consider the commutative diagram

\xymatrix{0 \ar[r] & H_{1}(C) \ar[d]_{_{HT_{H_{1}}}} \ar[r] & G_{A}[p](C) \ar[d]^{_{HT_{G_{A}[p]}}} \\ 0 \ar[r] & ω_{H_{1}^{∨}} \ar[r] & ω_{G_{A}[p]^{∨}}}

with exact rows. We have ω_{G_{A}[p]^{∨}} = ω_{G_{A}^{∨}}/pω_{G_{A}^{∨}}. Note that H_{1} is an Oort-Tate group scheme and hence isomorphic to Spec(O_{C}[Y]/(Y^{p} − aY)) for some a ∈ O_{C} with v(a) = deg(H_{1}) and H_{1}^{∨} is isomorphic to Spec(O_{C}[Y]/(Y^{p} − bY)) with ab = p. Fix a generator s ∈ H_{1}(C). By choosing generators the inclusion ω_{H_{1}^{∨}} → ω_{G_{A}[p]^{∨}} may be written as O_{C}/bO_{C} → O_{C}/pO_{C} where the map is multiplication by a, and HT_{H_{1}^{∨}}(s) has valuation v(a)/(p − 1). Since A is wordinary we have a HT_{H_{1}^{∨}}(s) = HT_{A[p]}(s) ∈ p^{w}ω_{A[p]^{∨}} and hence pv(a)/(p − 1) ≥ w, i.e. deg(H_{1}) ≥ (p − 1)w/p. By our assumption on w we deduce deg(H_{1}) > 1/(p + 1) and hence that H_{1} is the canonical subgroup. □

Remark 2.15. As emphasized by a referee, one may also bound the Hodge height of G_{A}, at least under stronger assumptions on w (recall that the Hodge height is the valuation of the Hasse invariant, truncated by 1). For example, using [FGL, Théorème II.1.1] and [AIP, Proposition 3.1.2], one sees that if \frac{1}{2} + \frac{1}{p−1} < w ≤ 1 and p ≥ 5, then the Hodge height of G_{A} is ≤ 1 − w + \frac{1}{p−1}.

We will now, somewhat overdue, discuss the interpretation of w-ordinarity in terms of the Hodge-Tate period map. Define a coordinate z on ℙ^{1} by letting z correspond to the line spanned by ( \begin{matrix}x \\ y \end{matrix}) with z = −y/x and for w ∈ ℚ_{>0} define U_{w} ⊆ ℙ^{1} to be the locus |z| ≤ |p^{w}|. If w ≤ n, then this locus is K_{0}(p^{n})-stable. It is a rational subset of V_{1}. Now define

U_{∞,w} = π_{HT}^{−1}(U_{w}).

This is a K_{0}(p^{n})-stable open subspace of X_{∞}. In fact, U_{∞,w} is affinoid perfectoid since U_{w} is a rational subset of V_{1} ⊆ ℙ^{1} and therefore U_{∞,w} is a rational subset of V_{1} since π_{HT} is adic. The conclusion follows from [Sch1, Theorem 6.3(ii)]. Note that, directly from the definition, the (C, O_{C})-points of U_{∞,w} are exactly the pairs (A, α) for which α is w-ordinary.

In this article we will want to vary w in order to capture all weights when defining overconvergent modular forms. A minor disadvantage of the loci U_{∞,w} is that they are stable under different compact open subgroups as w varies. We will instead define related loci that have the advantage that they all stable under the action of K_{0}(p) (independent of w).

Definition 2.16. Let w ∈ ℚ_{>0} and let A be a w-ordinary false elliptic curve over C. Let α be a trivialisation of T_{p}G. We say that α is strict if (α mod p)|_{ℤ/pℤ⊕0} trivializes the pseudocanonical subgroup.

Note that if α is w-ordinary then it is strict. Given (A, α) with A w-ordinary and α strict, the orbit of (A, α) in X_{∞} under K_{0}(p) consists exactly of the pairs (A, β) for which β is strict. We define ℙ_{w}^{1} ⊆ V_{1} to be the rational subsets whose (C, O_{C})-points correspond to

{z ∈ O_{C} | ∃s ∈ pℤ_{p} : |z − s| ≤ |p^{w}|}.

One checks directly that if γ ∈ K_{0}(p) and |z|, |s| < 1, then |γ^{∨}z −γ^{∨}s| = |z −s|. Thus ℙ_{w}^{1} is stable under K_{0}(p) and moreover ℙ_{w}^{1} = U_{w}.K_{0}(p). We then define

X_{∞,w} = π_{HT}^{−1}(ℙ_{w}^{1}).

By the discussion above these are K_{0}(p)-stable and a (C, O_{C})-point (A, α) of X_{∞} is in X_{∞,w} if and only if A is w-ordinary and α is strict.

Theorem 2.17. Let w ∈ ℚ_{>0}.

(1) There is a unique affinoid open rigid subspace X_{w} ⊆ X whose (C, O_{C})- points are exactly the pairs (A, H) with A a w-ordinary false elliptic curves (with K^{p}-level structure) and H its pseudocanonical subgroup.

(2) Let q : X_{∞} → X denote the projection map. Then X_{∞,w} = q^{−1}(X_{w}).

(3) The sets (X_{w})_{w} form a cofinal set of open neighbourhoods of the closure of the ordinary-multiplicative locus in X.

Proof. Write, for any level K_{p} ⊆ GL_{2}(ℤ_{p}), q_{K_{p}} for the projection map X_{∞} → X. We prove (1) and (2) together. Indeed, we define X_{w} := q(X_{∞,w}). Since q is pro-étale, X_{w} is open and quasicompact ([Sch3], Lemma 3.10(iv)) and the characterization of the (C, O_{C})-points of X_{w} follows directly from the characterization of the (C, O_{C})-points of X_{∞,w}. Note that uniqueness is clear since quasicompact open subsets of quasiseparated rigid analytic varieties (viewed as adic spaces) are determined by their classical points (cf. [Sch1, Theorem 2.21]). This also finishes the proof (1), except that we need to show that X_{w} is affinoid.

By the definition of the inverse limit topology we may find m and a quasicompact open subset W ⊆ X_{K(p^{m})} such that q_{K(p^{m})}^{−1}(W) = X_{∞,w}. We claim that W = q_{K(p^{m})}(X_{∞,w}). Indeed, both sets are determined by their (C, O_{C})-points and q_{K(p^{m})} is surjective on (C, O_{C})-points. Thus it remains to see that if r is the natural map X_{K(p^{m})} → X, then r^{−1}(X_{w}) = W, which we can check on (C, O_{C})-points. This follows from the surjectivity of q_{K_{p}} on (C, O_{C})-points for arbitrary K_{p} and the fact that any trivialization that maps down to the pseudocanonical subgroup is automatically strict (in the sense that one/any lift to a trivialization of the whole Tate module is strict). This proves (2). Finally we finish the proof of (1) by showing that X_{w} is affinoid. To see this we use X_{∞,w} = q^{−1}(X_{w}). Since X_{∞,w} is a rational subset of V_{1} we may find some large N and an affinoid S ⊆ X_{K(p^{N})} such that X_{∞,w} = q_{K(p^{N})}^{−1}(S) (this follows from Theorem 2.8 and the fact that rational subsets come from finite level). Then X_{w} is the quotient of S by the finite group K_{0}(p)/K(p^{N}) and hence affinoid by Corollary 6.26.

We now prove (3). As a qualitative result it follows by general topology arguments using the constructible topology, cf. [Sch4, Lemma III.3.8]. Here we deduce a quantitative version using the Lubin-Katz theory of the canonical subgroup. For a statement of the results we need phrased in terms of the degree function see Proposition 1.2.8 of [Kas] (as before this reference treats elliptic curves but the results carry over verbatim). Without loss of generality we restrict our attention to w = 1, 2, 3, .... On X_{1}, H_{1} is the canonical subgroup and

deg(H_{1}) ≥ (p − 1)/p. Let h = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) ∈ GL_{2}(ℚ_{p}). Then h(X_{∞,w}) = X_{∞,w+1} for

all w. Moreover, if (A, α) ∈ X_{K(p^{∞}),1}(C, O_{C}) and (A, α).h = (A/H^{⊕2}, β), then H is anticanonical and hence of degree (1 − deg(H_{1}))/p. Hence deg(G[p]/H) = 1 − (1 − deg(H_{1}))/p > 1/2 and G[p]/H is the canonical subgroup. Write δ_{n} for the degree of the canonical subgroup of (h^{n}).A. Then we get the recurrence relation δ_{n+1} = (p − 1)/p + δ_{n}/p and from δ_{1} ≥ (p − 1)/p we deduce that

δ_{n} ≥ δ′_{n} := \frac{p − 1}{p} ∑_{i=0}^{n−1} \frac{1}{p^{i}} = 1 − \frac{1}{p^{n}} .

We have therefore proved that if (A, H) ∈ X_{n}(C, O_{C}) then deg(H) ≥ δ′_{n}. As δ′_{n} → 1 as n → ∞, the result follows since the loci {(A, H) | deg(H) ≥ r} for r ∈ (0, 1) form a cofinal set of open neighbourhoods of the closure of the ordinary-multiplicative locus (which is the locus {(A, H) | deg(H) = 1}) . □

2.4. The fundamental period and a non-vanishing section. Recall the coordinate z on ℙ^{1} defined earlier. The action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℚ_{p}) is

z.g = g^{∨}z = \frac{az + c}{bz + d}.

Below, whenever we have a matrix g ∈ GL_{2} we will use a, b, c, d as above to denote its entries. Note that z defines a function in H^{0}(V_{1}, O_{ℙ^{1}}^{+}) and by composing with π_{HT} we obtain a function 𝔷 ∈ H^{0}(V_{1}, O_{X_{∞}}^{+}) which we will call the fundamental period. We will use the same notation to denote its restriction to X_{∞,w} for any w > 0; by definition 𝔷 ∈ H^{0}(X_{∞,w}, pℤ_{p} + p^{w}O_{X_{∞}}^{+}). Note that if γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K_{0}(p) then

γ^{∗}𝔷 = \frac{a𝔷 + c}{b𝔷 + d}

as functions on X_{∞,w}. Next we wish to trivialize O(1) over S = {z ≠ ∞} ⊆ ℙ^{1} by defining a non-vanishing section. Everything we write in this section is standard but we repeat it since we will need some explicit formulas. The section we are after is algebraic so we will momentarily work in the realm of algebraic geometry. Explicitly it is given by the following formula: The line bundle O(1) has a geometric incarnation with total space (GL_{2} × 𝔸^{1})/B where B denotes the lower triangular Borel in GL_{2} and acts on 𝔸^{1} by g.x = d^{−1}x (this equivariant structure may differ from the canonical one, but it is the one corresponding to the Hodge-Tate sequence on X_{∞}). Global sections correspond to functions f : GL_{2} → 𝔸^{1} satisfying f(gh) = d(h)^{−1}f(g) for g ∈ GL_{2}, h ∈ B. There is a morphism of equivariant vector bundles

(GL_{2}/B) × 𝔸^{2} → (GL_{2} × 𝔸^{1})/B,

given by

(g, (\begin{matrix}x \\ y \end{matrix})) ↦ (g, dx − by) .

Here the left hand side has the GL_{2}-action g.(h, v) = (gh, (g^{∨})^{−1}v). Then the global section of (GL_{2}/B) × 𝔸^{2} given by the constant function g ↦ e_{2} maps to global section of O(1) given by the function

s : GL_{2} → 𝔸^{1};
s(g) = −b(g).

Note that s = 0 if and only g ∈ B so s is invertible on S. Let us now return to the rigid analytic world and let W ⊆ S be an open subset such h^{∨}(W) ⊆ W for some h ∈ GL_{2}(C). Then by direct calculation

(sh^{∨})(g) = d(g)b(h) − b(g)d(h).

Thus

\frac{s ∘ h^{∨}}{s} (g) = \frac{d(h)b(g) − b(h)d(g)}{b(g)}.

This is a function on W. Taking g = g_{z} = ( \begin{matrix}0 & −1 \\ 1 & z \end{matrix} ) we see that

\frac{s ∘ h^{∨}}{s} (z) = b(h)z + d(h).

We may then pull s back via π_{HT} to get (compatible) non-vanishing sections 𝔰 ∈ H^{0}(X_{∞,w}, ω) for all w satisfying

\frac{γ^{∗}𝔰}{𝔰} = b𝔷 + d

for all γ ∈ K_{0}(p). We remark that 𝔰 is one of the ”fake” Hasse invariants constructed in [Sch4].

2.5. A perfectoid definition of overconvergent modular forms. We will give definitions of sheaves of overconvergent modular forms with prescribed small or affinoid weight. Recall the forgetful morphism q : X_{∞,w} → X_{w} ⊆ X.

Definition 2.18. Let U be a weight and let w ∈ ℚ_{>0} be such that w ≥ 1 + s_{U}. We define a sheaf ω_{U,w}^{†} on X_{w} by

ω_{U,w}^{†}(U) = {f ∈ O_{X_{∞,w}}(q^{−1}(U))ˆ⊗A_{U} | γ^{∗}f = χ_{U}(b𝔷 + d)^{−1}f ∀γ ∈ K_{0}(p)}

where U ⊆ X_{w} is a qcqs open subset.

We remark that, since b𝔷 + d ∈ ℤ_{p}^{×}(1 + p^{w}O_{X_{∞,w}}^{+}), the assumption w ≥ 1 + s_{U} ensures that χ_{U}(b𝔷 + d) is well defined (by Proposition 2.6). Define

Δ_{0}(p) = \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(ℤ_{p}) ∩ GL_{2}(ℚ_{p}) | c ∈ pℤ_{p}, d ∈ ℤ_{p}^{×}\}.

This is a submonoid of GL_{2}(ℚ_{p}) containing K_{0}(p), and it stabilizes X_{∞,w} for all w. We can form a sheaf F on X_{∞,w} by F(U_{∞}) = O_{X_{∞,w}}(U_{∞})ˆ⊗A_{U} for U_{∞} ⊆ X_{∞,w} qcqs (not necessarily of the form q^{−1}(U)), and we may equip it with a Δ_{0}(p)-equivariant structure by the rule

γ ·_{U} − : F(U_{∞}) → F(γ^{−1}U_{∞});
γ ·_{U} f = χ_{U}(b𝔷 + d)γ^{∗}f

for γ ∈ Δ_{0}(p). Then ω_{U,w}^{†} = (q_{∗}F)^{K_{0}(p)} and we obtain an action of the double cosets K_{0}(p)γK_{0}(p) for γ ∈ Δ_{0}(p) on ω_{U,w}^{†}, given by the following standard formula:

[K_{0}(p)γK_{0}(p)] : ω_{U,w}^{†}(V) → ω_{U,w}^{†}(U);

[K_{0}(p)γK_{0}(p)].f = ∑_{i} χ_{U}(b_{i}𝔷 + d_{i})γ_{i}^{∗}f

where U ⊆ X_{w} is qcqs, V ⊆ X_{w} is the image of U under [K_{0}(p)γK_{0}(p)] viewed as a correspondence on X_{w}, and K_{0}(p)γK_{0}(p) = ∐_{i} γ_{i}K_{0}(p) is a coset decomposition. This gives us the Hecke action at p for ω_{U,w}^{†}. As in the complex case, we may also view this action as an action by correspondences. We give a few remarks on this for Hecke operators at primes ℓ ≠ p below; the action at p is similar. The statements analogous to Proposition 2.19 that are needed follow from Proposition 4.7.

For the Hecke actions at ℓ ≠ p, let us momentarily introduce the tame level into our notation, writing ω_{U,w,K^{p}}^{†} for the sheaf on X_{w,K^{p}} ⊂ X_{K_{0}(p)K^{p}}. The construction of Hecke operators is then immediate from the following proposition.

Proposition 2.19. (1) For any g ∈ G(𝔸_{f}^{p}) with g : X_{w,K^{p}} →^{∼} X_{w,g^{−1}K^{p}g} the associated isomorphism of Shimura curves, there is a canonical isomorphism g^{∗}ω_{U,w,g^{−1}K^{p}g}^{†} ≅ ω_{U,w,K^{p}}^{†}.

(2) For any inclusion K_{2}^{p} ⊂ K_{1}^{p} of tame levels with π : X_{w,K_{2}^{p}} → X_{w,K_{1}^{p}} the associated finite étale projection of Shimura curves, there is a canonical isomorphism

π_{∗}ω_{U,w,K_{2}^{p}}^{†} ≅ ω_{U,w,K_{1}^{p}}^{†} ⊗_{O_{X_{w,K{p \atop 1}}}} π_{∗}O_{X_{w,K{p \atop 2}}}

In particular there is a canonical O_{X_{w,K{p \atop 1}}} -linear trace map

π_{∗}ω_{U,w,K_{2}^{p}}^{†} → ω_{U,w,K_{1}^{p}}^{†}

Proof. (1) follows immediately from the definition of the sheaves together with the fact that g^{∗}𝔷 = 𝔷; the latter follows from [Sch4, Theorem IV.1.1(iv)].

For (2) we consider the pullback diagram

\xymatrix{X_{∞,w,K_{2}^{p}} \ar[r]^{q_{2}} \ar[d]^{π_{∞}} & X_{w,K_{2}^{p}} \ar[d]^{π} \\ X_{∞,w,K_{1}^{p}} \ar[r]^{q_{1}} & X_{w,K_{1}^{p}}}

of adic spaces. Given a rational subset U ⊂ X_{w,K_{1}^{p}}, we then have

O(q_{2}^{−1}π_{1}U) = O(π_{∞}^{−1}q_{1}^{−1}U)
≅ O(q_{1}^{−1}U)ˆ⊗_{O(U)}O(π^{−1}U)
≅ O(q_{1}^{−1}U) ⊗_{O(U)} O(π^{−1}U)

where in the final line we use the fact that O(π^{−1}U) is a finite projective O(U)-Banach module. Applying −ˆ⊗A_{U} and making use of Lemma 6.7, we get a canonical isomorphism

O(q_{2}^{−1}π_{1}^{−1}U)ˆ⊗A_{U} ≅ (O(q_{1}^{−1}U)ˆ⊗A_{U}) ⊗_{O(U)} O(π^{−1}U).

Since π_{∞}^{∗}𝔷 = 𝔷, this isomorphism is equivariant for the χ_{U}-twisted action of K_{0}(p). Passing to K_{0}(p)-invariants for the twisted action, the left-hand side then becomes ω_{U,w,K_{2}^{p}}^{†}(π^{−1}U) = (π_{∗}ω_{U,w,K_{2}^{p}}^{†})(U), while the right-hand side

becomes ω_{U,w,K_{1}^{p}}^{†}(U) ⊗_{O(U)} (π_{∗}O_{X_{w,K{p \atop 2}}} )(U), so (2) follows. □

We now define our spaces of overconvergent modular forms.

Definition 2.20. Let U be a weight and let w ∈ ℚ_{>0} be such that w ≥ 1 + s_{U}.

(1) We define the space of w-overconvergent modular forms M_{U}^{†,w} of weight

U by

M_{U}^{†,w} = H^{0}(X_{w}, ω_{U,w}^{†} ⊗_{O_{Xw}} Ω_{X_{w}}^{1}).

(2) We define the space of overconvergent modular forms M_{U,w}^{†} of weight

U by

M_{U}^{†} = \underrightarrow{lim}_{w} M_{U}^{†,w}.

The spaces M_{U}^{†,w} are ℚ_{p}-Banach spaces, and M_{U}^{†} is an LB-space. Using the functoriality of Ω_{X}^{1} one may define Hecke operators on M_{U,w}^{†} in the same way that we did for ω_{U,w}^{†}.

Remark 2.21. The reason we have formulated the definition in this way is because it fits naturally with our discussion of the overconvergent Eichler-Shimura map in §4-5. It would be more fitting with our desire to be “explicit” to consider the space

H^{0}(X_{w}, ω_{U+2,w}^{†})

where U + 2 is the weight (R_{U}, z ↦ χ_{U}(z)z^{2}). Note that this shift by 2 is forced upon us by our convention to associate k ∈ ℤ with the character z ↦ z^{k−2}. Since ω is trivialized over X_{∞,w} by 𝔰 one sees that ω_{U+2,w}^{†} ≅ ω_{U,w}^{†} ⊗_{O_{Xw}} ω^{2}. By the Kodaira-Spencer isomorphism Ω_{X}^{1} ≅ ω^{2} we then have that ω_{U+2,w}^{†} ≅ ω_{U,w}^{†} ⊗_{O_{Xw}} Ω_{X_{w}}^{1}. Thus the two definitions give the same spaces. However, it is well known that the Kodaira-Spencer isomorphism fails to be equivariant for the natural Hecke actions on both sides. Indeed it is customary in the theory of modular forms to renormalize the natural Hecke action on ω^{⊗k} by multiplying T_{ℓ} by ℓ^{−1} (more precisely one multiplies the action of any double coset by det^{−1}). The Kodaira-Spencer isomorphism is then Hecke-equivariant. Thus, by defining M_{U,w}^{†} the way we have we can use the natural Hecke actions; there is no need to normalize.

For a discussion of how the normalized Hecke action on ω^{⊗k} corresponds to the natural Hecke action on ω^{⊗k−2} ⊗_{O_{X}} Ω_{X}^{1} , see [FC, p. 257-258].

2.6. Locally projective of rank one. In this subsection we prove that ω_{U,w}^{†} is locally projective of rank one as a sheaf of O_{X_{w}} ˆ⊗A_{U}-modules.

First we prove a general lemma. Let A_{∞} be a uniform ℚ_{p}-Banach algebra equipped with an action of a profinite group G by continuous homomorphisms. Let A = A_{∞}^{G}. We record the following easy facts:

Proposition 2.22. A is a closed subalgebra of A_{∞}, hence carries an induced structure of a uniform ℚ_{p}-Banach algebra, and A^{∘} = (A_{∞}^{∘})^{G}.

We then have:

Lemma 2.23. Keep the above notations and assumptions.

(1) Let M be any profinite flat ℤ_{p}-module (in particular M could be a small ℤ_{p}-algebra). We equip M with the trivial G-action. Then (A_{∞} ˆ⊗M)^{G} = Aˆ⊗M.

(2) Let V be a Banach space over ℚ_{p} (in particular V could be a reduced affinoid ℚ_{p}-algebra topologically of finite type). Equip V with the trivial G-action. Then (A_{∞} ˆ⊗_{ℚ_{p}}V)^{G} = Aˆ⊗_{ℚ_{p}}V.

Proof. To prove (1), we choose a pseudobasis (e_{i})_{i∈I} of M and follow the computation in Proposition 6.4 to see that the natural map A^{∘} ˆ⊗M → A_{∞}^{∘} ˆ⊗M is the inclusion

∏_{i∈I} A^{∘}e_{i} ⊆ ∏_{i∈I} A_{∞}^{∘}e_{i}

with the action of G on the right hand side being coordinate-wise. Thus the natural map Aˆ⊗M → A_{∞} ˆ⊗M is the inclusion of bounded sequences in A indexed by I into bounded sequences in A_{∞} indexed by I, with the G-action on the latter being coordinate-wise. The desired statement now follows from the definition of A as A_{∞}^{G}. The proof of (2) is similar, using an orthonormal basis instead of a pseudobasis.

□

We record the following fact, which is certainly implicit in [Sch3] (our main reason for recording it is that some assumption on the sheaf F seems to be needed).

Lemma 2.24. Let Y be a rigid analytic variety, let G be a profinite group and assume that Y_{∞} ∈ Y_{proét} is a Galois G-cover of Y. Let U ∈ Y_{proét} be quasicompact and quasiseparated and set U_{∞} := U ×_{Y} Y_{∞}; this is a Galois G-cover of U. Let F be a sheaf on Y_{proét} that comes via pullback from Y_{ét}. Then F(U) = F(U_{∞})^{G}.

Proof. Recall that, by the definitions, we may find a system of open normal subgroups G_{j} of G such that G ≅ \underleftarrow{lim}_{j} G/G_{j}, and a compatible system of Galois G/G_{j}-covers Y_{j} of Y such that Y_{∞} = \underleftarrow{lim}_{j} Y_{j} is a pro-étale presentation. Pick a pro-étale presentation U = \underleftarrow{lim}_{i} U_{i}; we get U_{∞} = \underleftarrow{lim}_{i,j} U_{i} ×_{Y} Y_{j}. Note that U_{∞} is quasicompact and quasiseparated by [Sch3, Proposition 3.12(v)]. We may then compute

F(U_{∞})^{G} = (\underrightarrow{lim}_{i,j} F(U_{i} ×_{Y} Y_{j}))^{G}
= \underrightarrow{lim}_{i,j} (F(U_{i} ×_{Y} Y_{j})^{G})
= \underrightarrow{lim}_{i,j} F(U_{i}) = F(U).

Here we have used [Sch3, Lemma 3.16] for the first and fourth equality and that U_{i} ×_{Y} Y_{j} → U_{i} is a Galois G/G_{j}-cover for the third equality. For the second equality we use that F(U_{i} ×_{Y} Y_{j}) → F(U_{i′} ×_{Y} Y_{j′}) is injective for large enough i, since U_{i′} ×_{Y} Y_{j′} → U_{i} ×_{Y} Y_{j} is an étale cover for large enough i (by the definition of a pro-étale presentation), and that direct limits commute with taking invariants if the maps in the direct system are eventually injective. □

Remark 2.25. An elaboration of this argument allows one to deduce the full “Cartan–Leray spectral sequence”

E_{2}^{pq} = H_{cts}^{p}(G, H^{q}(U_{∞}, F)) ⟹ H^{p+q}(U, F)

from the Čech-to-derived functor spectral sequence, under the same assumptions on U and F (here H^{q}(U_{∞}, F) is given the discrete topology). This is implicit in [Sch3] and used repeatedly there (though it should be noted that it is the Čech-to-derived functor spectral sequence itself that is referred to as the Cartan–Leray spectral sequence in [Sch3], following SGA 4).

Lemma 2.26. Let K_{p} ⊆ GL_{2}(ℤ_{p}) be an open compact subgroup and let U ⊆ X_{K_{p}} be an open subset. Put U_{∞} = q_{K_{p}}^{−1}(U) ⊆ X_{∞}. Then O_{X_{Kp}}^{+} (U) = O_{X_{∞}}^{+}(U_{∞})^{K_{p}}, and hence O_{X_{Kp}} (U) = O_{X_{∞}}(U_{∞})^{K_{p}}.

Proof. We work on the pro-étale site of X_{K_{p}}. We may assume that U is quasicompact; the general case follows by gluing. U_{∞} → U is a perfectoid object of X_{K_{p},proét} which is Galois with group K_{p}, and Lemma 2.24 implies that (O_{X_{Kp}}^{+} /p^{m})(U) = (O_{X_{Kp}}^{+} /p^{m})(U_{∞})^{K_{p}} for all m. Taking the inverse limit

we get, by definition, that Ô_{X_{Kp}}^{+} (U) = Ô_{X_{Kp}}^{+} (U_{∞})^{K_{p}}. By [Sch3, Corollary

6.19] we have Ô_{X_{Kp}} (U) = O_{X_{Kp}} (U). It then follows from [Sch3, Lemma 4.2(ii),(v)] that Ô_{X_{Kp}}^{+} (U) = O_{X_{Kp}}^{+} (U), which finishes the proof (note that

Ô_{X_{Kp}}^{+} (U_{∞}) = O_{X_{∞}}^{+}(U_{∞}) ). □

Now let U ⊆ X_{w} be any rational subset, U_{n} its preimage in X_{K(p^{n})} and put U_{∞} = q^{−1}(U) ⊆ X_{∞,w}. Note that U_{n} → U is finite étale and so O_{X_{K(p^{n})}}(U_{n}) is a finite projective O_{X} (U)-module since O_{X} (U) is Noetherian. Suppose that ω|_{U} is free, and choose a nowhere vanishing section (i.e. generator) η_{U} ∈ H^{0}(U, ω). Define t_{U} ∈ O_{X_{∞}}(U_{∞}) by the equality

𝔰 = t_{U} · q^{∗}η_{U}.

Proposition 2.27. We have γ^{∗}t_{U} = (b𝔷 + d)t_{U} for any γ ∈ K_{0}(p), and t_{U} is a unit. Furthermore, for any m ∈ ℤ_{≥1} we may choose some large n = n(m) and elements

t_{U}^{(n)} ∈ 1 + p^{m}O_{X_{∞}}^{+}(U_{∞}),
s_{U,n} ∈ O_{X_{K(p^{n})}}(U_{n})^{×},

such that t_{U} = t_{U}^{(n)}s_{U,n}.

Proof. Since neither 𝔰 nor q^{∗}η_{U} vanish, t_{U} is a unit, and

(b𝔷 + d)t_{U}q^{∗}η_{U} = (b𝔷 + d)𝔰 = γ^{∗}𝔰 = (γ^{∗}t_{U})q^{∗}η_{U} since γ^{∗}q^{∗}η_{U} = q^{∗}η_{U}, hence γ^{∗}t_{U} = (b𝔷 + d)t_{U}. For the second sentence, first choose M, N ∈ ℤ_{≥0} such that |p^{M}| ≤ |p^{N}t_{U}| ≤ 1 (possible by quasicompactness of U_{∞} and invertibility of t_{U}). Since O_{X_{∞}}^{+}(U_{∞}) = (\underrightarrow{lim} O_{X_{K(p^{n})}}^{+}(U_{n}))^{∧} (padic completion) we may find an n = n(m) and s ∈ O_{X_{K(p^{n})}}^{+}(U_{n}) such that

p^{N}t_{U} − s ∈ p^{M+m}O_{X_{∞}}^{+}(U_{∞}). Then we set s_{U,n} = p^{−N}s and t_{U}^{(n)} = t_{U}/s_{U,n} and these do the job. □

This has several consequences - in particular, since s_{U,n} is fixed by K(p^{n}) and t_{U}^{(n)} is p-adically close to 1, the element χ_{U}(t_{U}^{(n)}) ∈ (O_{X_{∞}}(U_{∞})ˆ⊗A_{U})^{×} is welldefined (for m large enough) and satisfies γ^{∗}(χ_{U}(t_{U}^{(n)})) = χ_{U}(b𝔷 + d)χ_{U}(t_{U}^{(n)}) for all γ ∈ K(p^{n}). Recall that

ω_{U,w}^{†}(U) := {f ∈ O_{X_{∞}}(U_{∞})ˆ⊗A_{U} | γ^{∗}f = χ_{U}(b𝔷 + d)^{−1}f ∀γ ∈ K_{0}(p)} .

Thus, given any f ∈ ω_{U,w}^{†}(U), the element f · χ_{U}(t_{U}^{(n)}) ∈ O_{X_{∞}}(U_{∞})ˆ⊗A_{U} lies in (O_{X_{∞}}(U_{∞})ˆ⊗A_{U})^{K(p^{n})}, which is equal to O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U} by Lemma 2.23 and Lemma 2.26.

Furthermore, for any γ ∈ G_{n} := K_{0}(p)/K(p^{n}), we have

γ^{∗}(f · χ_{U}(t_{U}^{(n)})) = χ_{U}(b𝔷 + d)^{−1} · f · χ_{U}(γ^{∗}t_{U}^{(n)})
= f · χ_{U}(t_{U}^{(n)}) · χ_{U} (\frac{γ^{∗}t_{U}^{(n)}}{(b𝔷 + d)t_{U}^{(n)}} )

= f · χ_{U}(t_{U}^{(n)}) · χ_{U} (\frac{s_{U,n}}{γ^{∗}s_{U,n}} )
= f · χ_{U}(t_{U}^{(n)}) · χ_{U}(j_{U,n}(γ))^{−1}

where j_{U,n} is the cocycle

j_{U,n}(γ) : G_{n} → ℤ_{p}^{×} · (1 + p^{min(w,m)}O_{X_{K(p^{n})}}(U_{n})^{∘})
γ ↦ \frac{γ^{∗}s_{U,n}}{s_{U,n}} .

In summary, we find that the map f ↦ f · χ_{U}(t_{U}^{(n)}) defines an O_{X} (U)-module isomorphism of ω_{U,w}^{†}(U) onto the subspace of functions f_{0} ∈ O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U} fixed by the twisted action

γ : O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U} → O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U}
f_{0} ↦ χ_{U}(j_{U,n}(γ)) · γ^{∗}f_{0}

of G_{n}. Since G_{n} is a finite group, the usual idempotent

e_{n} = \frac{1}{|G_{n}|} ∑_{γ∈G_{n}} χ_{U}(j_{U,n}(γ))^{−1}γ

in the group algebra (O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U})[G_{n}] defines an O_{X} (U)ˆ⊗A_{U}-module splitting of the inclusion ω_{U,w}^{†}(U) ⊆ O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U}. This realizes ω_{U,w}^{†}(U) as a direct summand of a finite projective O_{X} (U)ˆ⊗A_{U}-module, and therefore ω_{U,w}^{†}(U) is finite projective over O_{X} (U)ˆ⊗A_{U} as desired. We may now prove the main result of this section:

Theorem 2.28. We have ω_{U,w}^{†} = Loc(ω_{U,w}^{†}(X_{w})) and ω_{U,w}^{†}(X_{w}) is a finite projective O_{X} (X_{w})ˆ⊗A_{U}-module of rank 1. Moreover, ω_{U,w}^{†} is étale locally free.

Proof. To prove that ω_{U,w}^{†} = Loc(ω_{U,w}^{†}(X_{w})) and that it is locally projective of finite rank it suffices, by Theorem 6.20, to prove that ω_{U,w}^{†} is a coherent O_{X} ˆ⊗A_{U}-module (it is then locally projective by the above). To do this, we work locally using the U above. We wish to show that for V ⊆ U (without loss of generality assume V rational) the natural map

(O_{X} (V)ˆ⊗A_{U}) ⊗_{(O_{X} (U)ˆ⊗A_{U})} ω_{U,w}^{†}(U) → ω_{U,w}^{†}(V)

is an isomorphism. To see this, note that by applying Lemma 6.7 twice the natural map

(O_{X} (V)ˆ⊗A_{U}) ⊗_{(O_{X} (U)ˆ⊗A_{U})} (O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U}) → O_{X_{K(p^{n})}}(V_{n})ˆ⊗A_{U}

is an isomorphism (note that O_{X} (V)⊗_{O_{X} (U)}O_{X_{K(p^{n})}}(U_{n}) ≅ O_{X_{K(p^{n})}}(V_{n})). This isomorphism matches up the idempotents on both sides, giving us the desired isomorphism. To prove that the rank is one note that O_{X} (U) → O_{X_{K(p^{n})}}(U_{n}) is Galois with group G_{n} and by applying Lemma 6.7 twice we may deduce that O_{X} (U)ˆ⊗A_{U} → O_{X_{K(p^{n})}}(U_{n})ˆ⊗A_{U} is also Galois with group G_{n}. Moreover, the twisted action whose invariants give ω_{U,w}^{†}(U) is a Galois descent datum, so we see that ω_{U,w}^{†}(U) is the descent of a rank 1 free module, hence rank 1 projective as desired. Finally, note that this also proves that ω_{U,w} becomes trivial over U_{n}, which gives the final statement of the theorem. □

The techniques of this proof also yield the following result.

Lemma 2.29. Let U = (A_{U}, χ_{U}) be a weight, and let i : A_{U} → A_{Z} be a surjection such that Z = (A_{Z}, χ_{Z} = i ∘ χ_{U}) is also a weight and such that ker(i) is generated by a regular element x ∈ A_{U}. Then we have a natural exact sequence of sheaves

0 → ω_{U,w}^{†} →^{·x} ω_{U,w}^{†} → ω_{Z,w}^{†} → 0

on X_{w}, and an exact sequence of global sections

0 → H^{0}(X_{w}, ω_{U,w}^{†}) →^{·x} H^{0}(X_{w}, ω_{U,w}^{†}) → H^{0}(X_{w}, ω_{Z,w}^{†}) → 0.

Proof. There is certainly a (not necessarily exact) sequence of sheaves

0 → ω_{U,w}^{†} →^{·x} ω_{U,w}^{†} → ω_{Z,w}^{†} → 0,

and we check exactness of this sequence on a basis of suitably small open subsets U ⊂ X_{w} as in the proof of Theorem 2.28. By assumption we have a short exact sequence

0 → A_{U} →^{·x} A_{U} → A_{Z} → 0

of A_{U}-modules. Tensoring this sequence over A_{U} with A_{U} ˆ⊗O_{X_{K(p^{n})}}(U_{n}), we obtain by the flatness of A_{U} ˆ⊗O_{X_{K(p^{n})}}(U_{n}) over A_{U} (proved in the same way as Lemma 6.13(2)) a short exact sequence

0 → A_{U} ˆ⊗O_{X_{K(p^{n})}}(U_{n}) →^{·x} A_{U} ˆ⊗O_{X_{K(p^{n})}}(U_{n}) → A_{Z} ˆ⊗O_{X_{K(p^{n})}}(U_{n}) → 0.

Applying the idempotent e_{n} as above gives a short exact sequence

0 → ω_{U,w}^{†}(U) →^{·x} ω_{U,w}^{†}(U) → ω_{Z,w}^{†}(U) → 0

as desired. Taking cohomology, we note that H^{1}(X_{w}, ω_{U,w}^{†}) = 0 by Theorem 2.28 and Proposition 6.16, and the lemma follows. □

2.7. Comparison with other definitions of overconvergent modular forms. In this section we take V = (S_{V}, χ_{V}) to be an open affinoid weight, and we will compare our definition of overconvergent modular forms of weight V with those in the literature (all known to be equivalent). More specifically we will compare it to that of [Pil], trivially modified to our compact Shimura curves.

We now recall the definition of the ”Pilloni torsor”. This is the object denoted by F_{n}^{×} in [Pil] but we will use the notation T (n, v). For any n ≥ 1 and any v < \frac{p−1}{p^{n}}, T (n, v) is a rigid space equipped with a smooth surjective morphism pr : T (n, v) → X (v), where X (v) ⊆ X is the locus where the Hodge height is ≤ v. For any point x ∈ X (v)(C, O_{C}), the (C, O_{C})-points in the fiber pr^{−1}(x) consists of the differentials η ∈ ω_{G_{A}} that reduce to an element in the image of the Hodge-Tate map (H_{n}^{∨})^{gen} → ω_{H_{n}}, where H_{n} is the canonical subgroup of G of level n and (H_{n}^{∨})^{gen} ⊆ H_{n}^{∨} denotes the subset of generators. The set pr^{−1}(x) is a torsor for the group ℤ_{p}^{×}(1 + p^{n− \frac{p^{n}−1}{p−1} Hdg(x)}O_{C}), and we think of T (n, v) as an open subspace of T , where T is the total space of e(0^{∗}Ω_{A^{univ}/X (v)}^{1}) (here 0 : X (v) → A^{univ} is the zero section; this is a line bundle on X (v)). We remark that the canonical action of ℤ_{p}^{×} on T preserves T (n, v) (as should be clear from the description of the fibres above). This induces an action on functions.

If we fix w = n − \frac{p^{n}−1}{p−1}v, then X (v) ⊆ X_{w}; this follows for example from [AIP, Proposition 3.2.1]. We will compare our construction of w-overconvergent modular forms over X (v) to the notion defined in [Pil]. For the purpose of the comparison, a Pilloni form of weight V over a quasi-compact open U ⊆ X (v) is defined to be an element f ∈ O_{T (n,v)}(pr^{−1}(U))ˆ⊗_{ℚ_{p}}S_{V} (i.e a function on pr^{−1}(U) × V ⊆ T (n, v) × W. Here we abuse notation and write V also for the image of the natural map into W induced by V) such that z.f = χ_{V}^{−1}(z)f for all z ∈ ℤ_{p}^{×}.

Now fix U ⊆ X (v) as used in the previous section, such that we have a nowhere vanishing η_{U} ∈ ω(U). We freely use the notation of the previous section with one exception: we use r instead of the already used letter n. Thus we have functions t_{U}^{(r)} ∈ 1 + p^{m}O_{X_{∞}}^{+}(U_{∞}) and s_{U,r} ∈ O_{X_{K(p^{r})}}(U_{r})^{×} such that

𝔰 = t_{U}^{(r)}s_{U,r}η_{U}.

Let T_{U,r}(n, v) = T (n, v) ×_{U} U_{r}. It is an open subset of T_{U,r} := T ×_{U} U_{r}. It inherits commuting actions of G_{r} := K_{0}(p)/K(p^{r}) and ℤ_{p}^{×}.

Lemma 2.30. Assume that m ≥ n and r ≥ n. Then s_{U,r}η_{U} trivializes T_{U,r}(n, v).

Proof. We think of s_{U,r}η_{U} as a section U_{r} → T_{U,r} and we wish to show that the image lands inside T_{U,r}(n, v), for which it is enough to argue on geometric points. Take a (C, O_{C})-point x̄ and lift it to a (C, O_{C})-point x of X_{∞}. Then we see that

(s_{U,r}η_{U})(x̄) = ((t_{U}^{(r)})^{−1}𝔰)(x).

If x̄ = (A, ᾱ) ∈ Y_{K(p^{r})}^{gd} we put x = (A, α). Then 𝔰(A, α) = HT_{A}(α(e_{2})) by definition and this maps to a generator of H_{n}^{∨} via the canonical map G[p^{n}] ≅ G[p^{n}]^{∨} → H_{n}^{∨}, so 𝔰(A, α) lies in the fibre of T_{U,r}(n, v) over (A, ᾱ). This proves that 𝔰(x) lies in the fibre of T_{U,r}(n, v) over x̄ for all x. The result follows since t_{U}^{(r)} is small. □

The assumption m ≥ n and r ≥ n will be in force throughout the rest of this section so that the Lemma applies. Recall what we proved in the process of proving Theorem 2.28: w-overconvergent modular forms f of weight V over U identifies, via the map f ↦ f_{0} := χ_{V}(t_{U}^{(r)})f, with functions f_{0} ∈ O_{X (p^{r})}(U_{r})ˆ⊗_{ℚ_{p}}S_{V} such that

γ^{∗}f_{0} = χ_{V}(j_{U,r}(γ))^{−1}f_{0}

for all γ ∈ G_{r}.

Proposition 2.31. (1) The space of Pilloni forms of weight V over U is isomorphic, via pullback, to the space of functions g on T_{U,r}(n, v) × V such that z.g = χ_{V}(z)^{−1}g for all z ∈ ℤ_{p}^{×} and γ^{∗}g = g for all γ ∈ G_{r}.

(2) The space of w-overconvergent modular forms of weight V over U is isomorphic to the space of functions g_{0} on T_{U,r}(n, v) × V such that z.g_{0} = g_{0} for all z ∈ ℤ_{p}^{×} and γ^{∗}g_{0} = χ_{V}(j_{U,r}(γ))^{−1}g_{0} for all γ ∈ G_{r}. The isomorphism is the map f ↦ f_{0} above composed with pullback from U_{r} × V to T_{U,r}(n, v) × V.

Proof. (1) follows from the fact that pr^{−1}(U)×V is the quotient of T_{U,r}(n, v)×V by G_{r}. The proof of (2) is a similar descent. □

To identify the two spaces, it remains to go from one kind of function on T_{U,r}(n, w) × V to the other. The key lies in trivializing the cocycle χ_{V}(j_{U,r}(γ)). Note that s_{U,r}η_{U}, as a nowhere vanishing section of T_{U,r} also canonically defines a function on T_{U,r} minus the zero section which we will denote by (s_{U,r}η_{U})^{∨}. The same applies to η_{U} itself and we have (s_{U,r}η_{U})^{∨} = s_{U,r}^{−1}η_{U}^{∨}. By restriction we obtain functions on T_{U,r}(n, v).

Lemma 2.32. (s_{U,r}η_{U})^{∨} ∈ ℤ_{p}^{×}.(1 + p^{w}O_{T_{U,r}}^{+}(T_{U,r}(n, v))).

Proof. This follows directly from Lemma 2.30. □

This implies that we may apply χ_{V} to (s_{U,r}η_{U})^{∨}, and we may therefore define a O_{X (v)}(U)ˆ⊗_{ℚ_{p}}S_{V}-module isomorphism

Φ : O_{T_{U,r}}(T_{U,r}(n, v))ˆ⊗_{ℚ_{p}}S_{V} → O_{T_{U,r}}(T_{U,r}(n, v))ˆ⊗_{ℚ_{p}}S_{V}

which is simply multiplication by χ_{V}((s_{U,r}η_{U})^{∨}):

Φ(h) = χ_{V}((s_{U,r}η_{U})^{∨})h.

Theorem 2.33. The image of the space of functions in part (1) of Proposition 2.31 under Φ is the space of functions in part (2) of Proposition 2.31. Moreover, the induced isomorphism between the space of w-overconvergent modular forms of weight V over U and the space of Pilloni forms of weight V over U is independent of all choices and hence the isomorphisms for varying U glue together to an isomorphism of sheaves over X_{w}.

Proof. For the first part, the key thing to notice is that

j_{U,r}(γ) = \frac{(s_{U,r}η_{U})^{∨}}{γ^{∗}(s_{U,r}η_{U})^{∨}}

and hence that

χ_{V}(j_{U,r}(γ))^{−1} = \frac{χ_{V}(γ^{∗}(s_{U,r}η_{U})^{∨})}{χ_{V}((s_{U,r}η_{U})^{∨})}.

The rest is then a straightforward computation. For the second part we remark that the composite isomorphism from w-overconvergent modular forms to Pilloni forms is given by

f ↦ χ_{V}(t_{U}^{(r)})χ_{V}((s_{U,r}η_{U})^{∨})^{−1}f

and then descending the right hand side to pr^{−1}(U)×V. Morally, the right hand side is equal to χ_{V}(𝔰^{∨})^{−1}f (recall that t_{U}^{(r)}s_{U,r}η_{U} = 𝔰) and hence independent of the choices made. To turn this into a rigorous argument is straightforward but tedious and notationally cumbersome. We leave the details to the interested reader. □

3. Overconvergent modular symbols

We will recall some material on overconvergent modular symbols in the form we need. Most of these constructions are probably well known with the exception of certain filtrations defined in [Han2].

3.1. Basic definitions and the filtrations. Let A^{s} be the affinoid ring over ℚ_{p} defined by

A^{s} = {f : ℤ_{p} → ℚ_{p} | f analytic on each p^{s}ℤ_{p} − coset}.

We let A^{s,∘} denote the subring of powerbounded elements of A^{s}. Given a weight U, consider the module

A_{U}^{s,∘} = A^{s,∘} ˆ⊗_{ℤ_{p}}A_{U}^{∘}

where the completion is the p-adic completion. Recall the Amice basis (e_{j}^{s})_{j≥0} of A^{s,∘} from Theorem 2.5. Using it we may write A_{U}^{s,∘} as

A_{U}^{s,∘} = ˆ⨁_{j≥0}A_{U}^{∘}e_{j}^{s}

and hence view elements of A_{U}^{s,∘} as functions ℤ_{p} → A_{U}^{∘}. For any s ≥ s_{U} and any d ∈ ℤ_{p}^{×}, c ∈ pℤ_{p}, x ↦ χ_{U}(cx+d) defines an element of A_{U}^{s,∘} (by calculations very similar to those in the proof of Proposition 2.6), and we then consider A_{U}^{s,∘} endowed with the right Δ_{0}(p)-action

(f ·_{U} γ)(x) = χ_{U}(cx + d)f (\frac{ax + b}{cx + d});

one checks without too much trouble that f ·_{U} γ ∈ A_{U}^{s,∘}. We set D_{U}^{s,∘} = Hom_{A_{U}^{∘}} (A_{U}^{s,∘}, A_{U}^{∘}) and D_{U}^{s} = D_{U}^{s,∘}[\frac{1}{p}], with the dual left action. It is the continuous A_{U}[\frac{1}{p}]-dual of the A_{U}[\frac{1}{p}]-Banach module A_{U}^{s} := A_{U}^{s,∘}[\frac{1}{p}]. Note that

D_{U}^{s,∘} = Hom_{ℤ_{p}}(A^{s,∘}, A_{U}^{∘}).

The Amice basis gives an orthonormal A_{U}[\frac{1}{p}]-basis of A_{U}^{s} and induces an isomorphism

D_{U}^{s} ˜ ⟶ ∏_{j≥0} A_{U}^{∘}

given by

μ ↦ (μ(e_{j}^{s}))_{j≥0}.

Proposition 3.1. Let U = (R_{U}, χ_{U}) be a small weight and let s ≥ 1 + s_{U}. (1) D_{U}^{s,∘} admits a decreasing Δ_{0}(p)-stable filtration by sub-R_{U}-modules

D_{U}^{s,∘} = Fil^{0}D_{U}^{s,∘} ⊃ Fil^{1}D_{U}^{s,∘} ⊃ ⋯ ⊃ Fil^{i}D_{U}^{s,∘} ⊃ ⋯

such that each quotient D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘} is a finite abelian group of exponent p^{k}, the group K(p^{s+k}) acts trivially on D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}, and D_{U}^{s,∘} ≅ \underleftarrow{lim}_{k} D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}.

(2) D_{U}^{s,∘}, with the topology induced by the submodules (Fil^{k}D_{U}^{s,∘})_{k≥0}, is a profinite flat ℤ_{p}-module.

Proof. We will only recall how the filtrations are constructed and refer to [Han2, §2.2] for the details (note that (2) is a consequence of (1)). The module Fil^{k}D_{U}^{s,∘} is defined to be the kernel of the natural map

D_{U}^{s,∘} → D_{U}^{s−1,∘}/𝔞_{U}^{k}D_{U}^{s−1,∘},

where we recall that 𝔞_{U} is our fixed choice of ideal of definition for the profinite topology on R_{U}. □

Let k ≥ 2 be an integer and let A be a ring. We let ℒ_{k}(A) denote the module of polynomials A[X]^{deg≤k−2} with left M_{2}(A)-action

(δ ·_{k} p)(X) = (d + bX)^{k−2}p (\frac{c + aX}{d + bX} ) ,

and set in particular ℒ_{k} = ℒ_{k}(ℚ_{p}) and ℒ_{k}^{∘} = ℒ_{k}(ℤ_{p}). By direct calculation, the map

ρ_{k} : D_{k}^{s,∘} → ℒ_{k}^{∘}
μ ↦ ∫ (1 + Xx)^{k−2}μ(x) =

= ∑_{j=0}^{k−2} ( \begin{matrix}k − 2 \\ j \end{matrix} ) μ(x^{j})X^{j}

is Δ_{0}(p)-equivariant.

Definition 3.2. The integration map in weight k, denoted i_{k}, is the Δ_{0}(p)- equivariant map i_{k} : D_{U}^{s,∘} → ℒ_{k}^{∘} defined by i_{k} = ρ_{k} ∘ σ_{k}.

3.2. Slope decompositions. We will recall material from [AS] and [Han1] in order to define slope decompositions on the spaces of overconvergent modular symbols that we are interested in. We will mildly abuse notation by writing X(ℂ) for the complex Shimura curve of level K := K^{p}K_{0}(p) (viewed as a Riemann surface). Any K_{0}(p)-module M defines a local system on X(ℂ) which we will also denote by M. If M in addition is a Δ_{0}(p)-module then we get induced Hecke actions as well. The spaces of overconvergent modular symbols that we are interested in are

H^{1}(X(ℂ), D_{U}^{s})

for open weights U. We will give these spaces slope decompositions using the methods of [Han1]. Almost everything goes through verbatim and we will content ourselves with a brief discussion. Throughout this section, h will denote a non-negative rational number. For definitions and generalities on slope decompositions that we will use we refer to [JN, §2.2] (we remark that the notion introduced in [AS, §4] needs slight tweaking for the purpose of constructing the whole eigencurve, as opposed to just local pieces).

Let us start by recalling the general setup from [Han1, §2.1]. We have the functorial adelic (co)chain complexes C_{•}^{ad}(K, −) and C_{ad}^{•}(K, −) whose (co)homology functorially computes homology and cohomology of local systems attached to Δ_{0}(p)-modules, respectively. Fix once and for all a choice of triangulation of X(ℂ). This choice gives us functorial complexes C_{•}(K, −) and C^{•}(K, −), called Borel-Serre complexes, which are chain homotopic to C_{•}^{ad}(K, −) and C_{ad}^{•}(K, −) respectively. We fix such a chain homotopy once and for all. The key features of the Borel-Serre complexes are that there are non-negative integers r(i) such that r(i) = 0 for i < 0 and for i sufficiently large, and such that for any R[Δ_{0}(p)]-module M (where R is any commutative ring),

C_{i}(K, M) ≅ M^{r(i)}

functorially as R-modules (and similarly for C^{•}(K, −), with the same integers r(i)). Therefore the total complex ⨁_{i} C_{i}(K, M) inherits properties of M, such as being Banach if R is a ℚ_{p}-Banach algebra (and also orthonormalizability). In this case, we have a canonical and functorial topological duality isomorphism

C^{•}(K, Hom_{R,cts}(M, P)) ≅ Hom_{R,cts}(C_{•}(K, M), P)

where M and P are Banach R-modules.

We now follow [Han1, §3.1] using the orthonormalizable A_{U}[\frac{1}{p}]-module A_{U}^{s}, where U = (A_{U}, χ_{U}) is an open weight. Our U_{p}-operator is given by the double coset K_{0}(p) ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) K_{0}(p) and the formalism gives us lifts to our Borel-Serre complexes that we denote by Ũ. We note that Ũ ∈ End_{A_{U} [ \frac{1}{p}]}(C_{•}(K, A_{U}^{s})) is compact and we denote its Fredholm determinant by F_{U}(X) ∈ A_{U}[\frac{1}{p}][[X]]. The proof of [Han1, Proposition 3.1.1] goes through for small weights to show that this definition is independent of s. Thus the existence of a slope ≤ hdecomposition of C_{•}(K, A_{U}^{s}) is equivalent to the existence of a slope ≤ hfactorization of F_{U}(X). If V is another open weight with V^{rig} ⊆ U^{rig}, then the relation A_{V}^{s} ≅ A_{U}^{s} ˆ⊗_{A_{U} [ \frac{1}{p}]}A_{V}[\frac{1}{p}] implies that the F_{U} glue to a power series F (X) ∈ O_{𝒲} (𝒲 )[[X]]. Also [Han1, Proposition 3.1.2] goes through in this more general setting: the slope ≤ h-subcomplex C_{•}(K, A_{U}^{s})_{≤h} of C_{•}(K, A_{U}^{s}) is independent of s (if it exists) and if V^{rig} ⊆ U^{rig} then we have a canonical isomorphism

C_{•}(K, A_{U}^{s})_{≤h} ⊗_{A_{U} [ \frac{1}{p}]} A_{V}[\frac{1}{p}] ≅ C_{•}(K, A_{V}^{s})_{≤h}.

We say that (U, h) is a slope datum if C_{•}(K, A_{U}^{s}) has a slope ≤ h-decomposition or equivalently if F_{U} has a slope ≤ h-factorization. We have the following version of [Han1, Proposition 3.1.3]:

Proposition 3.3. Assume that (U, h) is a slope datum and that V = (S_{V}, χ_{V}) is an open affinoid weight with V ⊆ U^{rig}. Then there is a canonical isomorphism

H_{∗}(X(ℂ), A_{U}^{s})_{≤h} ⊗_{A_{U} [ \frac{1}{p}]} S_{V} ≅ H_{∗}(X(ℂ), A_{V}^{s})_{≤h}

for any s ≥ s_{U}.

Proof. When U is affinoid this is exactly [Han1, Proposition 3.1.3], so assume without loss of generality that U = (A_{U}, χ_{U}) is small. The same proof will go through if we can verify that A_{U}[\frac{1}{p}] → S_{V} is flat. This is standard. One proof goes as follows: Consider the adic space Spa(A_{U}, A_{U}) and pick a rational subset V in its generic fibre which contains V. Then O(V) → S_{V} is flat since it corresponds to an open immersion of affinoids in rigid geometry, and A_{U} → O(V) is flat since it is a rational localization in the theory of adic spaces when the rings of definition are Noetherian. □

We may then give the analogue of the above proposition for H^{∗}(X(ℂ), D_{U}^{s}) (cf. [Han1, Proposition 3.1.5]).

Proposition 3.4. Assume that (U, h) is a slope datum. Then C^{•}(K, D_{U}^{s}), and hence H^{∗}(X(ℂ), D_{U}^{s}), admit slope ≤ h-decompositions. If furthermore V ⊆ U^{rig} is an open affinoid weight, then there are canonical isomorphisms

C^{•}(K, D_{U}^{s})_{≤h} ⊗_{A_{U} [ \frac{1}{p}]} S_{V} ≅ C^{•}(K, D_{V}^{s})_{≤h}

and

H^{∗}(X(ℂ), D_{U}^{s})_{≤h} ⊗_{A_{U} [ \frac{1}{p}]} S_{V} ≅ H^{∗}(X(ℂ), D_{V}^{s})_{≤h}.

Proof. Using the duality C^{•}(K, D_{U}^{s}) ≅ Hom_{A_{U} [ \frac{1}{p}],cts}(C_{•}(K, A_{U}^{s}), A_{U}[\frac{1}{p}]) and the flatness of A_{U}[\frac{1}{p}] → S_{V} the proof of [Han1, Proposition 3.1.5] goes through verbatim. □

4. Sheaves on the pro-étale site

In the next two sections we will often consider Shimura curves over ℂ_{p} as well as ℚ_{p}. Any rigid analytic variety we have defined may be base changed from ℚ_{p} to ℂ_{p}, and we will denote this base change by a subscript −_{ℂ_{p}}, e.g. X_{ℂ_{p}}. The space X_{ℂ_{p}} may be considered as an object of X_{proét} using the pro-étale presentation X_{ℂ_{p}} = \underleftarrow{lim}_{K} X_{K}, where K/ℚ_{p} is finite and X_{K} denotes the base change of X to K. Note that the slice X_{proét}/X_{ℂ_{p}} is equivalent to X_{ℂ_{p},proét} as sites (use [Sch3, Proposition 3.15] and [Sch1, Proposition 7.4]). We define X_{∞} := \underleftarrow{lim}_{K} X_{∞,K} (with K as above) where the inverse limit is taken in the category of perfectoid spaces. Equivalently, we may define it as (the perfectoid space corresponding to) the perfectoid object \underleftarrow{lim}_{K} X_{∞,K} in X_{proét}; it lives in X_{proét}/X_{ℂ_{p}} ≅ X_{ℂ_{p},proét}. Similar remarks apply to X_{w}, X_{∞,w} et cetera.

4.1. A handy lemma. Let X be a rigid analytic variety over Spa(ℚ_{p}, ℤ_{p}), G a profinite group, and X_{∞} a perfectoid space with X_{∞} → X a pro-étale G-covering of X, i.e. X_{∞} ∼ \underleftarrow{lim}_{j} X_{j} where (X_{j})_{j∈J} is an inverse system of rigid analytic varieties finite étale over X equipped with a right action of G, such that each X_{j} → X is a finite étale G/G_{j}-covering for G_{j} ⊆ G a cofinal sequence of normal open subgroups. We can and often will view X_{∞} as an object in the pro-étale site of X. Let M be a profinite flat ℤ_{p}-module equipped with a continuous left G-action. Define a sheaf on the pro-étale site of X by

(M ˆ⊗O_{X})(V) = (M ˆ⊗Ô_{X}(V ×_{X} X_{∞}))^{G} ,

where V is a qcqs object in X_{proét}. Here the right action of G on the tower (X_{i})_{i} induces a left action on Ô_{X}(V_{∞}), and the action indicated is the diagonal left action g · (m ⊗ f) = gm ⊗ g^{∗}f.

Lemma 4.1. Given any presentation M = \underleftarrow{lim}_{i} M_{i}, M_{i} = M/I_{i} as in Definition 6.3 with each I_{i} preserved by G, the sheaf M ˆ⊗O_{X} coincides with the sheaf

ℚ̂_{p} ⊗ℤ̂_{p} \underleftarrow{lim}_{i} (ν^{∗}M̃_{i} ⊗_{ℤ_{p}} O_{X}^{+}) ,

where M̃_{i} is the locally constant sheaf on X_{ét} associated with M_{i}, and we recall that ν : X_{proét} → X_{ét} is the canonical morphism of sites.

Proof. We may check this on affinoid perfectoid V ∈ X_{proét} since these form a basis of X_{proét} (and are qcqs). Set V_{∞} = V ×_{X} X_{∞}, this is also affinoid perfectoid (see e.g. the proof of [Sch3, Lemma 4.6]). Each M_{i} is finite abelian group of p-power exponent. The key observation is then that there is an almost equality

M_{i} ⊗_{ℤ_{p}} Ô_{X}^{+}(V_{∞}) =^{a} (ν^{∗}M̃_{i} ⊗_{ℤ_{p}} O_{X}^{+})(V_{∞})

using [Sch3, Lemma 4.10]. Taking G-invariants we see that, since V_{∞} → V is a G-cover,

(M_{i} ⊗_{ℤ_{p}} Ô_{X}^{+}(V_{∞}))^{G} =^{a} (ν^{∗}M̃_{i} ⊗_{ℤ_{p}} O_{X}^{+})(V)

by Lemma 2.24. Now take inverse limits and invert p, and use that these operations commute with taking G-invariants. □

4.2. Sheaves of overconvergent distributions. We introduce certain pro-étale sheaves that compute overconvergent modular symbols and carry a Galois action. When 𝕃 is a locally constant constructible sheaf on the étale site of a rigid analytic variety, we may pull it back to a sheaf ν^{∗}𝕃 on the proétale site. We will often abuse notation and denote ν^{∗}𝕃 by 𝕃 as well; by [Sch3, Corollary 3.17] there is little harm in this. For the rest of this section we will let U = (R_{U}, χ_{U}) be a small weight. Note that the K_{0}(p)-modules D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘} define locally constant constructible sheaves on the étale site X_{ét}.

Definition 4.2. Let U be a small weight and let s ≥ 1 + s_{U}. Set

V_{U}^{s,∘} = \underleftarrow{lim}_{k} H_{ét}^{1}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘})
= \underleftarrow{lim}_{k} H_{proét}^{1}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘})

and

V_{U,O_{ℂp}}^{s,∘} = \underleftarrow{lim}_{k} (H_{proét}^{1}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{ℂ_{p}}) .

We also define V_{U}^{s} = V_{U}^{s,∘}[\frac{1}{p}] and V_{U,ℂ_{p}}^{s} = V_{U,O_{ℂp}}^{s,∘} [\frac{1}{p}].

These objects carry natural global Galois actions. We have the following comparison:

Proposition 4.3. There is a canonical Hecke-equivariant isomorphism V_{U}^{s} ≅ H^{1}(X(ℂ), D_{U}^{s}).

Proof. Artin’s comparison theorem between étale and singular cohomology gives us canonical Hecke-equivariant isomorphisms

H_{ét}^{1}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ≅ H^{1}(X(ℂ), D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘})

for all k. Now take inverse limits and invert p. For this to do the job we need to be able to commute the inverse limit and the H^{1} on the right hand side. But we may do this since the relevant higher inverse limits vanish, by finiteness of D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘} and by applying [Sch3, Lemma 3.18]. □

We now introduce a sheaf on X_{proét} that computes overconvergent modular symbols.

Proposition 4.4. Let U be a small weight. There is a canonical Hecke- and Galois-equivariant isomorphism

V_{U,ℂ_{p}}^{s} ≅ H_{proét}^{1}(X_{ℂ_{p}}, OD_{U}^{s})

where OD_{U}^{s} is the sheaf on X_{proét} defined by

OD_{U}^{s} := (\underleftarrow{lim}_{k} ((D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{X_{ℂp}}^{+} )) [\frac{1}{p}] .

Proof. By [Sch3, Theorem 5.1]), we have an almost isomorphism

H_{proét}^{1}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{ℂ_{p}} ≅^{a} H_{proét}^{1}(X_{ℂ_{p}}, (D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{X_{ℂp}}^{+} )

of O_{ℂ_{p}}-modules. Passing to the inverse limit over k, note that on one hand, the higher inverse limits of any H_{proét}^{i}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘})⊗_{ℤ_{p}} O_{ℂ_{p}} vanish since H_{proét}^{i}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) is finite (by the finiteness properties of Fil^{•}D_{U}^{s,∘}), thus H_{proét}^{i}(X_{ℂ_{p}}, D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{ℂ_{p}} satisfy the Mittag-Leffler condition. On the other hand the higher inverse limit

R^{i} \underleftarrow{lim}_{k} ((D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} O_{X}^{+})

vanishes for i > 0. This follows from [Sch3, Lemma 3.18] upon noting that (in the notation of Lemma 4.1, with X = X_{ℂ_{p}}, X_{∞} = X_{∞,ℂ_{p}} and G = K_{0}(p)) objects of the form V_{∞} give a basis of X_{ℂ_{p},proét} satisfying the hypotheses of that lemma. Thus we may commute the inverse limit with taking cohomology on the right hand side. Inverting p we get the result. □

Let V = \underleftarrow{lim}_{i} V_{i} → X be a pro-étale presentation of an affinoid perfectoid object of X_{proét}, and let V_{∞} = V ×_{X} X_{∞}. This is still affinoid perfectoid and V_{∞} → V is pro-étale. The following is immediate from Lemma 4.1:

Lemma 4.5. Let U be a small weight. The sheaf OD_{U}^{s} admits the following explicit description on qcqs V ∈ X_{proét}:

OD_{U}^{s}(V) = (D_{U}^{s,∘} ˆ⊗Ô_{X} (V_{∞}))^{K_{0}(p)} .

We remark that the sheaf V_{∞} ↦ D_{U}^{s,∘} ˆ⊗Ô_{X} (V_{∞}) on X_{proét}/X_{∞} is Δ_{0}(p)- equivariant and the induced Hecke action on OD_{U}^{s} agrees with the natural action by correspondences.

4.3. Completed sheaves of overconvergent modular forms. Let q : X_{∞} → X be the natural projection as before. Recall that for any weight U and any w ≥ 1 + s_{U}, we have the sheaf ω_{U,w}^{†} of overconvergent modular forms of weight U on the analytic site of X_{w}, whose sections over a rational subset U ⊆ X_{w} are given by

ω_{U,w}^{†}(U) = {f ∈ O_{X_{∞}}(U_{∞})ˆ⊗A_{U} | γ^{∗}f = χ_{U}(b𝔷 + d)^{−1}f ∀γ ∈ K_{0}(p)} .

We may similarly define a sheaf ω_{U,w,ℂ_{p}}^{†} on the analytic site of X_{w,ℂ_{p}} by

ω_{U,w,ℂ_{p}}^{†}(U) = {f ∈ O_{X_{∞,ℂp}} (U_{∞})ˆ⊗A_{U} | γ^{∗}f = χ_{U}(b𝔷 + d)^{−1}f ∀γ ∈ K_{0}(p)}

with U ⊆ X_{w,ℂ_{p}} quasi-compact. We now define completed versions of these sheaves.

Definition 4.6. We define a sheaf ω_{U,w}^{†} on X_{w,proét} by

ω_{U,w}^{†}(V) = {f ∈ Ô_{X_{w}}(V_{∞})ˆ⊗A_{U} | γ^{∗}f = χ_{U}(b𝔷 + d)^{−1}f ∀γ ∈ K_{0}(p)} ,

where V ∈ X_{w,proét} is qcqs and V_{∞} = V ×_{X_{w}} X_{∞,w}. We let ω_{U,w,ℂ_{p}}^{†} denote the

restriction of ω_{U,w}^{†} to the slice X_{w,proét}/X_{w,ℂ_{p}} ≅ X_{w,ℂ_{p},proét}.

Let us write η for the canonical morphism of sites from the étale site to the analytic site, and put λ = η ∘ ν.

Proposition 4.7. There are canonical isomorphisms

ω_{U,w}^{†} ≅ λ^{∗}ω_{U,w}^{†} ⊗_{O_{Xw} ˆ⊗A_{U}} (Ô_{X_{w}} ˆ⊗A_{U})

and

ω_{U,w,ℂ_{p}}^{†} ≅ λ^{∗}ω_{U,w,ℂ_{p}}^{†} ⊗_{O_{X_{w,ℂp}} ˆ⊗A_{U}} (Ô_{X_{w,ℂp}} ˆ⊗A_{U}).

This has the following consequences:

(1) We have a canonical isomorphism

ω_{U,w}^{†} ≅ ν^{∗}ω_{U,w}^{†} ⊗_{O_{Xw} ˆ⊗A_{U}} (Ô_{X_{w}} ˆ⊗A_{U})

and similarly for the ℂ_{p}-sheaves, where we abuse the notation and write ω_{U,w}^{†} also for the étale sheaf η^{∗}ω_{U,w}^{†} ⊗_{O_{Xw,an} ˆ⊗A_{U}} (O_{X_{w,ét}} ˆ⊗A_{U}) (where we have written out ”an” and ”et” for clarity).

(2) ω_{U,w,ℂ_{p}}^{†} is the base change of ω_{U,w}^{†} from X_{w} to X_{w,ℂ_{p}} (see Lemma 6.21)

and hence ω_{U,w,ℂ_{p}}^{†} = Loc(ω_{U,w,ℂ_{p}}^{†}(X_{w,ℂ_{p}})).

(3) We have a canonical isomorphism ν_{∗}ω_{U,w,ℂ_{p}}^{†} ≅ ω_{U,w,ℂ_{p}}^{†} and

R^{1}ν_{∗}ω_{U,w,ℂ_{p}}^{†} = ω_{U,w,ℂ_{p}}^{†} ⊗_{O_{X_{w,ℂp}}} Ω_{X_{w,ℂp}}^{1} (−1).

Furthermore, R^{i}ν_{∗}ω_{U,w,ℂ_{p}}^{†} vanishes for i ≥ 2.

Proof. The isomorphisms are proven by repeating much of the proof of Theorem 2.28 (including the preliminary lemmas). We focus on the ℚ_{p}-sheaves; the case of the ℂ_{p}-sheaves is identical. Let V ∈ X_{w,proét} be an affinoid perfectoid with image U in X_{w} and assume without loss of generality that U is a rational subset of X_{w} and that ω|_{U} is trivial. We use the notation of the proof of Theorem 2.28 freely. Arguing in the same way we obtain isomorphisms

ω_{U,w}^{†}(V) ≅ (Ô_{X_{w}}(V_{n})ˆ⊗A_{U})^{G_{n}}

and

(ν^{∗}ω_{U,w}^{†} ⊗_{O_{Xw} ˆ⊗A_{U}} (Ô_{X_{w}} ˆ⊗A_{U}))(V) ≅

≅ (Ô_{X_{w}}(U_{n})ˆ⊗A_{U})^{G_{n}} ⊗_{Ô_{Xw} (U)ˆ⊗A_{U}} (Ô_{X_{w}}(V)ˆ⊗A_{U})

where V_{n} = U_{n} ×_{U} V and G_{n} acts by the twisted action f_{0} ↦ χ_{U}(j_{U,n}(γ))γ^{∗}f_{0}. Applying Lemma 6.7 twice one obtains a G_{n}-equivariant isomorphism

Ô_{X_{w}}(V_{n})ˆ⊗A_{U} ≅ (Ô_{X_{w}}(U_{n})ˆ⊗A_{U}) ⊗_{Ô_{Xw} (U)ˆ⊗A_{U}} (Ô_{X_{w}}(V)ˆ⊗A_{U}).

Taking invariants we obtain we desired isomorphism.

Assertion (1) then follows by transitivity of pullbacks. To prove (2), first evaluate the isomorphism for ω_{U,w}^{†} on U ⊆ X_{w,ℂ_{p}} to see that ω_{U,w,ℂ_{p}}^{†} restricted

to the analytic site of X_{w,ℂ_{p}} is the base change of ω_{U,w}^{†}, and then restrict the second isomorphism to the analytic site of X_{w,ℂ_{p}} to get the first part (2). The second then follows from Lemma 6.21.

To prove (3) we apply the projection formula to the isomorphism in (1) to get

R^{i}ν_{∗}ω_{U,w,ℂ_{p}}^{†} ≅ ω_{U,w,ℂ_{p}}^{†} ⊗_{(O_{X_{w,ℂp}} ˆ⊗A_{U})} R^{i}ν_{∗}(Ô_{X_{w,ℂp}} ˆ⊗A_{U}),

and the result then follows from Corollary 6.9. □

Corollary 4.8. The Leray spectral sequence

H_{ét}^{i}(X_{w,ℂ_{p}}, R^{j}ν_{∗}ω_{U,w,ℂ_{p}}^{†}) ⇒ H_{proét}^{i+j}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†})

induces a canonical Hecke- and Galois-equivariant isomorphism

H_{proét}^{1}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†}) ≅ H^{0}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†} ⊗_{O_{X_{w,ℂp}}} Ω_{X_{w,ℂp}}^{1} )(−1)

Proof. By Proposition 4.7 R^{j}ν_{∗}ω_{U,w,ℂ_{p}}^{†} is a locally projective O_{X_{w,ℂp}} ˆ⊗A_{U}- module for all j and hence has no higher cohomology by (the étale version of) Proposition 6.15. Therefore the Leray spectral sequence degenerates at the E_{2}-page, giving us isomorphisms

H_{ét}^{0}(X_{w,ℂ_{p}}, R^{j}ν_{∗}ω_{U,w,ℂ_{p}}^{†}) ≅ H_{proét}^{j}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†})

for all j. Now apply Proposition 4.7 again.

For Hecke-equivariance we focus on the Hecke operators away from p; the Hecke equivariance at p follows by the same arguments but is notationally slightly different. We will only check compatibility for the trace maps (compatibility for pullbacks follow from functoriality), and we put ourselves in the situation of Proposition 2.19. By the first isomorphisms in Proposition 4.7, the trace map t of Proposition 2.19(2) induces a trace map

t̂: f_{proét,∗}ω_{U,w,ℂ_{p},K_{2}^{p}}^{†} → ω_{U,w,ℂ_{p},K_{1}^{p}}^{†}.

We need to check that the diagram

\xymatrix{H_{proét}^{1}(X_{w,ℂ_{p},K_{2}^{p}}, ω_{U,w,ℂ_{p},K_{2}^{p}}^{†}) \ar[r] \ar[d]_{t̂} & H^{0}(X_{w,ℂ_{p},K_{2}^{p}}, ω_{U,w,ℂ_{p},K_{2}^{p}}^{†} ⊗ Ω^{1})(−1) \ar[d] \\ H_{proét}^{1}(X_{w,ℂ_{p},K_{1}^{p}}, ω_{U,w,ℂ_{p},K_{1}^{p}}^{†}) \ar[r] & H^{0}(X_{w,ℂ_{p},K_{1}^{p}}, ω_{U,w,ℂ_{p},K_{1}^{p}}^{†} ⊗ Ω^{1})(−1)}

commutes, where the left hand vertical arrow is induced by t̂ as in Lemma 4.9(2) and the right hand vertical arrow is induced by t in the obvious way. We do this in two steps. For the first step, we note that Lemma 4.9(3) gives a commutative diagram

\xymatrix{H_{proét}^{1}(X_{w,ℂ_{p},K_{2}^{p}}, ω_{U,w,ℂ_{p},K_{2}^{p}}^{†}) \ar[r] \ar[d]_{t̂} & H^{0}(X_{w,ℂ_{p},K_{2}^{p}}, R^{1}ν_{∗}ω_{U,w,ℂ_{p},K_{2}^{p}}^{†}) \ar[d]^{R^{1}ν_{∗}t̂} \\ H_{proét}^{1}(X_{w,ℂ_{p},K_{1}^{p}}, ω_{U,w,ℂ_{p},K_{1}^{p}}^{†}) \ar[r] & H^{0}(X_{w,ℂ_{p},K_{1}^{p}}, R^{1}ν_{∗}ω_{U,w,ℂ_{p},K_{1}^{p}}^{†}).}

For the second step, we claim that the diagram

\xymatrix{R^{1}ν_{∗}ω_{U,w,ℂ_{p},K_{2}^{p}}^{†} \ar[r] \ar[d]_{R^{1}ν_{∗}t̂} & ω_{U,w,ℂ_{p},K_{2}^{p}}^{†} ⊗ Ω^{1}(−1) \ar[d]^{t} \\ R^{1}ν_{∗}ω_{U,w,ℂ_{p},K_{1}^{p}}^{†} \ar[r] & ω_{U,w,ℂ_{p},K_{1}^{p}}^{†} ⊗ Ω^{1}(−1)}

commutes, where the horizontal arrows are the isomorphisms of Proposition 4.7(3); this follows upon combining the first isomorphism of Proposition 4.7, the pullback-pushforward adjunction of ν, and Lemma 4.10. □

Lemma 4.9. Let f : X → Y be a finite étale morphism of rigid analytic varieties.

(1) The functor f_{proét,∗} is exact.

(2) Suppose we are given abelian sheaves F and G on X_{proét} and Y_{proét}, respectively, together with a “trace” map t : f_{proét,∗}F → G of abelian sheaves on Y_{proét}. Then t induces a canonical map t : H_{proét}^{n}(X, F) → H_{proét}^{n}(Y, G) together with derived trace maps R^{i}ν_{∗}t : f_{ét,∗}R^{i}ν_{X,∗}F → R^{i}ν_{Y,∗}G of abelian sheaves on Y_{ét}.

(3) The maps in (2) are compatible with the Grothendieck spectral sequences for RΓ_{proét} = RΓ_{ét} ∘ Rν_{∗}, i.e. the maps t and R^{i}ν_{∗}t fit into compatible morphisms

\xymatrix{H_{ét}^{i}(X, R^{j}ν_{X,∗}F) \ar@{=>}[r] \ar[d]_{R^{j}ν_{∗}t} & H_{proét}^{i+j}(X, F) \ar[d]^{t} \\ H_{ét}^{i}(Y, R^{j}ν_{Y,∗}G) \ar@{=>}[r] & H_{proét}^{i+j}(Y, G)}

of pro-étale cohomology groups and of the spectral sequences computing them.

Proof. To prove right-exactness of f_{proét,∗} we need to check that it preserves surjections. For this we may work étale locally on Y, so we may pick an étale cover which splits f and the assertion is then trivial.

For (2) and (3), we argue as follows. By (1), the Leray spectral sequence gives an isomorphism H_{proét}^{n}(X, F) ≅ H_{proét}^{n}(Y, f_{proét,∗}F), so composing this with the evident map H_{proét}^{n}(Y, f_{proét,∗}F) → H_{proét}^{n}(Y, G) gives the claimed map on pro-étale cohomology. On the other hand, a direct calculation gives a natural isomorphism f_{ét,∗}ν_{X,∗} ≅ ν_{Y,∗}f_{proét,∗} as functors Sh(X_{proét}) → Sh(Y_{ét}). Indeed, both functors send a sheaf F to the sheaf associated with the presheaf U ↦ F(U ×_{Y} X) (where U ×_{Y} X is regarded as an element of X_{proét}). Since all these functors preserve injectives, we may pass to total derived functors, getting Rf_{ét,∗}Rν_{X,∗} ≅ Rν_{Y,∗}Rf_{proét,∗} as functors D^{+}(Sh(X_{proét})) → D^{+}(Sh(Y_{ét})). Since f_{proét,∗} and f_{ét,∗} are both exact functors, this becomes f_{ét,∗}Rν_{X,∗} ≅ Rν_{Y,∗}f_{proét,∗}. Applying this to F and composing with the evident map Rν_{Y,∗}f_{proét,∗}F → Rν_{Y,∗}G induced by t gives a map Rν_{∗}t : f_{ét,∗}Rν_{X,∗}F → Rν_{Y,∗}G, and we obtain the maps R^{i}ν_{∗}t upon passing to cohomology sheaves. Finally, (3) follows by applying RΓ_{Y,ét} to the map Rν_{∗}t, in combination with the isomorphism RΓ_{X,ét}Rν_{X,∗} ≅ RΓ_{Y,ét}f_{ét,∗}Rν_{X,∗}. □

Lemma 4.10. Maintain the setup of the previous lemma, and suppose X and Y are smooth and are defined over a complete algebraically closed extension C/ℚ_{p}. Take F = Ô_{X} and G = Ô_{Y}, and let t be the natural Ô_{Y} -linear trace map t̂r : f_{proét,∗}Ô_{X} → Ô_{Y}. Then R^{i}ν_{∗}t̂r : f_{ét,∗}R^{i}ν_{X,∗}Ô_{X} → R^{i}ν_{Y,∗}Ô_{Y} coincides with the composite map

f_{ét,∗}R^{i}ν_{X,∗}Ô_{X} ≅ f_{ét,∗}Ω_{X}^{i}(−i) ≅ Ω_{Y}^{i} (−i) ⊗_{O_{Y}} f_{ét,∗}O_{X} ⟶^{1⊗tr} Ω_{Y}^{i} (−i) ≅ R^{i}ν_{Y,∗}Ô_{Y} .

Proof. This follows from a careful reading of the proof of Lemma 3.24 of [Sch2]. Let us give a very brief outline of the argument. One considers a short exact sequence S_{X}

0 → ℤ̂_{p}(1) → \underleftarrow{lim}_{×p} O_{X}^{×} → O_{X}^{×} → 0

as in Lemma 3.24 of [Sch2]. We have also an analogous short exact sequence S_{Y} on Y and a norm map f_{proét,∗}S_{X} → S_{Y} on Y_{proét}. To both S_{X} and S_{Y} we can associate commutative diagrams as in [Sch2, Lemma 3.24] and one shows that they fit together into a commutative cube. Chasing through this cube gives us a lemma for i = 1. The general case follows from this as in [Sch2, Proposition 3.23] by taking exterior products. □

4.4. The overconvergent Eichler-Shimura map. Working with the proétale site of X_{w} has the advantage of allowing us to define the overconvergent Eichler-Shimura map at the level of sheaves explicitly, using the cover X_{∞,w} and the fundamental period 𝔷.

Fix a small weight U. The following proposition is the key step in the construction of the overconvergent Eichler-Shimura map.

Proposition 4.11. Let w ≥ 1 + s_{U} and s ≥ 1 + s_{U}. Let V_{∞} ∈ X_{w,proét}/X_{∞,w} be qcqs. We have a map

β_{U} : D_{U}^{s,∘} ˆ⊗Ô_{X_{w}}(V_{∞}) → R_{U} ˆ⊗Ô_{X_{w}}(V_{∞})

defined on pure tensors by β_{U} : μ⊗f ↦ μ(χ_{U}(1+𝔷x))f. Here we use Proposition 2.6, with B = Ô_{X_{w}}(V_{∞}), to be able to apply χ_{U} to 1 + 𝔷x for any x ∈ ℤ_{p}. This defines a morphism of sheaves on X_{w,proét}/X_{∞,w}. β_{U} satisfies the equivariance relation

β_{U}(γ^{∗}h) = χ_{U}(b𝔷 + d)γ^{∗}β_{U}(h)

for any γ ∈ Δ_{0}(p), V_{∞} ∈ X_{w,proét} qcqs and h ∈ D_{U}^{s,∘} ˆ⊗Ô_{X_{w}}(γ^{−1}V_{∞}). In particular, pushing forward to X_{w,proét} and passing to K_{0}(p)-invariants, β_{U} induces an R_{U}-linear and Hecke equivariant map

δ_{U} : OD_{U}^{s} → ω_{U,w}^{†}

of abelian sheaves on X_{w,proét}.

Here we are interpreting χ_{U}(1 + 𝔷x) as an element of A_{U}^{s,∘} ˆ⊗Ô_{X_{w}}(V_{∞}), and extending μ by linearity and continuity to an R_{U} ˆ⊗Ô_{X_{w}}(V_{∞})-linear map A_{U}^{s,∘} ˆ⊗Ô_{X_{w}}(V_{∞}) → R_{U} ˆ⊗Ô_{X_{w}}(V_{∞}).

Proof. First we check that β_{U} is well defined, i.e. that the description on pure tensors extends by continuity. To do this, note that the description on tensors define compatible maps

(D_{U}^{s−1,∘}/𝔞_{U}^{k}D_{U}^{s−1,∘}) ⊗_{ℤ_{p}} Ô_{X_{w}}(V_{∞}) → (R_{U}/𝔞_{U}^{k}) ⊗_{ℤ_{p}} Ô_{X_{w}}(V_{∞})

for all k. By the definition of the filtrations we then get compatible maps

(D_{U}^{s,∘}/Fil^{k}D_{U}^{s,∘}) ⊗_{ℤ_{p}} Ô_{X_{w}}(V_{∞}) → (R_{U}/𝔞_{U}^{k}) ⊗_{ℤ_{p}} Ô_{X_{w}}(V_{∞}).

Taking limits and inverting p we get the desired map. It is then enough to check equivariance on pure tensors h = μ ⊗ f. We compute

β_{U}(γh) = (γ ·_{U} μ)(χ_{U}(1 + 𝔷x))γ^{∗}f
= μ (χ_{U}(cx + d)χ_{U}(1 + 𝔷\frac{ax + b}{cx + d})) γ^{∗}f
= μ (χ_{U}(cx + d + 𝔷(ax + b))) γ^{∗}f
= μ (χ_{U}(b𝔷 + d)χ_{U}(1 + x\frac{a𝔷 + c}{b𝔷 + d})) γ^{∗}f
= χ_{U}(b𝔷 + d)μ (χ_{U}(1 + x\frac{a𝔷 + c}{b𝔷 + d})) γ^{∗}f
= χ_{U}(b𝔷 + d)γ^{∗} (μ (χ_{U}(1 + 𝔷x)) f)
= χ_{U}(b𝔷 + d)γ^{∗}β_{U}(h)

and arrive at the desired conclusion. We leave the Hecke equivariance away from p to the reader. □

The map δ_{U} is our version of the map δ_{k}^{∨}(w) from [AIS2, §4.2]. We may then define the overconvergent Eichler-Shimura map at the level of spaces in the same way as in [AIS2]:

Definition 4.12. Let U be a small weight. The weight U overconvergent Eichler-Shimura map ES_{U} is given as the composite

V_{U,ℂ_{p}}^{s} ≅

≅ H_{proét}^{1}(X_{ℂ_{p}}, OD_{U}^{s}) →^{res} H_{proét}^{1}(X_{w,ℂ_{p}}, OD_{U}^{s}) →^{δ_{U}} H_{proét}^{1}(X_{w,ℂ_{p}}, ω̂_{U,w,ℂ_{p}}^{†}) ≅

≅ H^{0}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†} ⊗ Ω_{X_{w,ℂp}}^{1} )(−1) =: M_{U,ℂ_{p}}^{†,w}(−1)

where the first map is induced by restricting cohomology classes along the inclusion X_{w} ⊆ X. ES_{U} is Hecke- and G_{ℚ_{p}}-equivariant by Propositions 4.3, 4.4, 4.11 and Corollary 4.8.

The above definition gives also a map ES_{κ} for any individual point κ ∈ W(\overline{ℚ}_{p}), since any such weight is both small and affinoid. Note also that the construction of ES_{U} is functorial in U: if we have a morphism U → U′ compatible with the characters, then we have a commutative diagram

\xymatrix{V_{U′,ℂ_{p}}^{s} \ar[r]^{ES_{U′}} \ar[d] & M_{U′,ℂ_{p}}^{†,w}(−1) \ar[d] \\ V_{U,ℂ_{p}}^{s} \ar[r]^{ES_{U}} & M_{U,ℂ_{p}}^{†,w}(−1)}

where the vertical maps are the natural ones. In particular, when U is a small open weight and κ ∈ U^{rig}, we have a commutative diagram

\xymatrix{V_{U,ℂ_{p}}^{s} \ar[r]^{ES_{U}} \ar[d] & M_{U,ℂ_{p}}^{†,w}(−1) \ar[d] \\ V_{κ,ℂ_{p}}^{s} \ar[r]^{ES_{κ}} & M_{κ,ℂ_{p}}^{†,w}(−1).}

4.5. Factorization for weights k ≥ 2. To gain some control of the overconvergent Eichler-Shimura map we will prove that it factors through the p-adic Eichler-Shimura map defined by Faltings ([Fal]) for integral weights k ∈ ℤ_{≥2}. The key step is to prove this factorization on the level of sheaves, which is the goal of this section. In our setup, this factorization turns out to be rather transparent.

Let V ∈ X_{proét} be qcqs and put V_{∞} = V ×_{X} X_{∞}. Let k ≥ 2. By Lemma 4.5 we know that

OD_{k}^{s}(V) = (D_{k}^{s} ˆ⊗_{ℚ_{p}} Ô_{X} (V_{∞}))^{K_{0}(p)}.

Let G^{univ} be the universal p-divisible group over X. We define T to be the relative Tate module of G^{univ} viewed as a sheaf on X_{proét}. Let ω be as before. We then define variants of those sheaves on X_{proét} by

T̂ = T ⊗ℤ̂_{p} Ô_{X};
ω̂ = λ^{∗}ω ⊗_{O_{X}} Ô_{X} .

Let us write V̂_{k} = Sym^{k−2}(T ) ⊗ℤ̂_{p} Ô_{X} = Sym^{k}(T̂) and identify ℒ_{k}^{∘} with

Sym^{k−2}(ℤ_{p}^{2}) via the isomorphism sending X^{i} to e_{1}^{i}e_{2}^{k−2−i} (0 ≤ i ≤ k − 2). This identifies the M_{2}(ℤ_{p})-action on ℒ_{k}^{∘} with the standard left action on Sym^{k−2}(ℤ_{p}^{2}).

Lemma 4.13. For any V ∈ X_{proét} qcqs, we have V̂_{k}(V) ≅ (ℒ_{k} ⊗_{ℚ_{p}} Ô_{X} (V_{∞}))^{K_{0}(p)}.

Proof. Follows from Lemma 4.1 upon noting that T (V_{∞}) ≅ ℒ_{1}^{∘} functorially in V and Δ_{0}(p)-equivariantly via the universal trivialization over X_{∞}. □

The above lemma and Lemma 4.5 gives us a map

OD_{k}^{s}(V) → V̂_{k}(V)

induced by the integration map

i_{k} : D_{k}^{s} → ℒ_{k}

defined in Definition 3.2.

Let 𝔰 ∈ H^{0}(X_{∞,w}, ω) be the non-vanishing section defined in §2.4. Recall that

ω_{k,w}^{†}(V) = {f ∈ Ô_{X_{w}}(V_{∞})|γ^{∗}f = (b𝔷 + d)^{−k}f} for V ∈ X_{w,proét} qcqs.

Lemma 4.14. We have

ω_{k,w}^{†} ≅ (ω^{⊗k−2})|_{X_{w}}

via the map

f ↦ f · 𝔰^{⊗k−2}

Proof. For V qcqs the map induces an isomorphism between ω_{k,w}^{†}(V) and the set {η ∈ ω^{⊗k−2}(V_{∞}) | γ^{∗}η = η} using the transformation rule for 𝔰. By an argument similar to that in the (first half of the) proof of Lemma 2.26 the latter is functorially isomorphic to ω^{⊗k−2}(V). □

Recall the linearized Hodge-Tate map

T̂ → ω̂

as a map of sheaves of X_{proét}. Taking (k − 2)-th symmetric powers we get a map

V̂_{k} → ω^{⊗k−2}.

Restricting to X_{w} and using Lemma 4.14 we get a map

v_{k} : V̂_{k} → (ω^{⊗k−2})|_{X_{∞,w}} ≅ ω_{k,w}^{†}

which we want to describe explicitly:

Lemma 4.15. Let k ≥ 2 and let V_{∞} ∈ X_{w,proét}/X_{∞} be qcqs. Define a map

ℒ_{k} ⊗_{ℚ_{p}} Ô_{X_{w}}(V_{∞}) → Ô_{X_{w}}(V_{∞})

by X^{i} ↦ 𝔷^{i} for 0 ≤ i ≤ k − 2. This is an Ô_{X} -linear and Δ_{0}(p)-equivariant morphism of sheaves on X_{w,proét}/X_{∞}. Pushing forward to X_{w,proét} and taking K_{0}(p)-invariants we get a map V̂_{k} → ω_{k,w}^{†}. Evaluating this on V ∈ X_{w,proét} qcqs and putting V_{∞} = V ×_{X} X_{∞,w}, we obtain a map

(ℒ_{k} ⊗_{ℚ_{p}} Ô_{X_{w}}(V_{∞}))^{K_{0}(p)} → ω_{k,w}^{†}(V).

which is equal to v_{k} (using Lemma 4.13).

Proof. We describe the map V̂_{k} → ω^{⊗k−2} and then use the isomorphism of the previous lemma. Over V_{∞}, the sheaf ω^{⊗k−2} is trivialized by 𝔰^{⊗k−2}, so v_{k} is morphism

ℒ_{k} ⊗_{ℚ_{p}} Ô_{X_{w}}(V_{∞}) → Ô_{X_{w}}(V_{∞})𝔰^{⊗k−2}.

From the definitions we see that it sends X^{i} to HT(α(e_{1}))^{i} HT(α(e_{2}))^{k−2−i} where α denotes the trivialization of T over V_{∞} and HT denotes the Hodge-Tate map. By definition, 𝔰 = HT(α(e_{2})) and over X_{w} we have the relation HT(α(e_{1})) = 𝔷 HT(α(e_{2})). Thus X^{i} is sent to 𝔷^{i}𝔰^{⊗k−2}. Finally observe that the isomorphism of Lemma 4.14 sends 𝔰^{⊗k−2} to 1. □

We can now prove the factorization theorem at the level of sheaves for our overconvergent Eichler-Shimura map.

Proposition 4.16. For k ≥ 2 we have a commutative diagram:

[figure: OD_{k}^{s}; μ ↦ μ((1 + 𝔷x)^{k−2}); μ ↦ μ((1 + Xx)^{k−2}); ω_{k,w}^{†}; V̂_{k}; X^{i} ↦ 𝔷^{i}]

Proof. It follows from our prior discussion and Lemma 4.15. □

Remark 4.17. This diagram, which is a diagram of sheaves over X_{w}, may be viewed as the restriction of the following diagram of sheaves over X:

[figure: OD_{k}^{s}; μ ↦ μ((1 + 𝔷x)^{k−2})𝔰^{⊗k−2}; μ ↦ μ((1 + Xx)^{k−2}); ω^{⊗k−2}; V̂_{k}; X^{i} ↦ 𝔷^{i}𝔰^{⊗k−2}]

using the isomorphism of Lemma 4.14. Note that although 𝔷 and 𝔰 do not extend to X_{∞}, the products 𝔷^{i}𝔰^{⊗k−2} do extend to X_{∞} for all 0 ≤ i ≤ k − 2, so the formulas in the diagram make sense. This will cause our overconvergent Eichler-Shimura maps to have very large kernels at classical weights k ≥ 2 (see §5.2).

5. The overconvergent Eichler-Shimura map over the eigencurve

In this section we prove our main results concerning the overconvergent Eichler-Shimura map. These results are analogous to the main results of [AIS2]. However, the payoff for working with arbitrary small open weights is that we may glue our overconvergent Eichler-Shimura maps for different small open weights into a morphism of coherent sheaves over the eigencurve. This allows us to work over rather arbitrary regions of weight space and with general slope cutoffs, as opposed to the rather special open discs inside the analytic part of weight space used in [AIS2].

5.1. Sheaves on the eigencurve. Let U be a small weight and let s ≥ 1+s_{U}. Recall we have defined the modules

V_{U}^{s} = H_{proét}^{1}(X_{ℂ_{p}}, D_{U}^{s});
V_{U,ℂ_{p}}^{s} ≅ H_{proét}^{1}(X_{ℂ_{p}}, OD_{U}^{s});
M_{U}^{†,w} = H^{0}(X_{w}, ω_{U,w}^{†} ⊗_{O_{Xw}} Ω_{X_{w}}^{1});

M_{U,ℂ_{p}}^{†,w} = H^{0}(X_{w,ℂ_{p}}, ω_{U,w,ℂ_{p}}^{†} ⊗_{O_{X_{w,ℂp}}} Ω_{X_{w,ℂp}}^{1} ).

In this subsection we spread the finite-slope pieces of these modules into sheaves on the whole eigencurve C = C_{K^{p}}, and we glue our Eichler-Shimura maps into a morphism of sheaves.

Recall the rigid analytic weight space W = Spf(ℤ_{p}[[ℤ_{p}^{×}]])^{rig}, with its universal character χ_{W} : ℤ_{p}^{×} → O(W)^{×}. If U is an open weight, we write O^{bd}(U^{rig}) for the ring of bounded functions on U^{rig}. This is a Banach ℚ_{p}-algebra equal to A_{U}[\frac{1}{p}] as a dense subring, and with a natural map O(W) → O(U^{rig}). Recall that we have defined the Fredholm determinant

F_{U}(T) = det(1 − ŨT |C_{•}(K, A_{U}^{s}))

in §3.2. It is an element of O(U^{rig})^{bd}{{T }}, the ring of entire power series with coefficients in O^{bd}(U^{rig}){{T }}, and we showed that it glues to a Fredholm series F (T) ∈ O(W){{T }} (see e.g. [Han1, Definition 4.1.1] for the definition of a Fredholm series). Now let V = (S_{V}, χ_{V}) be an open affinoid weight. Arguing as in [Pil, §5.2] we see that M_{V}^{†,w} satisfies Buzzard’s property (Pr) over S_{V} (i.e. it is a direct summand of an orthonormalizable S_{V}-module, see the paragraph before Lemma 2.11 of [Buz3]). Since U_{p} is compact we get a Fredholm determinant

G_{V}(T) = det(1 − U_{p}T |M_{V}^{†,w})

in O(V){{T }} and these glue to a Fredholm series G(T) ∈ O(W){{T }}. Let us set H(T) = F (T)G(T), this is also a Fredholm series. Recall ([Han1, §4.1]) that a Fredholm hypersurface is a closed subvariety of W × 𝔸^{1} cut out by a Fredholm series. Given a Fredholm series f we write 𝒵 (f) for the corresponding Fredholm hypersurface. Recall that, if h ∈ ℚ_{≥0} and W ⊆ W is open, then f has a slope ≤ h-decomposition in O^{bd}(W){{T }} if and only if the natural map 𝒵_{W,h}(f) := 𝒵 (f) ∩ (W × 𝔹[0, h]) → 𝔹[0, h] is finite flat, where 𝔹[0, h] ⊆ 𝔸^{1} is the closed disc around 0 of radius p^{h}. We say that (W, h) is slope-adapted for f if these equivalent conditions hold. We have the following key result:

Lemma 5.1. There exists a collection of pairs (V_{i}, h_{i}), with V ⊆ W open and affinoid and h_{i} ∈ ℚ_{≥0}, such that (V_{i}, h_{i}) is slope-adapted for H for all i and the (𝒵_{V_{i},h_{i}}(H))_{i} form an open (admissible) cover of 𝒵 (H). Moreover, for each i we may find a small open weight U_{i} such that V_{i} ⊆ U_{i}^{rig} and (U_{i}^{rig}, h_{i}) is slope-adapted for H.

Proof. See [Han2, Lemmas 2.3.1-2.3.4] (the first part is a theorem of Coleman-Mazur and Buzzard and is the key step in the ”eigenvariety machine”, cf. [Buz3, Theorem 4.6]). □

Fix collections V_{i}, U_{i} and h_{i} satisfying the conclusions of Lemma. For now, let h ∈ ℚ_{≥0} and let U be any small open weight such that (U^{rig}, h) is slope-adapted for H, and let V ⊆ U^{rig} be an open affinoid weight. Recall our overconvergent Eichler-Shimura map

ES_{U} : V_{U,ℂ_{p}}^{s} → M_{U,ℂ_{p}}^{†,w}(−1).

Since (U^{rig}, h) is slope-adapted for H (and hence for F and G) and ES_{U} is Hecke-equivariant we may take slope ≤ h-parts on both sides to obtain an R_{U}[\frac{1}{p}]-linear map

ES_{U,≤h} : V_{U,ℂ_{p},≤h}^{s} → M_{U,ℂ_{p},≤h}^{†,w}(−1)

of finite projective R_{U}[\frac{1}{p}]-modules. We may compose with this map the natural map M_{U,ℂ_{p},≤h}^{†,w}(−1) → M_{V,ℂ_{p},≤h}^{†,w}(−1) and tensor the source with S_{V} to obtain an S_{V}-linear map

ES_{V,≤h} : V_{V,ℂ_{p},≤h}^{s} ≅ V_{U,ℂ_{p},≤h}^{s} ⊗_{R_{U} [ \frac{1}{p}]} S_{V} → M_{V,ℂ_{p},≤h}^{†,w}(−1)

where the first isomorphism is that of Proposition 3.4.

Let us now recall the eigencurves for B^{×} of tame level K^{p} constructed out of overconvergent modular forms resp. overconvergent modular symbols, using the perspective of [Han1, §4]. Let Σ_{0} denote the finite set of primes ℓ for which K_{ℓ} ≅ G̸L_{2}(ℤ_{ℓ}) and let Σ = Σ_{0} ∪ {p}. We let 𝕋 denote the abstract Hecke algebra generated by commuting formal variables T_{ℓ}, S_{ℓ} for ℓ ∉ Σ and U_{p}. Then 𝕋 acts on all spaces of modular forms and modular symbols used in this article. We may construct two eigenvariety data ([Han1, Definition 4.2.1]) 𝔇_{M^{†}} = (W, 𝒵 (H), M^{†}, 𝕋, ψ_{M^{†}}) and 𝔇_{V} = (W, 𝒵 (H), V, 𝕋, ψ_{V}). Here M^{†} and V are the coherent sheaves on 𝒵 (H) obtained by gluing the \underrightarrow{lim}_{w} M_{V_{i},≤h_{i}}^{†,w} resp. the \underleftarrow{lim}_{s} V_{V_{i},≤h_{i}}^{s} over the 𝒵_{V_{i},h_{i}}(H) (see [Han1, Proposition 4.3.1] and the discussion following it for the construction of V; the construction of M^{†} is similar but easier). Given these, [Han1, Theorem 4.2.2] allows us to construct the eigenvarieties C_{M^{†}} and C_{V} together with coherent sheaves M^{†} on C_{M^{†}} resp. V on C_{V} (this is a mild abuse of notation; we will have no further reason to consider the sheaves with the same name over 𝒵 (H)). Since the following proposition is well known we only give a brief sketch of the proof:

Proposition 5.2. There is a canonical isomorphism C_{M^{†}} ≅ C_{V} compatible with the projection maps down to W and the actions of 𝕋.

Proof. This follows by applying [Han1, Theorem 5.1.2] twice (once in each direction), upon noting that both eigencurves are reduced and equi-dimensional (of dimension 1; they are ”unmixed” in the terminology of [Han1]). For these applications one should modify the eigenvariety data, replacing 𝒵 (H) in 𝔇_{M^{†}} with the support of M^{†} and similarly for 𝔇_{V}. The relevant very Zariski dense subsets are then constructed using the control/classicality theorems of Stevens and Coleman together with the usual Eichler-Shimura isomorphism (see [Han1, Theorem 3.2.5] for a general version of the control theorem for overconvergent modular symbols, and see e.g. [Joh, Theorem 4.16] for a proof of Coleman’s control theorem in the context of the compact Shimura curves used here). □

In light of this we will identify the two eigencurves and simply denote it by C. It carries two coherent sheaves M^{†} and V which are determined by canonical isomorphisms

M^{†}(C_{V_{i},h_{i}}) ≅ \underrightarrow{lim}_{w} M_{V_{i},≤h_{i}}^{†,w} ≅ M_{V_{i},≤h_{i}}^{†,w};

V(C_{V_{i},h_{i}}) ≅ \underleftarrow{lim}_{s} V_{V_{i},≤h_{i}}^{s} ≅ V_{V_{i},≤h_{i}}^{s}

for all i and sufficiently large w resp. s, where C_{V_{i},h_{i}} is the preimage of 𝒵_{V_{i},h_{i}}(H) in C. We form the sheaves M_{ℂ_{p}}^{†} := M^{†} ˆ⊗_{ℚ_{p}}ℂ_{p} and V_{ℂ_{p}} := Vˆ⊗_{ℚ_{p}}ℂ_{p}. They are determined by canonical isomorphisms

M_{ℂ_{p}}^{†}(C_{V_{i},h_{i}}) ≅ \underrightarrow{lim}_{w} M_{V_{i},ℂ_{p},≤h_{i}}^{†,w} ≅ M_{V_{i},ℂ_{p},≤h_{i}}^{†,w};

V_{ℂ_{p}}(C_{V_{i},h_{i}}) ≅ \underleftarrow{lim}_{s} V_{V_{i},ℂ_{p},≤h_{i}}^{s} ≅ V_{V_{i},ℂ_{p},≤h_{i}}^{s}

for any sufficiently large w resp. s. We may also naturally view these sheaves as coherent sheaves on the base change C_{ℂ_{p}} of C from ℚ_{p} to ℂ_{p}. They are then determined by the obvious modifications of the above canonical isomorphisms.

With these preparations we may now state the main theorem of this section which glues the maps ES_{V_{i},≤h_{i}}:

Theorem 5.3. There exists a canonical Hecke and Galois-equivariant morphism

ES : V_{ℂ_{p}} → M_{ℂ_{p}}^{†}(−1)

of coherent sheaves of O_{C} ˆ⊗_{ℚ_{p}}ℂ_{p}-modules on C (or coherent sheaves of O_{C_{ℂp}} - modules on C_{ℂ_{p}}), which glues (ES_{V_{i},≤h_{i}})_{i}.

Proof. The ES_{V_{i},≤h_{i}} induce maps over the cover (C_{V_{i},h_{i}})_{i} using the canonical isomorphisms. Checking that they glue is tedious but straightforward, using the functoriality of the maps ES_{U} in the small open weight U as well as the naturality of the construction of ES_{V,≤h} for V ⊆ U^{rig}, V open affinoid and U small open such that (U^{rig}, h) is slope-adapted for H. We leave the remaining details to the reader. □

The morphism ES is our main object of study in this section. Our main technique to study it is via specialization to non-critical classical points of C (recall that a classical point x is non-critical if h < k −1, where h is the slope and k is the weight of x). Let κ ∈ W(L) with L/ℚ_{p} finite and let (V, h) be slope-adapted with V connected and affinoid and κ ∈ V(L). By [Han1, Theorem 3.3.1] there is a second quadrant spectral sequence (with s big enough)

E_{2}^{pq} = Tor_{−p}^{O_{W} (V)}(H^{q}(K, D_{V}^{s})_{≤h}, L) ⟹ H^{p+q}(K, D_{κ}^{s})_{≤h}

which degenerates at the E_{2}-page since O_{W}(V) is a Dedekind domain. This gives us a short exact sequence

0 → V(C_{V,h}) ⊗_{O_{W} (V)} L → H^{1}(K, D_{κ}^{s})_{≤h} → Tor_{1}^{O_{W} (V)}(H^{2}(K, D_{V}^{s})_{≤h}, L) → 0.

Now assume that x ∈ C(L′) is a classical non-critical point of weight k = κ (with L′/L a finite field extension). After localizing the above short exact sequence at x it becomes an isomorphism

(V(C_{V,h}) ⊗_{O_{W} (V)} L)_{x} ≅ (H^{1}(K, D_{k}^{s})_{≤h})_{x}.

This follows from the fact that x does not occur in H^{2}(K, D_{V}^{s})_{≤h}, which by [Han1, Proposition 4.5.2] follows from the fact that it does not occur in H^{2}(K, D_{k}^{s})_{≤h}, which in turn follows from the control theorem. Using the control theorem again, the right hand side of the isomorphism is canonically isomorphic to H_{ét}^{1}(X_{ℂ_{p}}, V_{k})_{x}. From this we deduce that the fiber of V at x is the largest semisimple quotient of the generalized eigenspace in H_{ét}^{1}(X_{ℂ_{p}}, V_{k}) associated with x.

By a similar but simpler analysis (using Lemma 2.29) one sees that the fibre of M^{†} at an arbitrary point x ∈ C(\overline{ℚ}_{p}) is equal to the largest semisimple quotient of the generalized eigenspace in H^{0}(X_{w}, ω_{κ,w}^{†} ⊗_{O_{Xw}} Ω_{X_{w}}^{1}) associated with x where w is sufficiently large. One may of course apply the control theorem if x is non-critical and classical to gain a further refinement. Similarly, the analogous statements apply to the same objects based changed from ℚ_{p} to ℂ_{p}.

We will need a few extras fact about C before we proceed to analyze ES.

Lemma 5.4. Let κ ∈ W(\overline{ℚ}_{p}) be a weight and let s ≥ s_{κ}. Then H^{0}(K, D_{κ}^{s}) = 0.

Proof. Write G^{1} for the closed subgroup of G of elements of reduced norm one. Under our fixed isomorphism G(ℤ_{p}) ≅ GL_{2}(ℤ_{p}) we have G^{1}(ℤ_{p}) ≅ SL_{2}(ℤ_{p}). To prove the lemma, it suffices to show that H^{0}(Γ, D_{κ}^{s}) = 0 for any congruence subgroup Γ ⊂ G^{1}(ℤ) contained in K_{0}(p). By the p-adic continuity of the Γ-action on D_{κ}^{s}, we have

H^{0}(Γ, D_{κ}^{s}) = H^{0}(Γ_{p}, D_{κ}^{s}),

where Γ_{p} denotes the p-adic closure of Γ in GL_{2}(ℤ_{p}). By [Rap, Lemma 2.7] and the Zariski density of Γ in G^{1} (see e.g. [PR, Theorem 4.10]), Γ_{p} contains an open subgroup of SL_{2}(ℤ_{p}), and in particular contains a nontrivial element

u = (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) for some a ∈ ℤ_{p}, a ≠ 0. Note that u acts on μ ∈ D_{κ}^{s} by

(u · μ)(f(x)) = μ(f(x + a)); in particular this action does not depend on κ. Since any element of H^{0}(Γ_{p}, D_{κ}^{s}) ⊆ D_{κ}^{s} = D^{s} ⊗_{ℚ_{p}} L (L the residue field of κ) is fixed by u, it now suffices to show that no nonzero element of D^{s} is fixed by u. To see this, note that the Amice transform

μ ↦ A_{μ}(T) = ∫ (1 + T)^{x}μ(x) ∈ ℚ_{p}[[T]]

defines a ℚ_{p}-linear injection of D^{s} (regarded as a ring under convolution) into a subring of ℚ_{p}[[T]] (cf. [Col, §1.8] for more detailed statements). An easy calculation shows that A_{u·μ}(T) = (1 + T)^{a}A_{μ}(T), so if u · μ = μ then ((1 + T)^{a} − 1)·A_{μ}(T) = 0, and since ℚ_{p}[[T]] is a domain this implies A_{μ}(T) = 0 and then μ = 0 as desired. □

Lemma 5.5. The sheaves M^{†} and V are torsion free.

Proof. We start with M^{†}. This is a local statement so we may work over some C_{V,h} coming from a slope-adapted (V, h) with V affinoid. Then O_{C}(C_{V,h}) is finite over O_{W}(V), so it suffices to show torsion freeness over O_{W}(V), which is clear by definition.

We now prove torsion-freeness for V. Working locally as above it is enough to show that H^{1}(K, D_{V}^{s})_{≤h} is a torsion-free O_{W}(V)-module, where V ⊆ W is affinoid and (V, h) is slope-adapted. Let κ ∈ V(\overline{ℚ}_{p}) be a weight, cutting out a maximal ideal 𝔪_{κ} and let L := O_{W}(V)/𝔪_{κ}. It is enough to show to that the 𝔪_{κ}- torsion vanishes for all κ. This torsion is equal to Tor_{1}^{O_{W} (V)}(H^{1}(K, D_{V}^{s})_{≤h}, L). Using the Tor-spectral sequence ([Han1, Theorem 3.3.1]) one sees that this Tor-group is a subquotient of H^{0}(K, D_{κ}^{s})_{≤h}, which vanishes by the previous Lemma. □

Next, assume that our tame level is of the form K_{1}(N) for some N with (N, dp) = 1. Let C^{nc} ⊂ C denote the set of non-critical classical crystalline points for which the roots of the pth Hecke polynomial are distinct.^{2} For each positive divisor M|N, let C_{M−new}^{nc} denote the set of points x in C^{nc} for which the tame Artin conductor of the Galois representation ρ_{x} associated to x is exactly Md. Note that the tame Artin conductor is sometimes just called the tame conductor; see the discussion in [Bel] before Lemma IV.4.3. Let C_{M−new} denote the Zariski-closure of C_{M−new}^{nc} in C. The following lemma was stated without proof for modular curves in [Han2].

Lemma 5.6. Assume that the tame level is of the form K_{1}(N). For any M|N, C_{M−new} is a union of irreducible components of C, and C = ∪_{M|N}C_{M−new}.

Proof. Adapting the proof of [Bel, Lemma IV.4.7], one shows that C_{M−new}^{nc} is an accumulation subset of C: each point x ∈ C_{M−new}^{nc} has a neighborhood basis of affinoids U ⊂ C for which U ∩ C_{M−new}^{nc} is Zariski-dense in U. This property implies that each irreducible component of the Zariski-closure C_{M−new} has positive dimension. Since C is equidimensional of dimension one, we deduce from [Con, Corollary 2.2.7] that C_{M−new} is a union of irreducible components of C. Since C^{nc} = ∪_{M|N}C_{M−new}^{nc} is a Zariski-dense accumulation subset of C, the remainder of the lemma is clear. □

5.2. Faltings’s Eichler-Shimura map. In this section we adapt Faltings’s construction ([Fal]) of the p-adic Eichler-Shimura morphism to our setting, and use it to give a precise description of ES_{k} at weights k ≥ 2. Fix k ∈ ℤ_{≥2}. Recall the morphism

V̂_{k} → ω^{⊗k−2}

defined in §4.5. It induces a map

H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}).

2This last condition is conjecturally automatic.

Note that we have an exact sequence

(1) 0 → H_{ét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1) → 0

coming from the spectral sequence H_{ét}^{i}(X_{ℂ_{p}}, R^{j}ν_{∗}ω^{⊗k−2}) ⇒ H_{proét}^{i+j}(X_{ℂ_{p}}, ω^{⊗k−2}), which degenerates at the E_{2}-page by an argument similar to (but simpler than) the proof of Proposition 4.7.

Proposition 5.7. The composite map

H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1)

is surjective, and the kernel is isomorphic to H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(k) as a Hecke- and Galois module.

Proof. Recall the Hodge-Tate sequence

0 → ω^{⊗−1}(1) → T̂ → ω̂ → 0

for G^{univ}. Since the morphism V̂_{k} → ω^{⊗k−2} comes from the map T̂ → ω̂ by taking (k − 2)-th symmetric powers, we see that its kernel Ker_{k} sits in a short exact sequence

0 → ω^{⊗2−k}(k − 2) → Ker_{k} → Q → 0

where Q is simply defined to be the quotient. We get exact sequences

H_{proét}^{1}(X_{ℂ_{p}}, Ker_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2});
H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k)) → H_{proét}^{1}(X_{ℂ_{p}}, Ker_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, Q).

We have an isomorphism H_{proét}^{1}(X_{ℂ_{p}}, V̂_{j}) ≅ H_{ét}^{1}(X_{ℂ_{p}}, Sym^{j} T ) ⊗_{ℤ_{p}} ℂ_{p} for all j ≥ 0. From the Hodge-Tate sequence one deduces that Q carries a filtration with non-zero graded pieces ω^{⊗i} for i = 4−k, 6−k, ..., k−4 (up to twists). Using this filtration and the sequences (1) with k−2 replaced by i = 4−k, 6−k, ..., k−4 together with the usual Eichler-Shimura isomorphism for quaternionic Shimura curves one sees that H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) and H_{proét}^{1}(X_{ℂ_{p}}, Q) do not have any Hecke eigenvalues in common. Thus, looking at the second exact sequence above, we see that any generalized Hecke eigenvector in H_{proét}^{1}(X_{ℂ_{p}}, Ker_{k}) that is not killed by the map H_{proét}^{1}(X_{ℂ_{p}}, Ker_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) must come from H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)).

Hence we have an exact sequence

H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)) → H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}).

By the exact sequence (1) and looking at Hecke eigenvalues again we get an exact sequence

H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)) → H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1).

We now apply a similar argument to H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)). Replacing k − 2 with 2 − k in (1) we have the exact sequence

0 → H_{ét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}) → H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗2−k} ⊗ Ω_{X_{ℂp}}^{1} )(−1) → 0.

Arguing with Hecke eigenvalues as above and using Serre duality we get a sequence

(2) 0 → H_{ét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)) → H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1) → 0

which is exact in the middle. By Serre duality

H_{ét}^{1}(X_{ℂ_{p}}, ω^{⊗2−k}(k − 2)) ≅ H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(k − 2).

Counting dimensions, we see that (2) is a short exact sequence as desired. □

We have a diagram

[figure: H_{proét}^{1}(X_{ℂ_{p}}, OD_{k}^{s}); ES_{k}; H_{proét}^{1}(X_{w,ℂ_{p}}, ω_{k,w}^{†}); ∼; H_{ét}^{0}(X_{w,ℂ_{p}}, ω_{k,w}^{†} ⊗ Ω_{X_{w,ℂp}}^{1} )(−1); H_{proét}^{1}(X_{ℂ_{p}}, V̂_{k}); H_{proét}^{1}(X_{ℂ_{p}}, ω^{⊗k−2}); H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1)]

which commutes by Remark 4.17 and the functoriality of the remaining maps. Note that the right vertical map is injective by analytic continuation. Its image is, by definition, the space of classical modular forms. We then have:

Proposition 5.8. Let k ≥ 2. The image of ES_{k} is contained in the space of classical modular forms. Moreover ES_{k} is surjective on slope ≤ h-parts when h < k − 1.

Proof. The first statement is clear from the diagram. To see the second, note that both the left and right vertical maps in the diagram above are isomorphisms on slope ≤ h-parts when h < k − 1 by the control theorems of Stevens and Coleman. To conclude, use that the map H^{1}(X_{ℂ_{p}}, V̂_{k}) → H_{ét}^{0}(X_{ℂ_{p}}, ω^{⊗k−2} ⊗ Ω_{X_{ℂp}}^{1} )(−1) is surjective by Proposition 5.7. □

Note that the kernel of ES_{k} is big: it is infinite-dimensional with finite codimension.

5.3. Results. In this section we deduce some properties of our overconvergent Eichler-Shimura map ES and the sheaves V and M^{†}. The results are very similar to those of [AIS2, §6], but we are able to prove them in a more global form. Let us denote by C^{sm} the smooth locus of C. It contains the étale locus C^{ét} of the weight map, and both these loci are Zariski open. Furthermore C^{ét} contains the set C^{nc} of non-critical classical crystalline points for which the roots of the p-Hecke polynomial are distinct. Let us explicitly record a lemma in rigid geometry that makes arguments involving Zariski density simpler. We will (sometimes implicitly) apply it in the remainder of this section.

Lemma 5.9. Let X be a smooth rigid space over a non-archimedean field K, and assume that (V_{n})_{n=1}^{∞} is an increasing cover of X by affinoids. Assume that S ⊆ X is a very Zariski dense set of points. Then we may find an increasing cover (U_{n}) of X by affinoids such that S ∩ U_{n} is very Zariski dense in U_{n} for all n.

Proof. We define U_{n} to be the Zariski closure of S ∩ V_{n} in V_{n}. By definition U_{n} is a union of components of V_{n}, hence affinoid, and it remains to show that X = ⋃_{n=1}^{∞} U_{n}. For this it suffices to show that, for fixed n and a fixed component C of V_{n}, there exists m ≥ n such that C ⊆ U_{m}. Fix n and C. C is a subset of a (global) component D of X, so by Zariski density we pick a point s ∈ S ∩ D. If m ≥ n, let C_{m} denote the component of V_{m} such that C ⊆ C_{m}. Define, for m ≥ n,

T_{m} = {x ∈ V_{m} | ∃k ≥ m : i_{km}(x) ∈ C_{k}}

where i_{km} : V_{m} → V_{k} is the inclusion map. Then T_{m} is a union of components and we claim that s ∈ T_{m} for some m. To see this, note that T_{m} = V_{m} ∩ T_{m+1} for all m ≥ n, so T := ⋃_{m≥n} T_{m} is an open subset of X with open complement. Hence if s ∉ T, s could not lie on the same component as C. Therefore there is some m for which s ∈ T_{m}, and hence some k for which s ∈ C_{k}, which implies that C ⊆ U_{k} by the very Zariski density of S. □

Remark 5.10. Note that C^{sm} may be covered by an increasing union of affinoids. To see this note first that this is true for C since it is finite over W ×𝔸^{1}, which is quasi-Stein. If we pick an increasing affinoid cover (V_{n}) of C, then C^{sm} = ⋃_{n} V_{n}^{sm} and V_{n}^{sm} is Zariski dense in V_{n}, with complement cut out by the functions f_{n,1}, ..., f_{n,k_{n}} say. We may choose ϵ_{n} ∈ p^{ℚ} tending to 0 such that

{|f_{n,1}|, ..., |f_{n,k_{n}}| < ϵ_{n}} ∩ V_{m} ⊆ {|f_{m,1}|, ..., |f_{m,k_{m}}| < 2^{n−m}ϵ_{m}}

for all m ≤ n, where we use a general fact that the sets ({|g_{i}| ≤ ϵ})_{ϵ∈p^{ℚ}} define a cofinal system of neighbourhoods of {|g_{i}| = 0}. Then we claim that the sets

U_{n} = {|f_{n,1}|, ..., |f_{n,k_{n}}| ≥ ϵ_{n}}

are affinoid and cover C^{sm}. Indeed, they cover C^{sm} by definition, and they are affinoid since they are rational subsets of the V_{n}.

With this we state a simple consequence of Proposition 5.8:

Theorem 5.11. View ES as a morphism of coherent sheaves on C_{ℂ_{p}}. Then ES is surjective outside a Zariski-closed subset of C_{ℂ_{p}} of dimension 0, and the support of coker(ES) is disjoint from the set of non-critical points.

Proof. Since coker(ES) is coherent its support is Zariski-closed. Hence the first statement follows from the second: if the support is not of dimension 0, then it must contain some component of C and hence non-critical points. To prove the second statement it suffices to prove that ES is surjective on fibres at noncritical points. Let x be such a point, with weight k. By the computation of the fibres in §5.1 and the computation of ES we see that it amounts to the Eichler-Shimura map ES_{k} restricted to maximal semisimple quotients of the generalized eigenspaces for x. Since x is non-critical this map is surjective by Proposition 5.8. □

Let us raise the following question:

Question: Is the support of coker ES precisely the set of x with weight k ∈ ℤ_{≥2} such that ES_{k} fails to be surjective on generalized eigenspaces for x?

Next let us state some results on the structure of V and M^{†} over C^{sm}. For any n ∈ ℤ_{≥1} we let τ(n) denote the number of positive divisors of n.

Theorem 5.12. Assume, for simplicity, that that the tame level K^{p} is of the form ∏_{ℓ∤disc(B)} K_{1}(N) × ∏_{ℓ|disc(B)} O_{B_{ℓ}}^{×} for some N ≥ 3. Let M ∈ ℤ_{≥1} be a divisor of N.

(1) V and M^{†} are locally free on C^{sm}. The rank of V over C_{M−new}^{sm} is 2τ(N/M) and the rank of M^{†} over C_{M−new}^{sm} is τ(N/M).

(2) Let ℓ be a prime not dividing Np · disc(B). Then V is unramified at ℓ and the trace of Frob_{ℓ} on V over C_{M−new}^{sm} is τ(N/M) · T_{ℓ}, where we view T_{ℓ} as a global function on C.

(3) Let x ∈ C_{N−new}^{sm}(\overline{ℚ}_{p}). Then the fibre of V at x is the Galois representation attached to x via the theory of pseudorepresentations.

Proof. First we prove (1). By Lemma 5.5 V and M^{†} are torsion free coherent sheaves over C^{sm}, which is smooth of pure dimension 1. Therefore V and M^{†} are locally free. To compute the ranks over C_{M−new}^{sm} it suffices to compute the dimension of the fibres at points in C^{nc} ∩ C_{M−new}^{sm}. This is a computation using Atkin-Lehner theory, the control theorems of Stevens resp. Coleman and the classical Eichler-Shimura isomorphism (see e.g [Bel, Lemma IV.6.3]).

To prove (2), note that the statements are true after specializing to points in C^{nc} (using Stevens’s control theorem and the well known structure of the Galois representations H_{ét}^{1}(X , V_{k})). By Zariski density of C^{nc} ∩ C_{M−new}^{sm} in C_{M−new}^{sm} we may then conclude. Note that, to check that it is unramified, one may reduce to the affinoid case by Lemma 5.9 and Remark 5.10, where it becomes a purely ring-theoretic statement. Localizing (ring-theoretically), one may then reduce to the free case, where one can check on matrix coefficients using Zariski density.

Finally, (3) follows from (2). □

Remark 5.13. The assumption on K^{p} in the above theorem is only to simplify the exposition. The theorem and the definition of the C_{M−new} may be adapted to arbitrary tame levels K^{p} = ∏_{ℓ≠p} K_{ℓ} with essentially the same proof, except for the explicit formulas for the ranks.

We remark that the statement that V is unramified at ℓ ∤ Np · disc(B) also follows from the same fact for the V_{U}^{s} (for U small open), which in turn follows from the construction upon noting that all members of the tower (X_{K_{p}})_{K_{p}} are analytifications of schemes over ℚ with good reduction at ℓ.

Finally, we prove that ES generically gives us Hodge-Tate filtrations/decompositions on V.

Theorem 5.14. We work over C^{sm}.

(1) The kernel K and image I of ES are locally projective sheaves of O_{C^{sm}} ˆ⊗_{ℚ_{p}}ℂ_{p}-modules, and may also be viewed as locally free sheaves on C_{ℂ_{p}}^{sm}.

(2) Let ϵ_{C^{sm}} be the character of G_{ℚ_{p}} defined by the composition

G_{ℚ_{p}} ⟶^{ϵ} ℤ_{p}^{×} ⟶^{χ_{W}} O_{W}^{×} ⟶ (O_{C^{sm}}^{×} ˆ⊗_{ℚ_{p}}ℂ_{p})^{×}

where ϵ is the p-adic cyclotomic character of G_{ℚ_{p}} and χ_{W} is the universal character of ℤ_{p}^{×}. Then the semilinear action of G_{ℚ_{p}} on the module K(ϵ_{C^{sm}}^{−1}) is trivial.

(3) The exact sequence

0 → K → V_{ℂ_{p}} → I → 0

is locally split. Zariski generically, the splitting may be taken to be equivariant with respect to both the Hecke- and G_{ℚ_{p}}-actions, and such a splitting is unique.

Proof. Part (1) is clear since V and M^{†} are locally free on C^{sm} and O_{C^{sm}} ˆ⊗_{ℚ_{p}}ℂ_{p} is locally a Dedekind domain. The second statement follows similarly.

For part (2) we use Lemma 5.9 and Remark 5.10 to reduce to the affinoid case with a very Zariski dense set of points in C^{nc}. Taking a further cover we may also assume that K is free (perhaps after making a finite extension, but this will not effect the rest of the argument). The statement now follows from the family version of Sen theory ([Se1], [Se2]) , using the fact that the Sen operator ϕ of K(ϵ_{C^{sm}}^{−1}) vanishes on points of C^{nc} and hence on C^{sm} by Zariski density of C^{nc} and analyticity of ϕ. The argument proceeds exactly as the proof of [AIS2, Theorem 6.1(c)], to which we refer for more details.

Finally we prove part (3), arguing more or less exactly as in the proof of [AIS2, Theorem 6.1(d)], to which we refer for more details. Once again we may work over an affinoid U ⊆ C^{sm} with a Zariski dense set of points U ∩ C^{nc}. We may without loss of generality assume that K, V_{ℂ_{p}} and I are free (once again, one might need to make a finite extension, but this does not effect the rest of the argument and we will ignore it). The short exact sequence is an extension which defines an element in H^{1}(G_{ℚ_{p}}, H) where

H := Hom_{O_{C}(U)ˆ⊗_{ℚp} ℂ_{p}}(I, K).

Let ϕ be the Sen operator of H. By work of Sen it is well-known that det(ϕ) kills H^{1}(G_{ℚ_{p}}, H). Specializing at points in U ∩ C^{nc} one sees that det(ϕ) does not vanish identically on any component of U. Thus, localizing with respect to det(ϕ), we find a Zariski open subset of U over which the extension is split as semilinear G_{ℚ_{p}}-representations. Moreover, if we further remove the finite set of points whose weight is −1 ∈ ℤ, then I and K have distinct Hodge-Tate weights fibre-wise (by using part (2) and that I has constant Hodge-Tate weight −1). Thus there can be no non-zero Galois equivariant homomorphisms between K and I. Since the Hecke action commutes with the G_{ℚ_{p}}-action, this implies that the G_{ℚ_{p}}-splitting must be Hecke stable as well. □

6. Appendix

In this appendix we define a ”mixed” completed tensor product that we will use in the main text, and prove some basic properties as well as a few technical results that we need.

6.1. Mixed completed tensor products. Let K be a finite extension of ℚ_{p} and let O be its ring of integers, with uniformizer ϖ. Our ”mixed completed tensor products” will be denoted by an unadorned ˆ⊗. The base field is assumed to be implicit in the notation. In the main text we will always take K = ℚ_{p} so this should not cause any problem. We starting by making the following definition:

Definition 6.1. Let M be a topological O-module.

(1) M is called linear-topological if there exists a basis of neighbourhoods of 0 consisting of O-submodules.

(2) We will say that M is a profinite flat O-module if M is flat over O (i.e torsion-free), linear-topological and compact.

Let us remark that if M is a profinite flat O-module then the topology on M is profinite, which justifies this terminology. We also remark that such M are exactly the O-modules which are projective and pseudocompact in the language of [SGA3, Exposé VII_{B}, §0]. Let us recall the following structure theorem for profinite flat O-modules, specialized from [SGA3, Exposé VII_{B}, 0.3.8]:

Proposition 6.2. M is a profinite flat O-module if and only if it is isomorphic to ∏_{i∈I} O equipped with the product topology, for some set I.

We note that a set of elements (e_{i})_{i∈I} in M such that M ≅ ∏_{i∈I} Oe_{i} is called a pseudobasis. In what follows ‘ϖ-adically complete” always means complete and separated.

Definition 6.3. Let M be a profinite flat O-module and let X be any Omodule. We define

X ˆ⊗M := \underleftarrow{lim}_{i}(X ⊗_{O} M/I_{i})

where (I_{i}) runs through any cofinal set of neighbourhoods of 0 consisting of ℤ_{p}-submodules.

By abstract nonsense this is independent of the choice of system of neighbourhoods of 0. Note that if A is an O-algebra and R is a small O-algebra, then Aˆ⊗R comes with a natural structure of a ring, as is seen by choosing the neighbourhoods to be ideals. To compute X ˆ⊗M in a useful form we fix a pseudobasis (e_{i})_{i∈I} and define, for J ⊆ I finite and n ≥ 1 an integer, M_{J,n} to be the submodule corresponding to

M_{J,n} := ∏_{i∈J} ϖ^{n}Oe_{i} × ∏_{i∉J} Oe_{i}.

This forms a system of neighbourhoods of 0. Then we have:

Proposition 6.4. The functor X ↦ X ˆ⊗M is isomorphic to the functor X ↦ ∏_{i∈I} X̂, where X̂ is the ϖ-adic completion of X.

Proof. We compute

X ˆ⊗M = \underleftarrow{lim}_{J,n} X ⊗_{O} M/M_{J,n}
= \underleftarrow{lim}_{J}(\underleftarrow{lim}_{n} X ⊗_{O} M/M_{J,n})
≅ \underleftarrow{lim}_{J}(\underleftarrow{lim}_{n} X ⊗_{O} ∏_{i∈J} O/ϖ^{n})

= \underleftarrow{lim}_{J} (∏_{i∈J}(\underleftarrow{lim}_{n} X ⊗_{O} O/ϖ^{n}))

= \underleftarrow{lim}_{J} (∏_{i∈J} X̂)
= ∏_{i∈I} X̂.
□

Corollary 6.5. We have (1) X ↦ X ˆ⊗M is exact on ϖ-adically complete X.

(2) If 0 → X → Y → Q → 0 is an exact sequence of O-modules where X is ϖ-adically complete and Q is killed by ϖ^{N} for some integer N ≥ 0, then Y is ϖ-adically complete and the natural map (X ˆ⊗M)[1/ϖ] → (Y ˆ⊗M)[1/ϖ] is an isomorphism.

Proof. (1) follows from exactness of products in the category of O-modules. For (2) note that Q is ϖ-adically complete and Qˆ⊗M is killed by ϖ^{N}, so the second statement will follow from the first and (1). To see that Y is ϖ-adically complete first note that ϖ^{N}Y ⊆ X (i.e. Q killed by ϖ^{N}) so

ϖ^{N+k}Y ⊆ ϖ^{k}X ⊆ ϖ^{k}Y

for all k ≥ 0 and hence we can compute Ŷ as \underleftarrow{lim}_{k} Y/ϖ^{k}X. Thus we get exact sequences

0 → X/ϖ^{k}X → Y/ϖ^{k}X → Q → 0

for all k and hence an exact sequence

0 → X = X̂ → Ŷ → Q

by left exactness of inverse limits. This sits in a natural diagram

\xymatrix{0 \ar[r] & X \ar[d] \ar[r] & Y \ar[d] \ar[r] & Q \ar[d] \ar[r] & 0 \\ 0 \ar[r] & X \ar[r] & Ŷ \ar[r] & Q}

with exact rows and the snake lemma now implies that the map Y → Ŷ is an isomorphism as desired. □

In particular we can now define −ˆ⊗M on K-Banach spaces (this is why we are calling it a ”mixed” completed tensor product). It is canonically independent of the choice of unit ball, but we also make a more general definition.

Definition 6.6. Let V be a K-vector space and let V^{∘} ⊆ V be an O_{K}- submodule such that V^{∘}[1/ϖ] = V. Then we define

V ˆ⊗M := (V^{∘} ˆ⊗M)[1/ϖ]

where the choice of V^{∘} is implicit (the choice will in general affect the result).

If V is naturally a Banach space we will always take V^{∘} to be an open and bounded O-submodule, and Corollary 6.5(2) implies that V ˆ⊗M is independent of the choice. We finish with a lemma needed in the main text.

Lemma 6.7. Let S be a K-Banach algebra and let V be a finite projective Smodule and U a Banach S-module. Let M be a profinite flat O_{K}-module. Then we have a natural isomorphism V ⊗_{S} (U ˆ⊗M) ≅ (V ⊗_{S} U)ˆ⊗M. If V and U in addition are Banach S-algebras and M is a small O_{K}-algebra, then this is an isomorphism of rings.

Proof. Pick an open bounded O_{K}-subalgebra S_{0} of S and pick open bounded S_{0}-submodules V_{0} ⊆ V and U_{0} ⊆ U. Moreover, pick O_{K}-submodules (I_{i}) of M that form a basis of neighbourhoods of 0. Then we have natural maps

V_{0} ⊗_{S_{0}} (\underleftarrow{lim}_{i}(U_{0} ⊗_{O_{K}} M/I_{i})) → V_{0} ⊗_{S_{0}} U_{0} ⊗_{O_{K}} M/I_{i}

for all i which give us a natural map

V_{0} ⊗_{S_{0}} (U_{0} ˆ⊗M) → (V_{0} ⊗_{S_{0}} U_{0})ˆ⊗M.

Now invert ϖ to get the desired map. If V and U are Banach S-algebras we may choose everything so that V_{0} and U_{0} are subrings, and if M is a small O_{K}-algebra we may choose the I_{i} to be ideals, hence we see that this is a ring homomorphism. To see that it is an isomorphism note that if V is in addition free we may choose V_{0} to be a finite free S_{0}-module. The assertion is then clear. In general, write V as a direct summand of a finite free S-module. □

6.2. Mixed completed tensor products on rigid-analytic varieties. Let X be a quasicompact and separated smooth rigid analytic variety over an algebraically closed and complete extension C of K, of pure dimension n. Here K is a finite extension of ℚ_{p} as in the previous subsection and we use the same notation O = O_{K}, ϖ et cetera. In this subsection we aim to prove:

Proposition 6.8. We have a canonical isomorphism

R^{i}ν_{∗}(Ô_{X} ˆ⊗M) ≅ (R^{i}ν_{∗}Ô_{X})ˆ⊗M

of sheaves on X_{ét}.

Here U ↦ (Ô_{X} ˆ⊗M)(U) = Ô_{X}(U)ˆ⊗M for U ∈ X_{proét} qcqs defines a sheaf on X_{proét} and the completed tensor product is computed using Ô_{X}^{+}(U) ⊆ Ô_{X}(U). The former is ϖ-adically complete by [Sch3, Lemma 4.2(iii)], so we have exactness of −ˆ⊗M and hence a sheaf. We use Im((R^{i}ν_{∗}Ô_{X}^{+})(U) → (R^{i}ν_{∗}Ô_{X})(U)) to compute the completed tensor product on the right hand side and claim that this is a sheaf, but these statements are not obvious and we will prove them below.

We note the following important corollary to the above proposition.

Corollary 6.9. We have

R^{i}ν_{∗}(Ô_{X} ˆ⊗M) ≅ Ω_{X}^{i}(−i) ⊗_{O_{X}} (O_{X} ˆ⊗M)

as sheaves on X_{ét}.

Proof. This follows from Proposition 6.8, Lemma 6.7 and the canonical isomorphism R^{i}ν_{∗}(Ô_{X}) ≅ Ω_{X}^{i}(−i) (Proposition 3.23 of [Sch2]) together with the finite projectivity of the latter sheaf. □

We will need several lemmas before we can prove Proposition 6.8.

Lemma 6.10. The presheaf U ↦ H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) on X_{ét} is a sheaf, where O_{X}^{+}(U) has its ϖ-adic topology and the trivial ℤ_{p}^{n}-action.

Proof. This follows from the computation in [Sch3, Lemma 5.5]; H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) is isomorphic to ⋀^{i} O_{X}^{+}(U)^{n} in a way compatible with completed tensor products. □

Put T^{n} = Spa(C⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩, O_{C}⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩). We let T̃^{n} be the inverse limit over n ≥ 0 of Spa(C⟨X_{1}^{±1/p^{n}}, ..., X_{n}^{±1/p^{n}}⟩, O_{C}⟨X_{1}^{±1/p^{n}}, ..., X_{n}^{±1/p^{n}}⟩). This is an affinoid perfectoid (see [Sch3, Example 4.4]).

Lemma 6.11. Let U ∈ X_{ét} and and assume that there is a map U → T^{n} which is a composition of rational embeddings and finite étale maps. Pull back the affinoid perfectoid ℤ_{p}^{n}-cover T̃^{n} to obtain an affinoid perfectoid ℤ_{p}^{n}-cover Ũ of U. Then:

(1) There is a canonical injection

H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) → H^{i}(U_{proét}, Ô_{X}^{+})

with cokernel killed by p.

(2) H^{i}(U_{proét}, Ô_{X}) ≅ H_{cts}^{i}(ℤ_{p}^{n}, O_{X}(U)).

(3) H^{i}(U_{proét}, Ô_{X}^{+} ˆ⊗M)^{a} ≅ (H^{i}(U_{proét}, Ô_{X}^{+})ˆ⊗M)^{a} (where −^{a} denotes the associated almost O_{C}-module).

(4) H^{i}(U_{proét}, Ô_{X} ˆ⊗M) ≅ H^{i}(U_{proét}, Ô_{X})ˆ⊗M.

Proof. We start with (1). The Cartan–Leray spectral sequence (cf. Remark 2.25) gives us

H_{cts}^{p}(ℤ_{p}^{n}, H^{q}(Ũ_{proét}, O_{X}^{+}/ϖ^{m})) ⟹ H^{p+q}(U_{proét}, O_{X}^{+}/ϖ^{m})

for all m. Since the H^{q}(Ũ_{proét}, O_{X}^{+}/ϖ^{m}) are almost zero for all q ≥ 1 ([Sch3, Lemma 4.10], see the proof of (i) and (v) ) we conclude that

H^{p}(U_{proét}, O_{X}^{+}/ϖ^{m})^{a} = H^{p}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(Ũ))^{a}.

Next, note that the inverse system

(H^{p}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(Ũ))_{m}

has surjective transition maps by the proof of [Sch3, Lemma 5.5]. Since the inverse system (O_{X}^{+}/ϖ^{m})_{m} has almost vanishing higher inverse limits on the pro-étale site (by an easy application of the almost version of [Sch3, Lemma 3.18]) we conclude that there is an almost isomorphism

\underleftarrow{lim} H^{p}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(Ũ)^{a} ≅ H^{p}(U_{proét}, Ô_{X}^{+})^{a}.

By [Sch3, Lemma 5.5] we have

⋀^{p}(O_{X}^{+}/ϖ^{m})(U)^{n} ≅ H^{p}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(U)) → H^{p}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(Ũ))

with cokernel killed by p, compatibly in m; in fact the proof shows that the injection is split. It remains to verify that \underleftarrow{lim} H^{q}(ℤ_{p}^{n}, (O_{X}^{+}/ϖ^{m})(U)) ≅ H_{cts}^{q}(ℤ_{p}^{n}, O_{X}^{+}(U)). This is standard; one may e.g. argue as in the proof of [NSW, Theorem 2.5.7] (note that the relevant \underleftarrow{lim}^{1} vanishes by the isomorphism above). This finishes the proof of (1).

For (2), use (1) and invert ϖ - this commutes with taking cohomology by quasicompactness.

For (3) we view −ˆ⊗M as the functor of taking self products over some index set I. We remark that H^{i}(U_{proét}, Ô_{X}^{+})^{a} is ϖ-adically complete by part (1), [Sch3, Lemma 5.5] and the almost version of Corollary 6.5(3). Thus we are left with verifying that cohomology for Ô_{X}^{+} almost commutes with arbitrary products. Writing ∏_{i∈I} as \underleftarrow{lim}_{J} ∏_{i∈J} for all J ⊆ I finite we are left with showing that the relevant higher inverse limits vanish, which is easy using the Mittag-Leffler condition and [Sch3, Lemma 3.18].

Finally we invert ϖ in (3) to deduce (4). □

Lemma 6.12. The sheaf R^{i}ν_{∗}Ô_{X} is equal to the sheaf U ↦ H_{cts}^{i}(ℤ_{p}^{n}, O_{X}(U)), and (R^{i}ν_{∗}Ô_{X})ˆ⊗M may be computed using H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) ⊆ R^{i}ν_{∗}Ô_{X}(U).

Proof. We may work étale locally on X, so assume without loss of generality that there is a map X → T^{n} which is a composition of rational embeddings and finite étale maps. By abstract nonsense R^{i}ν_{∗}Ô_{X} is the sheafification of the presheaf U ↦ H^{i}(U_{proét}, Ô_{X}). Let us write F for this presheaf. By part (1) of the previous Lemma there is an exact sequence

0 → H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) → H^{i}(U_{proét}, Ô_{X}^{+}) → Q^{i}(U) → 0

where Q^{i}(U) is simply defined as the quotient. This is an exact sequence of presheaves on X_{et}^{w}. Moreover Q^{i}(U) is killed by p for all U. Write G for the presheaf U ↦ H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)), F^{+} for U ↦ H^{i}(U_{proét}, Ô_{X}^{+}) and Q for U ↦ Q^{i}(U). Then the exact sequence above sheafifies to

0 → G → (F^{+})^{sh} = R^{i}ν_{∗}Ô_{X}^{+} → Q^{sh} → 0

( −^{sh} for sheafification) since G is already a sheaf by Lemma 6.10, and Q^{sh} is killed by p since Q is. Inverting ϖ we get that H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U))[1/ϖ] ≅ R^{i}ν_{∗}Ô_{X}(U). For the other assertion of the lemma, note that Im(R^{i}ν_{∗}Ô_{X}^{+}(U) → R^{i}ν_{∗}Ô_{X}(U)) coincides with H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) under the identification H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U))[1/ϖ] ≅ R^{i}ν_{∗}Ô_{X}(U), and is therefore a sheaf (as was asserted in Proposition 6.8). □

Proof of Proposition 6.8: Lemma 6.12 gives us that

((R^{i}ν_{∗}Ô_{X})ˆ⊗M)(U) = (H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U))ˆ⊗M)[1/ϖ].

On the other hand R^{i}ν_{∗}(Ô_{X} ˆ⊗M) is the sheaf associated with the presheaf

U ↦ H^{i}(U_{proét}, Ô_{X} ˆ⊗M)

by abstract nonsense. Working locally again, assume that there is a map U → T^{n} as above. We may then compute

H^{i}(U_{proét}, Ô_{X} ˆ⊗M) = H^{i}(U_{proét}, Ô_{X})ˆ⊗M
= (Im(H^{i}(U_{proét}, Ô_{X}^{+}) → H^{i}(U_{proét}, Ô_{X}))ˆ⊗M)[1/ϖ]
= (H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U))ˆ⊗M)[1/ϖ]

using Lemma 6.11 (4) and the proof of Lemma 6.12. Therefore U ↦ H^{i}(U_{proét}, Ô_{X} ˆ⊗M) is actually defines sheaf when restricting to the basis of such U (since U ↦ H_{cts}^{i}(ℤ_{p}^{n}, O_{X}^{+}(U)) is a sheaf), so it must equal R^{i}ν_{∗}(Ô_{X} ˆ⊗M) on these U. This gives the desired isomorphism. □

6.3. Analogues of the theorems of Tate and Kiehl. Let X be a quasicompact and quasiseparated rigid analytic variety over some complete nonarchimedean field K of characteristic zero. For simplicity let us work with the site given by the basis of the topology consisting of quasicompact open subsets together with finite covers (we will specialize this further eventually). Given a small ℤ_{p}-algebra R, we may form the sheaf R(U) = O_{X}(U)ˆ⊗R on X (U is qc open). In this section we are going to prove analogues of the theorems of Tate and Kiehl for O_{X} for R under the assumption that X is affinoid and that K is discretely valued. The assumption that K is discretely valued makes it easy to prove various flatness assertions. We suspect that this assumption can be dropped with some more work but we will not need the extra generality. We will however prove a few statements that we need without this assumption on K. For simplicity we work with the topology on X given by rational subsets and finite covers. We start with some basic properties.

Lemma 6.13. Let X be affinoid and assume that K is discretely valued. (1) For every rational U ⊆ X, R(U) is Noetherian.

(2) If V ⊆ U ⊆ X are rational, then the map R(U) → R(V) is flat.

(3) If U is rational and (U_{i})_{i} is a finite rational cover of U, then the natural map R(U) → ∏_{i} R(U_{i}) is faithfully flat.

Proof. To prove (1), choose surjections O_{K}⟨X_{1}, …, X_{m}⟩ ↠ O_{X}(U), ℤ_{p}[[T_{1}, …, T_{n}]] ↠ R and use the image of ℤ_{p}⟨X_{1}, …, X_{m}⟩ to compute O_{X}(U)ˆ⊗R (this is valid by the open mapping theorem). Then we get a surjection

O_{K}⟨X_{1}, …, X_{m}⟩ˆ⊗ℤ_{p}[[T_{1}, …, T_{n}]] ↠ O_{X}(U)ˆ⊗R

and by direct computation the left hand side is

(O_{K}⟨X_{1}, …, X_{m}⟩[[T_{1}, …, T_{n}]])[1/p]

, which is Noetherian (since K is discretely valued), hence the right hand side is Noetherian.

Next we prove (2). Pick functions f_{1}, …, f_{n}, g such that V = {|f_{1}|, …, |f_{n}| ≤ |g| ≠ 0}. We factor R(U) → R(V) as

O_{X}(U)ˆ⊗R → O_{X}(U)[1/g] ⊗_{O_{X}(U)} (O_{X}(U)ˆ⊗R) → O_{X}(V)ˆ⊗R.

The first morphism is flat since O_{X}(U) → O_{X}(U)[1/g] is, so it suffices to show that the second morphism is flat. It is obtained from

O_{X}^{∘}(U)[f_{1}/g, …, f_{n}/g] ⊗_{O_{X}^{∘}(U)} (O_{X}^{∘}(U)ˆ⊗R) → O_{X}^{∘}(V)ˆ⊗R

by inverting p, so it suffices to show that this map is flat. But the right hand side is the completion of the left hand side with respect to the ideal generated by the images of the generators of the maximal ideal of R. Since the ring on the left hand side is Noetherian (it is finitely generated over O_{X}^{∘}(U)ˆ⊗R, which is Noetherian by the same proof as in (1)), we conclude that the map is flat.

For (3) we use a more geometric argument. By part (2) the map

O_{X}(U)ˆ⊗R → ∏_{i} O_{X}(U_{i})ˆ⊗R

is flat. To see that it is faithfully flat, it suffices to show that it induces a surjection on spectra of maximal ideals, by [Bou, §§I.3.5, Proposition 9]. To simplify notation, put S = O_{X}(U) and S_{i} = O_{X}(U_{i}). Since we are only interested in maximal ideals, we will momentarily work with classical rigid spaces rather than adic spaces in this proof (only). R, S^{∘} and the S_{i}^{∘} are formally of finite type over O_{K}, and the unadorned completed tensor products S^{∘} ˆ⊗R etc. are completed tensor products over O_{K} in the category of topological O_{K}-algebras formally of finite type. Recall Berthelot’s generic fibre functor (−)^{rig} (see e.g. [dJ, §7]). By [dJ, Proposition 7.2.4(g)] the generic fibre functor commutes with fibre products, so the morphism

⨆_{i} U_{i} ×_{Sp K} Spf(R)^{rig} = ⨆_{i} Spf(S_{i}^{∘} ˆ⊗R)^{rig} →

→ Spf(S^{∘} ˆ⊗R)^{rig} = U ×_{Sp K} Spf(R)^{rig}

is surjective (it is a finite open cover). By [dJ, Lemma 7.1.9] we deduce that we have a surjection

⨆_{i} MaxSpec(S_{i} ˆ⊗R) = MaxSpec (∏_{i} S_{i} ˆ⊗R) → MaxSpec(S ˆ⊗R),

which is what we wanted to prove. □

Having established this Lemma we now view R as a sheaf of Banach algebras on X by picking the unit ball of R(U) to be R^{∘}(U) = O_{X}^{∘}(U)ˆ⊗R. Then all restriction maps are norm-decreasing, hence continuous, and if we view R as a profinite flat ℤ_{p}-module and choose a pseudobasis with index set I, the Banach space structure on R(U) is the same as the natural Banach space structure on the set of bounded sequences in O_{X}(U) indexed by I (of course the multiplication is very different from the pointwise multiplication on the latter). We will follow [AW, §5] (which in turn follows [FvdP, §4.5]) closely in the proof of our analogue of Kiehl’s theorem.

Definition 6.14. Let U ⊆ X be rational and let M be a finitely generated R(U)-module. For every V ⊆ U rational, we define

Loc(M)(V) := R(V) ⊗_{R(U)} M.

This is a presheaf of R-modules of U.

Proposition 6.15 (Tate’s theorem). For every rational U ⊆ X, Ȟ^{i}(𝔘, R) = 0 for all i ≥ 1 and all finite rational covers 𝔘 of U, i.e. the augmented Čech complex for 𝔘 is exact. As a consequence, H^{i}(U, R) = 0 for all i ≥ 1.

Proof. The second statement is a direct consequence of the first by a well known theorem of Cartan. For the first, pick a pseudobasis (e_{i})_{i∈I} of R, considered as a profinite flat ℤ_{p}-module. For a cover 𝔘 as above, write C_{aug}^{•}(𝔘, −) for the augmented Čech complex functor. Then, using the pseudobasis, we have

C_{aug}^{•}(𝔘, R) = C_{aug}^{•}(𝔘, R^{∘})[1/p] ≅ (∏_{i∈I} C_{aug}^{•}(𝔘, O_{X}^{∘})) [1/p].

Note that the cohomology groups Ȟ^{i}(𝔘, O_{X}^{∘}) are bounded p-torsion for i ≥ 1 by Tate’s theorem and the open mapping theorem, hence by exactness of products and the above displayed equation so are the cohomology groups Ȟ^{i}(𝔘, R^{∘}) for i ≥ 1. Thus Ȟ^{i}(𝔘, R) vanish as desired. □

Proposition 6.16. Assume that K is discretely valued. Then functor Loc defines a full exact embedding of abelian categories from the category of finitely generated R(U)-modules into the category of R-modules on U. Moreover, Ȟ^{i}(𝔘, Loc(M)) = 0 and H^{i}(U, Loc(M)) = 0 for all i ≥ 1 and all finite rational covers 𝔘 of U for all finitely generated R(U)-modules M. If K is not discretely valued, Loc(M) is still a sheaf with vanishing higher cohomology groups if M is a finitely generated projective module.

Proof. We start with the sheaf/vanishing assertions. Again, the statement about derived functor cohomology follows from that of Čech cohomology by Cartan’s theorem, and the assertion about Čech cohomology implies that Loc(M) is a sheaf. Pick a cover 𝔘. Then by Proposition 6.15 the complex C_{aug}^{•}(𝔘, R) is exact. If M is projective, then it is flat and hence C_{aug}^{•}(𝔘, R) ⊗_{R(U)} M = C_{aug}^{•}(𝔘, Loc(M)) is exact. If K is discretely valued, then by the flatness properties of R all terms are flat R(U)-modules and we may deduce that C_{aug}^{•}(𝔘, R) ⊗_{R(U)} M = C_{aug}^{•}(𝔘, Loc(M)) is exact for arbitrary finitely generated M. In either case, it follows that Loc(M) is a sheaf with vanishing higher Čech cohomology.

Now let K be discretely valued. To see that the functor is fully faithful, note that Loc(M) is generated by global sections as an R-module. For exactness, one checks that if f : M → N is a morphism of finitely generated R(U)-modules, then ker(Loc(f)) = Loc(ker(f)) and similarly for images and cokernels (in particular, images and cokernels in this sub-abelian category turn out to be equal to the presheaf images and cokernels). □

Remark 6.17. Propositions 6.15 and 6.16 also hold for the étale site of X, with the same proofs with obvious modifications.

We may now introduce coherent R-modules.

Definition 6.18. Assume that K is discretely valued. For a cover 𝔘 of U as above a sheaf F of R-modules on U is called 𝔘-coherent if for every U_{i} ∈ 𝔘 there exists an R(U_{i})-module M_{i} such that F|_{U_{i}} ≅ Loc(M_{i}).

F is said to be coherent if it is 𝔘-coherent for some 𝔘.

The equivalence of this definition and the usual definition is standard, cf. e.g. [AW, §5.2].

Before proving Kiehl’s theorem we need a simple lemma. Pick f ∈ O_{X}(X) and consider the standard cover of X consisting of X(f) = {|f| ≤ 1} and X(1/f) = {|f| ≥ 1}. We denote the intersection by X(f, 1/f). We let s denote the restriction map O_{X}(X(1/f)) → O_{X}(X(f, 1/f)).

Lemma 6.19. With notation as above:

(1) The image of s is dense.

(2) The image of the restriction map R(X(1/f)) → R(X(f, 1/f)) is dense for the natural Banach algebra structure on the target.

Proof. Statement (1) is well known, but since there seems to be a little confusion in the literature let us give the one-sentence proof: s is the completion of the identity map on O_{X}(X)[1/f] with respect to the group topology generated by (ϖ^{n}O_{X}^{∘}(X)[1/f])_{n≥0} on the source and the group topology generated by (ϖ^{n}O_{X}^{∘}(X)[f, 1/f])_{n≥0} on the target.

To prove (2), pick a pseudobasis (e_{i})_{i∈I} for R. Then the map is the natural map from bounded I-sequences in O_{X}(X(1/f)) to bounded I-sequences in O_{X}(X(f, 1/f)), which has dense image since s does. □

We may now follow the proof of Kiehl’s theorem from [AW, §5.3-5.5]. One firstly establishes an auxiliary lemma ([AW, Lemma 5.3]), which only uses the Banach algebra structure and vanishing of cohomology (in our setting this is Proposition 6.15). Using this auxiliary lemma one proves [AW, Theorem 5.4] on surjections of certain maps (upon noting that, in their notation, it is erroneously asserted in §5.3 that s_{1} rather than s_{2} has dense image, see Lemma 6.19). Then Ardakov and Wadsley establish a weak form of Kiehl’s theorem for the covering {X(f), X(1/f)} ([AW, Corollary 5.4]), using [AW, Theorem 5.4] and properties of Loc functor, which is Proposition 6.16 in our setting. From this, [AW, Theorem 5.5] follows as well. Following their arguments, we get

Theorem 6.20 (Kiehl’s Theorem). Assume that K is discretely valued and let F be a coherent R-module on X. Then there exists a finitely generated R(X)-module M such that F ≅ Loc(M). Moreover, if F is locally projective (i.e. there is a cover 𝔘 such that F(U) is projective for all U ∈ 𝔘), then M is projective.

Proof. For the last part, use that R(X) → ∏_{U∈𝔘} R(U) is faithfully flat and that projectivity for finitely presented modules may be checked after a faithfully flat base extension. □

We record a simple base change lemma.

Lemma 6.21. Assume that K is discretely valued and that C is the completion of an algebraic closure \overline{K} of K. Let X/K be an affinoid rigid space and let M be a finitely generated projective R(X) module. Let X_{C} be the base change of X to C and let f : X_{C} → X be the natural map, and let R_{C} be the sheaf U ↦ O_{X_{C}} ˆ⊗R on X_{C}. Then we have natural isomorphisms M ⊗_{R(X)} R_{C}(X) ≅ M ⊗_{K} C and

f^{−1}(Loc(M)) ⊗_{f^{−1}R} R_{C} ≅ Loc(M ⊗_{K} C).

Proof. The proof of the first assertion is similar to the proof of Lemma 6.7. Using this, the second assertion follows straight from the definitions upon noting that f is open, which makes it easy to compute f^{−1}. □

Remark 6.22. When working with affinoid weights in the main part of the paper we will often need analogues of the results of this section (as well as Lemma 6.7) when one replaces −ˆ⊗R by −ˆ⊗_{ℚ_{p}}S in the definition of R, where S is a reduced ℚ_{p}-Banach algebra of topologically of finite type. In this case these results are classical results in rigid geometry, or straightforward consequences of such results.

6.4. Quotients of rigid spaces by finite groups. In this section we show that if X is a rigid space and G is a finite group acting on X from the right such that X has a cover by G-stable affinoid open subsets, then the quotient X/G exists in the category of rigid spaces. If X = Spa(A, A^{∘}) is affinoid, then the quotient X/G := Spa(A^{G}, (A^{G})^{∘}) exists in the category of affinoid rigid spaces by [BGR, §6.3.3 Proposition 2]. The purpose of this section is to show that this construction glues. In fact, we show that the quotient is a quotient in Huber’s ambient category 𝒱 ([Hub, §2]), hence in the category of adic spaces. This is presumably well known, but we have not been able to find a reference.

We start with some general discussion on quotients. Let \underline{X} = (X, O_{X}, (v_{x})_{x∈X}) ∈ 𝒱 and let G be a finite group acting on \underline{X} from the right. Consider the quotient topological space Y = X/G and let π : X → Y be the quotient map. We may consider the sheaf of rings O_{Y} := (π_{∗}O_{X})^{G} on Y. For y ∈ Y we pick x ∈ X such that π(x) = y and let v_{y} denote the valuation on O_{Y,y} given by composing v_{x} with the natural map O_{Y,y} → O_{X,x}. This is independent of the choice of x (and representatives for the valuations v_{x}). Then we have:

Lemma 6.23. With notation as above, the sheaf O_{Y} := (π_{∗}O_{X})^{G} is a sheaf of complete topological rings on Y and (Y, O_{Y} , (v_{y})_{y∈Y}) is in 𝒱 and is the categorical quotient of X by G in 𝒱 .

Proof. We prove that O_{Y} is a sheaf of complete topological rings, the rest is then immediate from the definitions. First, note that O_{Y} is a sheaf since invariants are left exact. Second, note that taking invariants preserve completeness (for example, argue with nets). Finally, taking invariants preserves topological embeddings, so O_{Y} is a sheaf of complete topological rings. □

Now let K be a complete non-archimedean field with ring of integers O_{K}, and fix an element ϖ ∈ O_{K} with 0 < |ϖ| < 1. Let B be a K-algebra of topologically finite type, and assume that B carries a left action over K by a finite group G. Then A := B^{G} is a K-algebra of topologically finite type and the inclusion A → B is a finite homomorphism by [BGR, §6.3.3 Proposition 2]. We wish to show that Y := Spa(A, A^{∘}) is the quotient of X := Spa(B, B^{∘}) in 𝒱 by verifying the conditions of Lemma 6.23.

Proposition 6.24. Y = X/G as topological spaces.

Proof. On topological spaces π clearly factors through X/G. π is closed since it is finite. We know from commutative algebra (the going-up theorem) that MaxSpec(B) ↠ MaxSpec(A) as sets, hence π is surjective on classical points and is therefore surjective since it is closed and classical points are dense. Since π is closed and surjective it then suffices to show that the fibers of π are Gorbits, i.e. that Y = X/G as sets.

To show this, we first note that π fits into a G-equivariant commutative diagram

\xymatrix{X \ar[r]^{π} \ar[d] & Y \ar[d] \\ Spv(B) \ar[r]^{π′} & Spv(A)}

with injective vertical arrows; here, following Huber, Spv(R) denotes the set of all equivalence classes of valuations on a given ring R. It now suffices to show that the fibers of π′ are G-orbits. To see this, choose any valuation y ∈ Spv(A), with support 𝔭 = 𝔭_{y} = Ker y ∈ Spec(A). By [Bou, §§V.2.2 Theorem 2(i)], G acts transitively on the set of supports 𝔮_{x} ∈ Spec(B) of valuations x ∈ π^{′−1}(y). Therefore, choosing any 𝔮 = 𝔮_{x} of this form, we need to show that the stabilizer G_{𝔮} of 𝔮 ∈ Spec(B) acts transitively on the set of valuations x ∈ Spv(Frac(B/𝔮)) extending the given valuation y ∈ Spv(Frac(A/𝔭)). By [Bou, §§V.2.2 Theorem 2(ii)], Frac(B/𝔮) is a finite normal extension of Frac(A/𝔭) and G_{𝔮} surjects onto Aut(Frac(B/𝔮)/Frac(A/𝔭)), so this follows from [ZS, §VI.7 Corollary 3].

□

Corollary 6.25. Y = (X/G, (π_{∗}O_{X})^{G}, (v_{xG})) and is therefore the quotient of Spa(B, B^{∘}) by G in 𝒱 .

Proof. We know that Y = X/G on topological spaces and it suffices to verify that O_{Y} = (π_{∗}O_{X})^{G} since then the equality of the valuations follows directly. There is a natural map O_{Y} → (π_{∗}O_{X})^{G} and hence it suffices to prove that this is an isomorphism on rational subsets {|f_{1}|, …, |f_{n}| ≤ |h| ≠ 0} on Y, i.e. that

B⟨f_{1}/h, …, f_{n}/h⟩^{G} = B^{G}⟨f_{1}/h, …, f_{n}/h⟩.

To prove this, note that we have an exact sequence

0 → B^{G} → B ⟶^{a↦(ga−a)_{g∈G}} ∏_{g∈G} B_{(g)}

of finite B^{G}-modules. Note also that B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ is flat over B^{G} and that

B⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ = B ˆ⊗_{B^{G}}B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ = B ⊗_{B^{G}} B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩,

since B is module-finite over B^{G}. Then applying − ⊗_{B^{G}} B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ to the exact sequence above gives

0 → B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ → B⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ ⟶^{a↦(ga−a)_{g∈G}} ∏_{g∈G} B⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩_{(g)}.

Since the kernel of the third arrow here is B⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩^{G}, we get

B^{G}⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩ = B⟨\frac{f_{1}}{h} , …, \frac{f_{n}}{h} ⟩^{G}

as desired. □

Corollary 6.26. Let X be a rigid space over K with a right action of a finite group G and assume that there is a cover of X by G-stable affinoids. Then the quotient X/G of X by G exists in 𝒱 and is a rigid space. Moreover, the natural map X → X/G is finite and if X is affinoid then X/G is affinoid.

Proof. Consider (X/G, (π_{∗}O_{X})^{G}, (v_{xG})) where π : X → X/G is the quotient map on topological spaces. If this is a rigid space then the remaining statements hold by Lemma 6.23 and Corollary 6.25. To see that it is a rigid space let (U_{i})_{i∈I} be a cover of G-stable affinoids. By G-stability π^{−1}(π(U_{i})) = U_{i} for all i and hence π(U_{i}) = U_{i}/G is open for all i. By Corollary 6.25 (U_{i}/G, (π_{∗}O_{U_{i}})^{G}, (v_{xG})) = (X/G, (π_{∗}O_{X})^{G}, (v_{xG}))|_{U_{i}} is an affinoid rigid space for all i, and we are done. □

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Przemysław Chojecki Instytut Matematyczny PAN ul. Sniadeckich 8 00-656 Warszawa Poland pchojecki@impan.pl David Hansen Department of Mathematics Columbia University 2990 Broadway New York NY 10027 hansen@math.columbia.edu

Christian Johansson DPMMS University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom hcj24@cam.ac.uk