Twists of Drinfeld–Stuhler Modular Varieties

Michael Spieß

Received: October 26, 2009 Revised: May 6, 2010

Abstract. Let A be a maximal (or more generally a hereditary) order in a central simple algebra over a global field F of positive characteristic. We show that certain modular scheme of A-elliptic sheaves– for different A – are twists of each other and deduce that the uniformization at ∞ and the Cherednik-Drinfeld uniformization for these varieties are equivalent.

2010 Mathematics Subject Classification: Primary 11G09; Secondary 11G18, 14G22 Keywords and Phrases: Drinfeld modular varieties, rigid analytic uniformization

Dedicated to A.A.Suslin on his 60th birthday

1 Introduction

In [Dr1], Drinfeld has introduced the analogues of Shimura varieties for GL_{d} over a global field F of positive characteristic. Following a suggestion of U. Stuhler the corresponding varieties for an inner form of GL_{d}, i.e. the group of invertible elements A^{∗} of a central simple algebra A of dimension d^{2} over F, have been introduced by Laumon, Rapoport and Stuhler in [LRS]. For d = 2 these are the analogues of Shimura curves. In this paper we show that some of these varieties (for different A) are twists of each other. Let us recall the latter in the simplest case (i.e. over ℚ and by neglecting level structure). Let D be an indefinite quaternion algebra over ℚ and D a maximal order in D. The Shimura curve S_{D} is the (coarse) moduli space corresponding to the moduli problem

(S → Spec ℤ) ↦ abelian surfaces over S with D-action.

By fixing an isomorphism D ⊗ ℝ ≅ M_{2}(ℝ) the group of units D^{∗} acts on the symmetric space H_{∞}: = ℙ_{/ℝ}^{1} −ℙ^{1}(ℝ) (the upper and lower half plane) through linear transformations. The curve S_{D} ⊗_{ℚ} ℝ admits the following concrete description

S_{D} ⊗_{ℚ} ℝ = D^{∗}\H_{∞}.  (1)

If p is a prime number which is ramified in D then there is a similar explicite description over ℚ_{p}. For that let \overline{D} be the definite quaternion algebra over ℚ given by the local data \overline{D} ⊗ ℚ_{ℓ} ≅ D ⊗ ℚ_{ℓ} for all prime numbers ℓ different from p and \overline{D} ⊗ ℚ_{p} ≅ M_{2}(ℚ_{p}). Let \overline{D} denote a maximal ℤ[\frac{1}{p}]-order in \overline{D} and denote by ℚ_{p}^{nr} the quotient field of the ring of Witt vectors of W (\overline{𝔽}_{p}). The Theorem of Cherednik-Drinfeld asserts that

S_{D} ⊗_{ℚ} ℚ_{p} = \overline{D}^{∗}\(H_{p} ⊗_{ℚ_{p}} ℚ_{p}^{nr}).  (2)

(see [Ce], [Dr2] or [BC]). Here \overline{D}^{∗} acts on ℚ_{p}^{nr} via γ ↦ Frob_{p}^{− ord_{p}(Nrd(γ))} and on the p-adic upper half plane H_{p}: = ℙ_{/ℚ_{p}}^{1} − ℙ^{1}(ℚ_{p}) via linear transformations. Now let F be a global field of positive characteristic, i.e. F is the function field of a smooth proper curve X over a finite field 𝔽_{q}. The analogues of Shimura curves over F are the moduli spaces of A-elliptic sheaves as introduced in [LRS]. In this paper we generalize this notion slightly by making systematically use of hereditary orders. Let ∞ ∈ X be a fixed closed point. For simplicity we assume in the introduction that deg(∞) = 1. Let A be a central simple F -algebra of dimension d^{2} and let A be a locally principal hereditary O_{X}- order in A. The condition locally principal means that the radical Rad(A_{x}) of A_{x}: = A ⊗_{O_{X}} Ô_{X,x} is a principal ideal for every closed point x ∈ X. There exists a positive integer e = e_{x}(A) such that Rad(A_{x})^{e} is the ideal A_{x}ϖ_{x} generated by a uniformizer ϖ_{x} of X at x. The number e_{x}(A) divides d for al l x and is equal to 1 for almost all x. We assume in the following that e_{∞}(A) = d. If A is unramified at ∞ then this amounts to require that A_{∞} is isomorphic to the subring of matrices in M_{d}(Ô_{X,∞}) which are upper triangular modulo ϖ_{∞}. Roughly, an A-elliptic sheaf with pole ∞ is a locally free A-module of rank 1 together with a meromorphic A-linear Frobenius having a simple pole at ∞ and a simple zero. The precise definition is as follows. An A-elliptic sheaf over an 𝔽_{q}-scheme S is a pair E = (E, t) consisting of a locally free right A ⊠ O_{S}-module of rank 1 and an injective homomorphism of A ⊠ O_{S}-modules

t : (id_{X} × Frob_{S})^{∗}(E ⊗_{A} A(−\frac{1}{d}∞)) ⟶ E

such that the cokernel of t is supported on the graph Γ_{z} ⊆ X ×_{Spec 𝔽_{q}} S of a morphism z : S → X (called the zero) and is – when considered as a sheaf on S ≅ Γ_{z} – a locally free O_{S}-module of rank d. Here A(−\frac{1}{d}∞) denotes the two-sided ideal in A given by A(−\frac{1}{d}∞)_{x} = A_{x} for all x ≠ ∞ and A(−\frac{1}{d}∞)_{∞} = Rad(A_{∞}). This definition differs, but, as will be proved in the appendix, is equivalent to the one given in [LRS]^{1}. Unlike in loc. cit. we do not require the zero z to be disjoint from the pole ∞ nor from the closed points which are ramified in A. Also we allow ∞ to be ramified in A. For an arbitrary effective divisor I on X there is the notion of a level-I-structure on E. We will show (Theorem 4.11) that the moduli stack of A-elliptic sheaves with level-I-structure Eℓℓ_{A,I}^{∞} is a Deligne- Mumford stack which is locally of finite type and of relative dime nsion d − 1 over X − I. If I ≠ 0, it is a smooth and quasiprojective scheme over X′ − I where X′ denotes the complement of set of closed points x ∈ X with e_{x}(A) > 1. Let B be another central simple F -algebra of dimension d^{2} and assume that there exists a closed point 𝔭 ∈ X − {∞} such that the local invariants of B are given by inv_{∞}(B) = inv_{∞}(A) + \frac{1}{d}, inv_{𝔭}(B) = inv_{𝔭}(A) − \frac{1}{d} and inv_{x}(B) = inv_{x}(A) for all x ≠ ∞, 𝔭. Let B be a locally principal hereditary O_{X}-order in B with e_{x}(B) = e_{x}(A) for all x. Our main result is that the moduli stack Eℓℓ_{B,I}^{𝔭} is a twist of Eℓℓ_{A,I}^{∞}. To state this more precisely we assume for simplicity that deg(𝔭) = 1 and I = 0 (see 4.24 and 4.25 for the general statement). We have

Eℓℓ_{B}^{𝔭} ≅ (Eℓℓ_{A}^{∞} ⊗_{𝔽_{q}} \overline{𝔽}_{q})/ < w_{𝔭} ⊗ Frob_{q} > .  (3)

Here w_{𝔭} is a certain modular automorphism of Eℓℓ_{A}^{∞} (in the case d = 2 it is the analogue of the Atkin-Lehner involution at p for a modular or a Shimura curve). We explain briefly our strategy for proving (3). We consider invertible A-B-bimodules L together with a meromorphic Frobenius Φ having a simple zero at ∞ and simple pole at 𝔭. More precisely, for an 𝔽_{q}-scheme S, we consider pairs L = (L, Φ) where L is an invertible A ⊠ O_{S}-B ⊠ O_{S}-bimodule and Φ is an isomorphism of bimodules

Φ : (id_{X} × Frob_{S})^{∗}(L ⊗_{A} A(−\frac{1}{d}𝔭)) ⟶ L ⊗_{A} A(−\frac{1}{d}∞).

These will be called invertible Frobenius bimodules of slope D = \frac{1}{d}∞ − \frac{1}{d}𝔭 and their moduli space will be denoted by SE_{A,B}^{D}. We will show in section 4.4 that it is a torsor over Spec 𝔽_{q} of the finite group of modular automorphisms of Eℓℓ_{A}^{∞} and compute it explicitely (it is instructive to view SE_{A,B}^{D} as an analogue of the moduli space of supersingular elliptic curves with a fixed ring of endomorphisms). In section 4.5 we construct a canonical tensor product Eℓℓ_{A}^{∞} ×SE_{A,B}^{D} → Eℓℓ_{B}^{𝔭}, (E, L) ↦ E ⊗_{A} L. The isomorphism (3) is then a simple consequence. From the global result (3) we deduce that the uniformization at ∞ and the analogue of the Cherednik-Drinfeld uniformization for the moduli spaces Eℓℓ_{A,I}^{∞} are equivalent. In fact an analogue of the uniformization result (1) for Eℓℓ_{A,I}^{∞} has been proved by Blum and Stuhler in [BS] (in case where the level I is prime to ∞). On the other hand Hausberger has shown in [Hau] (again under the assumption that ∞ does not divide the level I) that there is also an analogue of the Cherednik-Drinfeld theorem. We describe briefly the contents of each section. In section 2 and 3 we discuss hereditary orders in central simple algebras over local fields and global function fields. We show in particular that any hereditary order is Morita equivalent to a (locally) principal hereditary order. This is the reason why it suffices to consider A-elliptic sheaves for locally principal A. In section 4 we introduce A-elliptic sheaves and study their moduli spaces and sections 2.6 and 4.4 are devoted to invertible Frobenius bimodules. In section 3.3 we introduce the notion of a special A-module. If E = (E, t) is an A-elliptic sheaf then Coker(t) is special. The stack Coh_{A,sp} of special A-modules plays a key role in the study of the bad fibers of the characteristic morphism char : Eℓℓ_{A,I}^{∞} → X in section 4.3. In fact Coh_{A,sp} is an Artin stack and char admits a canonical factorization Eℓℓ_{A,I}^{∞} → Coh_{A,sp} → X. We shall show that the first map is smooth and the second semistable. In section 4.5 we construct the tensor product of an Aelliptic sheaf (with level-I-structure) and an invertible Frobenius bimodules (with level-I-structure) and prove our main result (Theorems 4.24 and 4.25). Finally, in section 4.6 we discuss the application to uniformization of Eℓℓ_{A,I}^{∞} by Drinfeld’s symmetric spaces and its coverings. Acknowledgements. I thank E. Lau and T. Zink for helpful conversations. Part of this work has been done during a stay at the Max-Planck-Institute in Bonn in the Winter 2004/05, so I am grateful for the hospitality.

1In [LRS], the authors work with hereditary orders A with A_{∞} ≅ M_{d}(Ô_{X,∞}) and parabolic structures at ∞ on E instead.

Documenta Mathematica · Extra Volume Suslin (2010) 595–654

Notation As an orientation for the reader we collect here a few basic notations which are used in the entire work. However most notations listed below will be introduced again somewhere in this work. For a scheme S we let |S| be the set of closed points of S. The category of S-schemes is denoted by Sch /S. If S = Spec k for a field k then we also write Sch /k. The algebraic closure of a field k is denote by \overline{k}. If k is finite then k_{n} ⊂ \overline{k} denotes the extension of degree n of k. In chapters 3, 4 and in 5.2, X denotes a smooth proper curve over some base field k. In chapter 3, k is an arbitrary perfect field of cohomological dimension 1, whereas in chapter 3 k is the finite field 𝔽_{q}. The function field of X is denoted by F. For Y, Z ∈ Sch /k we write X × Y for their product over k. For a closed point x ∈ X we denote by k(x) its residue field and by deg(x) the degree [k(x) : k]. If S is a k(x)-scheme, then x_{S} will denote the morphism S → Spec k(x) ↪ X. If S = Spec k′ is a field then we also write x_{k′} instead of x_{Spec k′}. For a non-zero effective divisor I on X, we denote the corresponding closed subscheme of X by I as well. If M is a sheaf of O_{X}-modules then we use M_{I} for M ⊗_{O_{X}} O_{I}. In chapter 4, for S ∈ Sch /𝔽_{q} we denote by Frob_{S} its Frobenius endomorphism (over 𝔽_{q}). In the case where S = Spec k′ for some algebraic extension field k′ of 𝔽_{q} we also sometimes write Frob_{q} for Frob_{Spec k′} and Frobenius in the Galois group G(k′/𝔽_{q}). If S ∈ Sch /𝔽_{q} and E is a sheaf of O_{X×S}-modules then ^{τ} E denotes the sheaf (id_{X} × Frob_{S})^{∗}(E). We denote by 𝔸 the Adele ring of F and for a finite set of closed points T of X we let 𝔸^{T} denote the Adele ring outside of T.

Contents

1 Introduction 595

2 Local theory of hereditary orders 599 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 599 2.2 Structure theory . . . . . . . . . . . . . . . . . . . . . . . . . . 600 2.3 Étale base change . . . . . . . . . . . . . . . . . . . . . . . . . . 603 2.4 Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 604 2.5 Maximal tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 2.6 Local theory of invertible Frobenius bimodules . . . . . . . . . 610

3 Global theory of hereditary orders 612 3.1 Morita equivalence. . . . . . . . . . . . . . . . . . . . . . . . . . 613 3.2 Locally free A-modules . . . . . . . . . . . . . . . . . . . . . . . 614 3.3 Special A-modules . . . . . . . . . . . . . . . . . . . . . . . . . 618

4 The moduli space of A-elliptic sheaves 621 4.1 A-elliptic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 621 4.2 Level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 4.3 The coarse moduli scheme . . . . . . . . . . . . . . . . . . . . . 628 4.4 Invertible Frobenius bimodules . . . . . . . . . . . . . . . . . . 630 4.5 Twists of moduli spaces of A-elliptic sheaves . . . . . . . . . . 637 4.6 Application to uniformization . . . . . . . . . . . . . . . . . . . 641

5 Appendix 646 5.1 Commutative subalgebras in semisimple algebras . . . . . . . . 646 5.2 A-elliptic sheaves according to Laumon-Rapoport-Stuhler … 647

2 Local theory of hereditary orders

2.1 Basic definitions

Let X be a scheme and A a quasi-coherent sheaf of O_{X}-algebras. We denote by Mod_{A} the category of sheaves of right A-modules. Let B be another quasicoherent O_{X}-algebra. An A-B-bimodule I is an O_{X}-module with a left A- and right B-action which are compatible with the O_{X}-action. A and B are said to be (Morita) equivalent (notation: A ≃ B) if there exists a quasi-coherent A-B-bimodule I and a quasi-coherent B-A-bimodule J such that the following equivalent conditions hold:

(i) There exists bimodule isomorphisms

I ⊗_{B} J ⟶ A, J ⊗_{A} I ⟶ B.

(ii) The functors

· ⊗_{A} I : Mod_{A} ⟶ Mod_{A}, · ⊗_{B} J : Mod_{B} ⟶ Mod_{A}

are equivalences of categories and mutually quasi-invers. In this case I and J are called invertible bimodules and J is called the inverse of I. The group of isomorphism classes of invertible A-A-bimodules will be denoted by Pic(A). Now assume that X is a Dedekind scheme that is a one-dimensional connected regular noetherian scheme with function field K, i.e. Spec K → X is the generic point. Let A be a central simple algebra over K. An O_{X}-order in A is a sheaf of O_{X}-algebras A with generic fiber A which is coherent and locally free as an O_{X}-module. If B is an O_{X}-order in another central simple K-algebra then it is easy to see that an invertible A-B-bimodule is a coherent and locally free O_{X}-module. The O_{X}-order A in A is called maximal if for any open affine U = Spec R ⊆ X the set of sections Γ(U, A) is a maximal R-order in A. A is called hereditary if its sections Γ(U, A) over any open affine U = Spec R ⊆ X is a hereditary R-order in A that is any left ideal in Γ(U, A) is projective (equivalently any right ideal is projective; compare ([Re], (10.7)). Let E be a locally free O_{X}- module of finite rank which has a left or right A-action compatible with the O_{X}-action. Then the set of sections of E over any affine open U = Spec R ⊆ X are a projective Γ(U, A)-module. If X is affine, i.e. the spectrum of a Dedekind ring O we usually identify A with its sections Γ(X, A). An O-lattice is a finitely generated torsionfree (hence projective) O-module. A (left or right) A-lattice is a (left or right) A-module which is an O-lattice. By ([Re], (10.7)) A is hereditary if and only if every (left or right) A-lattice is projective.

2.2 Structure theory

Let O be a henselian discrete valuation ring with maximal ideal 𝔭 and residue field k = O/𝔭. Let ϖ ∈ 𝔭 be a fixed prime element. We will recall the structure theory of hereditary O-orders in central simple K-algebras (a reference for what follows is [Re], section 39). Since we are only interested in applications to the case where O is the henselisation or completion of a local ring in a global field we will assume for simplicity that k is perfect and of cohomological dimension ≤ 1. Let A be a hereditary O-order in a central simple K-algebra A of dimension n^{2}. Its Jacobson radical will be denoted by 𝔓 = 𝔓_{A}. By ([Re], 39.1 and exercise 6 on p. 365) 𝔓 is an invertible two-sided ideal and any other twosided invertible fractional ideal is an integral power of 𝔓. Let B be a central simple K-algebra equivalent to A and B be a maximal order in B. We denote its radical by 𝔐 = 𝔓_{B}. Let I be an invertible A-B-bimodule. Its inverse is J: = Hom_{K}(I, K). Let I be a A-B-stable lattice in I, i.e. aI, Ib⊆I for all a ∈ A, b ∈ B. Such a lattice exists. In fact if L⊆I is any O-lattice then the O-module generated by the set {axb | a ∈ A, x ∈ L, b ∈ B} is a A-B-stable lattice. There exists a positive integer t – called the type of A – such that 𝔓^{t}I = I𝔐 (see [Re], 39.18 (i)). It is also equal to the number of isomorphism classes of indecomposable left (or right) A-lattices. If M is an indecomposable left A-lattice then {𝔓^{i}M| i = 0, 1, …, t − 1} is a full set of representatives of the set of indecomposable left A-lattices. For i ∈ ℤ we set I_{i}: = 𝔓^{−i}I and J_{i}: = Hom_{O}(I_{−i}, O). The sequences {I_{i} | i ∈ ℤ} and {J_{i} | i ∈ ℤ} satisfy the following conditions:

(i) 𝔓I_{i} = I_{i−1}, I_{i}𝔐 = I_{i−t}, J_{i}𝔓 = J_{i−1}, 𝔐J_{i} = J_{i−t} for all i ∈ ℤ.

(ii) Let A_{i}: = {x ∈ A | xI_{i}⊆I_{i}} = {x ∈ A | J_{−i}x⊆J_{−i}}. Then A_{1}, …, A_{t} are the different maximal orders containing A and we have A = A_{1} ∩…∩ A_{t} (note that A_{i} = A_{j} if i ≡ j mod t). The lattice I_{i} is an invertible A_{i}-B-bimodule with inverse J_{−i}. Note that A_{i} = A_{j} if i ≡ j mod t.

(iii) Let \overline{A}: = A/𝔓, \overline{B}: = B/𝔐 and let

\overline{A}^{(i)} : = Im(\overline{A} → End\overline{B}(I_{i}/I_{i−1})) ≅ Im(\overline{A} → End\overline{B}(J_{−i}/J_{−i−1}))

for i = 1, …, t. Then, considered as a \overline{A}^{(i)}-\overline{B}-bimodule, I_{i}/I_{i−1} is invertible with inverse J_{−i}/J_{−i−1}. We have

\overline{A} ≅ \overline{A}^{(1)} × … × \overline{A}^{(t)}

and \overline{A}^{(i)} ≅ M_{n_{i}}(k′) for i = 1, …, t. Here k′ is the center of \overline{B} and n_{i} = rank\overline{B}(I_{i}/I_{i−1}). The numbers (n_{1}, …, n_{t}) are called the invariants of A. They are well-defined up to cyclic permutation.

Definition 2.1. The positive integer e = e(A) with 𝔓^{e} = ϖA will be called the index of A.

We will see below (Lemma 2.4) that e(A) does not change under finite étale base change. If d is the order of [A] in Br(F) (hence d = [k′ : k]) and t is the type of A then e = dt. Recall ([BF], p. 216) that A is said to be principal if every two-sided invertible ideal of A is a principal ideal or equivalently if there exists Π ∈ 𝔓 with AΠ = ΠA = 𝔓. For example A is principal if it is a maximal O-order in A or if e(A) = n. This is a consequence of the following characterization of principal orders.

Lemma 2.2. Let A be a hereditary order in a central simple K-algebra A of dimension n^{2}. The following conditions are equivalent.

(i) A is a principal order. (ii) If (n_{1}, …, n_{t}) are the invariants of A then n_{1} = … = n_{t}. (iii) Let M_{1}, …, M_{t} be a full set of representatives of the isomorphism classes of indecomposable right A-lattices. Then there exists an integer f ∈ ℕ such that

A ≅ (M_{1} ⊕ … ⊕ M_{t})^{f}

as right A-modules. In this case we have f = n_{1} = … = n_{t} and n = ef.

Proof. (i) ⇔ (ii) see ([BF], Theorem 1.3.2, p. 217).

(i) ⇔ (iii) Since A is principal if and only if A ≅ 𝔓 as right A-modules this follows from the fact that the map [M] ↦ [M𝔓] is a cyclic permutation of the set isomorphism classes of indecomposable right A-lattices ([Re], 39.23). For the last assertion note that if A is principal then on the one hand

dim_{k}(A/𝔓) = ∑_{j=1}^{t} dim_{k}(\overline{A}^{(j)}) = ∑_{j=1}^{t} dn_{j}^{2} = tdn_{i}^{2} = en_{i}^{2}

for i ∈ \{1, …, t\}. On the other hand since M_{j}/M_{j}𝔓 is an irreducible \overline{A}^{(j)}- module we have

dim_{k}(A/𝔓) = f ∑_{j=1}^{t} dim_{k}(M_{j}/M_{j}𝔓) = ftdn_{i} = fen_{i}

Therefore we get f = n_{i}. Finally because of

n^{2} = dim_{k}(A/ϖA) = ∑_{i=0}^{e−1} dim_{k}(𝔓^{i}/𝔓^{i+1}) = e dim_{k}(A/𝔓)

we obtain ef = n. □

Suppose that A is principal. We denote the subgroup of A^{∗} of elements x ∈ A^{∗} with xA = Ax by N(A). For x ∈ N(A) there exists a unique m ∈ ℤ with xA = 𝔓^{m} and we set v_{A}(x) = \frac{m}{e}. We have a commuative diagram with exact rows

1 ⟶ O^{∗} ⟶ K^{∗} ⟶^{v_{K}} ℤ ⟶ 0
↓  ↓  ↓
1 ⟶ A^{∗} ⟶ N(A) ⟶^{v_{A}} \frac{1}{e}ℤ ⟶ 0

where v_{K} denoted the normalized valuation of K and the vertical maps are the natural inclusions. Next we consider the special case where A = End_{K}(V) for a finite-dimensional K-vector space V (i.e. A is split). A lattice chain in V is a sequence of O-lattices L_{⋆} = {L_{i} | i ∈ ℤ} such that (i) L_{i} ⊆ L_{i+1} for all i ∈ ℤ.

(ii) There exists a positive integer e, the period of L_{⋆}, such that L_{i−e} = ϖL_{i} for all i ∈ ℤ.

The ring

A = End(L_{⋆}): = {f ∈ A | f(L_{i})⊆L_{i} ∀i ∈ ℤ}  (4)

is a hereditary O-order in A of index (= type) e with invariants n_{i} = dim_{k}(L_{i}/L_{i−1}). We have:

𝔓_{A}^{−m} = End^{m}(L_{⋆}): = {f ∈ A | f(L_{i})⊆L_{i+m} ∀i ∈ ℤ}.  (5)

Any hereditary O-order in A is of the form (4) for some lattice chain.

2.3 Étale base change

We keep the notation and assumption of the last section. Let A be a central simple algebra and A an O-order in A with radical 𝔓.

Lemma 2.3. The following conditions are equivalent:

(i) A is hereditary. (ii) There exists a two-sided invertible ideal 𝔐 in A such that A/𝔐 is semisimple and 𝔐^{e} = ϖA for some e ≥ 1. Moreover if 𝔐 is as in (ii) then 𝔐 = 𝔓.

Proof. (i) ⇒ (ii) follows from ([Re], (39.18) (iii) ) (for 𝔐 = 𝔓). (ii) ⇒ (i) In view of ([Re], (39.1)) it suffices to show that 𝔐 = 𝔓. The inclusion 𝔐⊇𝔓 is a consequence of the assumption that A/𝔐 is semisimple. The converse inclusion follows from ([Re], exercise 1). □

Lemma 2.4. Let K′/K be a finite unramified extension and O′ the integral closure of O in K′. Then A is hereditary (resp. principal) if only if A ⊗_{O} O′ is hereditary (resp. principal). In this case 𝔓 ⊗_{O} O′ is the radical of the latter.

Proof. We will prove only the statement for hereditary orders and leave the case of principal orders to the reader. If A is hereditary then 𝔐: = 𝔓 ⊗_{O} O′ satisfies the condition (ii) of Lemma 2.3. Hence A ⊗_{O} O′ is hereditary. To prove the converse let P be a left A-lattice. We have to show that

Hom_{A}(P, ·) : Mod_{A} → Mod_{O}

is an exact functor or – since O′ is a faithfully flat O-algebra – that

Hom_{A}(P, ·) ⊗_{O} O′ ≅ Hom_{A⊗_{O}O′}(P ⊗_{O} O′, · ⊗_{O} O′)

is exact. However the assumption implies that P ⊗_{O} O′ is a projective A⊗_{O} O′- module. □

2.4 Morita equivalence

Let A be a central simple algebra and A a hereditary O-order in A with radical 𝔓. If A′ is another O-order in A containing A then A′ is hereditary as well and 𝔓_{A′}⊆𝔓.

Lemma 2.5. Let A_{1}, …, A_{s} be a collection of O-orders in A containing A with radicals 𝔓_{1}, …, 𝔓_{s}. If A_{1} ∩ … ∩ A_{s} = A then 𝔓_{1} + … + 𝔓_{s} = 𝔓.

Proof. Clearly 𝔓_{1} +…+𝔓_{s}⊆𝔓. By Lemma 2.4 to prove equality we may pass to a finite unramified extension K′/K. Hence we can assume A = End_{K}(V) for some finite-dimensional K vector space V and that there exists a lattice chain L_{⋆} = {L_{i} | i ∈ ℤ} in V with period e = e(A) such that

A = {f ∈ A | f(L_{i})⊆L_{i} ∀i ∈ ℤ}, 𝔓 = {f ∈ A | f(L_{i})⊆L_{i−1} ∀i ∈ ℤ}.

Clearly it is enough to consider the case where s = e and A_{i} = {f ∈ A | f(L_{i})⊆L_{i}}, i = 1, …, e are the different maximal orders containing A. We proceed by induction on e so we can assume that e > 1 and that the radical

𝔓′ = {f ∈ A | f(L_{i})⊆L_{i−1} ∀ i ≢ 0, 1 mod e and f(L_{i})⊆L_{i−2} ∀ i ≡ 1 mod e},

of B: = A_{1} ∩ … ∩ A_{e−1} is = 𝔓_{1} + … + 𝔓_{e−1}. Let f ∈ 𝔓. Consider the diagram of k-vector spaces

\xymatrix{L_{1}/L_{0} \ar[r]^{\overline{f}} \ar@{^{(}->}[d] & L_{0}/L_{−1} \\ L_{e}/L_{0} \ar@{-->}[r]^{\overline{g}} & L_{0}/L_{−e} \ar@{->>}[u]}

where the vertical maps are induced by L_{1} ↪ L_{e} and id : L_{0} → L_{0} respectively and the upper horizontal map by f. There exists a dotted arrow \overline{g} making the diagram commutative. Let g ∈ Hom_{O}(L_{e}, L_{0}) = 𝔓_{e} be a “lift” of \overline{g}. Then g(x) ≡ f(x) mod L_{−1} for every x ∈ L_{1}. Therefore (f − g)(L_{1})⊆L_{−1} and (f − g)(L_{i})⊆f(L_{i}) + g(L_{e})⊆L_{i−1} + L_{0} = L_{i−1} for i = 2, …, e − 1 and consequently f − g ∈ 𝔓′. This proves 𝔓⊆𝔓′ + 𝔓_{e} = 𝔓_{1} + … + 𝔓_{e−1} + 𝔓_{e}.

□

Corollary 2.6. Let A be a hereditary O-order of type t with radical 𝔓 in the central simple K-algebra A and let A_{1}, …, A_{t} denote the different maximal orders containing A. Then

A_{1} + … + A_{t} = 𝔓^{−t+1}

is a two-sided invertible ideal.

Proof. By ([Re], section 39, exercise 10) and Lemma 2.5 above we have

𝔓 = 𝔓_{1} + … + 𝔓_{t} = 𝔓^{t}A_{1} + … + 𝔓^{t}A_{t} = 𝔓^{t}(A_{1} + … + A_{t})

hence A_{1} + … + A_{t} = 𝔓^{−t+1}. □

Let B be another central simple K-algebra which is equivalent to A and let B a maximal order in B with radical 𝔐. Let I be an invertible A-B-bimodule, J: = Hom_{K}(I, K) and let {I_{i} | i ∈ ℤ} and {J_{i} | i ∈ ℤ} be as in section 2.2.

Lemma 2.7. Consider I_{i} ⊗_{B} J_{j} (resp. J_{i} ⊗_{A} I_{j}) as a submodule of (I_{i} ⊗_{B} J_{j}) ⊗_{O} K = I ⊗_{B} J (resp. J ⊗_{A} I).

(a) ∑_{i+j=−t+1} I_{i} ⊗_{B} J_{j} ≅ A as an A-A-bimodule.

(b) J_{i} ⊗_{A} I_{j} is an invertible bimodule. If i + j = 0 then J_{i} ⊗_{A} I_{j} ≅ B (as B-B-bimodule). We have

J_{i+1} ⊗_{A} I_{j} = J_{i} ⊗_{A} I_{j+1} = \{ \begin{matrix}𝔐^{−1}(J_{i} ⊗_{A} I_{j}) & if i + j ≡ 0 mod r; \\ J_{i} ⊗_{A} I_{j} & if i + j ≢ 0 mod r \end{matrix}

Proof. (a) Under the identification I ⊗_{B} J = Hom_{K}(J, K) ⊗_{B} J = Hom_{B}(J, J) the submodule I_{i} ⊗_{B} J_{j} corresponds to Hom_{B}(J_{−i}, J_{j}). Hence if we fix an A-A-bimodule isomorphism I ⊗_{B} J ≅ A so that Hom_{B}(J_{0}, J_{0}) is mapped to A_{0} then for arbitrary i, j ∈ ℤ with i + j = 0 the module Hom_{B}(J_{−i}, J_{j}) is mapped to A_{i}. It follows ∑_{i+j=0} I_{i} ⊗_{B} J_{j} ≅ A_{1} + … + A_{r} hence together with Lemma 2.5 the assertion.

(b) The proof of the first two statements is similar and will be left to the reader. For the last statement note that

Coker(J_{i} ⊗_{A} I_{j} → J_{i+1} ⊗_{A} I_{j}) ≅ J_{i+1}/J_{i} ⊗_{A} I_{j} ≅ J_{i+1}/J_{i} ⊗\overline{A} I_{j}/I_{j−1}

By (iii) above we have

J_{i+1}/J_{i} ⊗\overline{A} I_{j}/I_{j−1} ≅ J_{i+1}/J_{i} ⊗_{\overline{A}}_{(j)} I_{j}/I_{j−1} ≅ \overline{B}

if i + j ≡ 0 mod t and J_{i+1}/J_{i} ⊗\overline{A} I_{j}/I_{j−1} = 0 if i + j ≢ 0 mod t. □

Corollary 2.8. The assignment

M ↦ {M ⊗_{A} I_{i} | i ∈ ℤ}

defines an equivalence between the category of right A-lattices and the category of increasing chains {M_{i} | i ∈ ℤ} of right B-lattices such that M_{i}𝔐 = M_{i−t} for all i ∈ ℤ. A quasi-inverse is given by

{M_{i} | i ∈ ℤ} ↦ ∑_{i+j=−t+1} M_{i} ⊗_{B} J_{j}.

Here the sum is taken inside of (⋃_{i∈ℤ} M_{i}) ⊗_{B} J.

Proposition 2.9. Let A_{1} and A_{2} be hereditary O-orders in central simple K-algebras A_{1} and A_{2}. The following conditions are equivalent:

(i) A_{1} and A_{2} are Morita equivalent. (ii) A_{1} and A_{2} are equivalent and A_{1} and A_{2} have the same index.

Proof. We will show only that (ii) implies (i). The proof of the converse is easier and will be left to the reader. Suppose that A_{1} and A_{2} have the same period. Let D be a central division algebra over K equivalent to A_{1} and A_{2} and D be the maximal O-order in D. For ν = 1, 2 we fix increasing sequences of A_{ν}-D- and D-A_{ν}-bimodules {I_{i}^{(ν)} | i ∈ ℤ} and {I_{i}^{(ν)} | i ∈ ℤ} as in 2.2. Put I^{(ν)} = ⋃_{i∈ℤ} I_{i}^{(ν)} and J^{(ν)} = ⋃_{i∈ℤ} J_{i}^{(ν)}. The assumption implies that X : = ∑_{i+j=−t+1} I_{i}^{(1)} ⊗_{D} J_{j}^{(2)} is an A_{1}-A_{2}-lattice (the summation takes

place in I^{(1)} ⊗_{D} J^{(2)}) and Y: = ∑_{i+j=−t+1} I_{i}^{(2)} ⊗_{D} J_{j}^{(1)} a A_{2}-A_{1}-lattice. By Corollary 2.8 above the assignment M ↦ M ⊗_{A_{1}} cX defines an equivalence between the category of right A_{1}-lattices and the category of right A_{2}-lattices. A quasi-inverse is given by N ↦ N ⊗_{A_{2}} Y. This implies that A_{1} and A_{2} are Morita equivalent. In fact using Lemma 2.7 it is easy to see that the X ⊗_{A_{2}} Y ≅ A_{1} and Y ⊗_{A_{1}} X ≅ A_{2}. □

Recall that a right A-lattice M is called stably free if there exists integers r ≥ 1, s ≥ 0 such that M^{r} ≅ A^{s}.

Lemma 2.10. Let A is a principal O-order of index e in a central simple Kalgebra of dimension n^{2}. Let M_{1}, …, M_{t} be representatives of isomorphism classes of indecomposable right A-lattices. For a right A-lattice M ≠ 0 the following conditions are equivalent.

(i) M is stably free. (ii) M ≅ (M_{1} ⊕ … ⊕ M_{t})^{r} for some positive integer r. (iii) D: = End_{A}(M) is a principal O-order of index e in a central simple K-algebra D. Moreover in this case A and D are Morita equivalent and M is an invertible D-A-bimodule. If rank_{O} M = rne then dim_{K}(D) = (er)^{2}.

Proof. The equivalence of (i) and (ii) follows immediately from Lemma 2.2. (ii) ⇔ (iii) By Lemma 2.4 we may pass to a finite unramified extension K′/K. Therefore we can assume that A = End_{K}(V) for an n-dimensional K-vector space V and A = End(L_{⋆}) for a lattice chain L_{⋆} with period e in V. There exists r_{1}, …, r_{e} ≥ 0 with

M ≅ L_{1}^{r_{1}} ⊕ … ⊕ L_{e}^{r_{1}}

Since Hom_{A}(L_{i}, L_{j}) ≅ 𝔭^{μ} with i − j ≤ μe < i − j + e we have

End_{A}(M) ≅ (\begin{matrix}M_{r_{1},r_{1}}(O) & M_{r_{1},r_{2}}(O) & … & M_{r_{1},r_{e}}(O) \\ M_{r_{2},r_{1}}(𝔭) & M_{r_{2},r_{2}}(O) & … & M_{r_{2},r_{e}}(O) \\ ⋮ & ⋮ & ... & ⋮ \\ M_{r_{e},r_{1}}(𝔭) & M_{r_{e},r_{2}}(𝔭) & … & M_{r_{e},r_{e}}(O) \end{matrix} )

By ([Re], 39.14) the order on the right is a hereditary order in M_{m}(K) where m = ∑_{i=1}^{e} r_{i}. Its index is = e if and only if r_{i} ≥ 1 for all i ∈ {1, …, e} and in this case the invariants are (r_{1}, …, r_{e}). The equivalence of (ii) and (iii) follows. The proof of the last assertion will be left to the reader. □

Corollary 2.11. Let A be as in 2.10 and let M be a stably free A-module. We have:

(a) rank_{O} M is a multiple of en.

(b) M is free if and only if rank_{O} M is a multiple of n^{2}. In particular if e = n then M is free.

Proof. If A ≅ M_{m}(D) where D is the central division algebra equivalent to A then rank_{O} M_{i} = md^{2} with d^{2} = dim_{K}(D). Hence if M ≅ (M_{1} ⊕ … ⊕ M_{t})^{r} for r ∈ ℕ then rank_{O} M = rtmd^{2} = ren. The second assertion is obvious.

□

Corollary 2.12. Let A and B be a principal orders in central simple Kalgebras A and B and assume that dim_{K}(A) = dim_{K}(B) = n^{2} and e(A) = e(B) = e. Let I be an A-B-bimodule. The following conditions are equivalent:

(i) I is an invertible A-B-bimodule. (ii) I is a free left A-module of rank 1. (iii) I is a free right B-module of rank 1.

Proof. (i) ⇒ (iii) We show first that I is a lattice. Let J be an inverse of I and _{B−tor}J its B-torsion (A-)submodule. Since _{B−tor}J ⊗_{A} I ↪ J ⊗_{A} I ≅ B we have _{B−tor}J ⊗_{A} I = 0 and therefore _{B−tor}J = _{B−tor}J ⊗_{A} I ⊗_{B} J = 0. For m ∈ J , m ≠ 0 we get Bm ≅ B as left B-module and therefore

I ≅ I ⊗_{B} Bm ↪ I ⊗_{B} J ≅ A.

Hence I is a lattice. Let D: = End_{B}(I) ⊇ A. Thus I is a D-B-bimodule and so I ⊗_{B} J ≅ A is a D-A-bimodule. But End_{A}(A_{A}) = A and therefore D = A. By 2.10 and 2.11, I is a free B-module of rank 1. (iii) ⇒ (i) By 2.10, D is a principal order of index e in a central simple Kalgebra D of dimension d^{2} and I is an invertible D-B-bimodule. Since D ⊇ A this implies D = A. □

Corollary 2.13. Let A be a hereditary O-order in central simple K-algebra A of index e. Then there exists a principal O-order D in a central simple Kalgebra which is Morita equivalent to A. In fact that D can be chosen such that rank_{O}(D) = e^{2}.

Proof. Let A′ be a principal O-order of index e in A′: = M_{e}(A) (since e^{2} divides dim_{K}(A′) and e is a multiple of the order of [A′] = [A] in Br(F) such an order clearly exists). By Proposition 2.9 A′ is Morita equivalent to A. The second assertion follows immediately from 3.11. □

2.5 Maximal tori

Let A be a central simple K-algebra of dimension n^{2} and A a hereditary Oorder in A with radical 𝔓. In this section we consider commutative étale O-subalgebras of A. Note that a commutative finite flat O-algebra T is étale if and only if Rad(T ) = ϖT .

Lemma 2.14. Let T be a commutative étale O-subalgebra of A. Then we have Rad(T ) = T ∩ 𝔓.

Proof. Since T is a direct product of local O-algebras T = ∏ T_{i} and Rad(T ) = ∏ Rad(T_{i}) it suffices to prove the assertion for each factor. Thus we may assume that T is a local ring. Hence Rad(T ) is the maximal ideal of T which implies T ∩ 𝔓 ⊆ Rad(T ). On the other hand, by the assumption, we have Rad(T ) = ϖT hence Rad(T ) ⊆ T ∩ 𝔓. □

A commutative étale O-subalgebra T of A is called maximal torus if rank_{O} T = n. It follows immediately from the structure theory for hereditary O-orders in central simple K-algebras ([Re], 39.14) that there exists a maximal torus in A. We have the following characterization of maximal tori:

Lemma 2.15. Let T be a commutative étale O-subalgebra of A. The following conditions are equivalent.

(i) T is a maximal torus. (ii) T is a maximal commutative étale O-subalgebras of A. (iii) T = Z_{A}(T ) = {x ∈ A | xt = tx ∀t ∈ T }. (iv) T / Rad(T ) is a maximal commutative separable k-subalgebra of A/𝔓.

Proof. The simple proof of the equivalence of the first three conditions will be left to the reader. (iii) ⇔ (iv) By 2.14 above we have Rad(T ) = T ∩ 𝔓 = ϖT . Thus it follows from Lemma 5.1 of the appendix that (iv) holds if and only if rank_{O} T = dim_{k}(T /ϖT ) = n. □

Lemma 2.16. (a) If k = 𝔽_{q} and A is a maximal order in A then A admits a maximal torus isomorphic to O_{n}, the ring of integers of the unramified extension of degree n of K.

(b) Let O′ be a finite étale local O-algebra and T be a maximal torus in A. Then T ⊗_{O} O′ is a maximal torus in A ⊗_{O} O′.

(c) For any two maximal tori T , T′ of A there exists a finite étale local Oalgebra O′ such that T ⊗_{O} O′ and T′ ⊗_{O} O′ are conjugated (by some a ∈ (A ⊗_{O} O′)^{∗}).

Proof. (a) and (b) are obvious. To prove (c) we may pass to a finite unramified extension of K if necessary so that A = End_{K}(V) and A = End(L_{⋆}) where V is a finite-dimensional K-vector space and L_{⋆} is a lattice chain in V. We may also assume that T ≅ O^{n} ≅ T′ where n = dim(V). Let e be the period of L_{⋆} and let \overline{L}_{i}: = L_{i}/L_{i−1}. Consider the \overline{A}: = A/𝔓-module \overline{L}: = ⨁_{i=1}^{e} \overline{L}_{i}. As a \overline{T} : = T / Rad(T )- and \overline{T}′: = T′/ Rad(T′)-module it is free of rank 1 (by Lemma 5.3 of the appendix). Hence there exists an isomorphism \overline{Θ} : \overline{T} → \overline{T}′ such that \overline{Θ}(t̄)x = t̄x for all t̄ ∈ \overline{T} , x ∈ \overline{L}. We choose a lifting Θ of \overline{Θ} i.e. an isomorphism of O-algebras Θ : T → T′ which reduces to \overline{Θ} modulo ϖ. Then for any i ∈ ℤ we h ave

Θ(t)x = tx  for all t ∈ T , x ∈ \overline{L}_{i}  (6)

Since L_{0} is a free T - and T′-module of rank 1 there exists f ∈ Aut_{O}(L_{0}) ⊆ A^{∗} such that f(tx) = Θ(t)f(x) for all t ∈ T , x ∈ L_{0}. Hence Θ(t) = ftf^{−1} for all t ∈ T and therefore T′ = fT f^{−1}. We claim that f ∈ A^{∗}, i.e. f(L_{i}) = L_{i} for all i ∈ ℤ. For that it is enough to see that f(L_{i}) ⊆ L_{i} for all i = 1, 2, … e and in fact for i = 1 (by induction). Note that f(L_{1}) ⊆ f(L_{e}) = ϖ^{−1}f(L_{0}) = L_{e}. Choose i ∈ {1, 2, …, e} minimal with f(L_{1}) ⊆ L_{i} and assume that i ≥ 2. Then f induces a nontrivial T -linear homomorphism f̄: \overline{L}_{1} ⟶ \overline{L}_{i} such that

f̄(tx) = Θ(t)f̄(x) = tf̄(x) for all t ∈ T , x ∈ \overline{L}_{i}.

On the other hand since \overline{L} is a free \overline{T} -module of rank 1 we have Hom_{T} (\overline{L}_{1}, \overline{L}_{i}) = 0, a contradiction. This proves f ∈ A^{∗}. □

We need the following two simple Lemmas in section 3.3.

Lemma 2.17. Suppose that A is principal and let T be a maximal torus in A. Let M be a A-lattice and put \overline{T} : = T / Rad(T ). The following conditions are equivalent.

(i) M is stably free. (ii) M/𝔓M is a free \overline{T} -module.

The proof will be left to the reader.

Lemma 2.18. Assume that A is principal and let T be a maximal torus in A. Let 0 → M′ → M → N → 0 be a short exact sequence of A-modules and assume that M is a stably free A-lattice and ϖN = 0. The following conditions are equivalent.

(i) M′ is stably free. (ii) N is a free \overline{T} -module.

Proof. By using the exact sequence

0 ⟶ Ker(N ⊗_{A} 𝔓 → N) → M′/𝔓M′ ⟶ M/𝔓M ⟶ N /𝔓N ⟶ 0

we see that

[M′/𝔓M′] = [M/𝔓M] + [Ker(N ⊗_{A} 𝔓 → N)] − [N /𝔓N]
= [M/𝔓M] + [N ⊗_{A} 𝔓] − [N]

in the Grothendieck group K_{0}(\overline{T} ). Note that [N] = [N ⊗_{A} 𝔓] if and only if N is a free \overline{T} -module. Hence (ii) is equivalent to the equality [M′/𝔓M′] = [M/𝔓M] in K_{0}(\overline{T} ). The assertion follows from 2.17. □

2.6 Local theory of invertible Frobenius bimodules

Let O be a henselian discrete valuation ring with quotient field K, maximal ideal (ϖ) = 𝔭 and residue field k = O/𝔭. We assume that k is finite of characteristic p. Let v_{K} be the normalized valuation of K. We denote by inv the canonical isomorphism Br(K) → ℚ/ℤ of class field theory. Let O′ be a finite étale local O-algebra with quotient field K′. By σ ∈ G(K′/K) we denote Frobenius isomorphism (i.e. σ(x) ≡ x^{♯(k)} mod 𝔭). For an O′-module M we write ^{σ}M for M ⊗_{O′,σ} O′ (or equivalently ^{σ}M = M with the new O′-action x · m = σ(x)m). Let A be a central simple K-algebra of dimension d^{2} and A a principal O-order in A with radical 𝔓 and index e = e(A) (note that we have e inv(A) = 0). Let M be a free right A_{O′}-module of rank 1 together with an isomorphism of A_{O′}-modules

ϕ : ^{σ}M𝔓^{m} ⟶ M

for some m ∈ ℤ. We set

B: = End_{A_{O′}} (M, ϕ) = {f ∈ End_{A_{O′}} (M) | ϕ ∘ f = ^{σ}f ∘ ϕ}.

Lemma 2.19. The O-algebra B is a principal order of index e in the central simple K-algebra B: = B_{K} of dimension d^{2}. We have

inv(B) = inv(A) + \frac{m}{e} mod ℤ  (7)

Proof. Let ϕ_{K′}: = ϕ ⊗_{O′} id_{K′} : ^{σ}(M_{K′}) → M_{K′}. By Lemma 2.10 the O′-algebra B′: = End_{A_{O′}} (M) is a principal O′-order of index e in B′: = End_{A_{K′}} (M_{K′}). Define a σ-linear isomorphism ψ : B′ → B′ by ψ(f): = ϕ_{K′}^{−1} ∘ f ∘ ϕ_{K′}. We have

B = {b ∈ B′ | ψ(b) = b} and B′ ≅ B_{O′}.

Together with Lemma 2.4 this implies the first statement. The proof of the second assertion will be left to the reader. □

Conversely suppose that we have given a second central simple K-algebra B of dimension d^{2} and a principal O-order B in B of index e. We also assume that [K′ : K] is a multiple of the order of [B ⊗ A^{opp}] in Br(K). Let m be any integer such that (7) holds.

Lemma 2.20. There exists an invertible B_{O′}-A_{O′}-bimodule M and an isomorphism of bimodules

ϕ : ^{σ}M𝔓^{m} ⟶ M.

Proof. By Proposition 2.9 the principal orders B_{O′} and A_{O′} are Morita equivalent. Let M be an invertible B_{O′}-A_{O′}-bimodule. Then ^{σ}M is invertible as well. Hence there exists an isomorphism ϕ′ : ^{σ}M𝔓^{m′} → M for some m′ ∈ ℤ. By 2.19 we have

inv(B) = inv(A) + \frac{m′}{e} mod ℤ

and therefore m ≡ m′ mod e. Put ϕ: = ϖ^{\frac{m′−m}{e}} ϕ′. □

For the rest of this section we assume that O is an 𝔽_{q}-algebra (q = p^{r} for some r ∈ ℤ) and let k′ be an (possibly infinite) algebraic extension of k whose degree (over k) is a multiple of e. Let O′: = O ⊗_{𝔽_{q}} k′ and σ: = id_{O} ⊗ Frob_{q} ∈ G(O′/O). For ρ ∈ Hom_{𝔽_{q}}(k, k′) ≅ Hom_{k′}(k ⊗_{𝔽_{q}} k′, k′) we denote the kernel of O′ → k ⊗_{𝔽_{q}} k′ → k′ by 𝔭′_{ρ} and we set O′_{ρ}: = O′_{𝔭′_{ρ}}. Then O′_{ρ} is a (pro-)finite (pro-)étale local O-algebra whose degree is a multiple of e and O′ ≅ ⨁_{ρ} O′_{ρ}. Similarly

A_{O′} = ⨁_{ρ} A′_{ρ} with A′_{ρ} = A_{O′_{ρ}}

and 𝔓_{O′} = Rad(A_{O′}) is equal to the product ∏_{ρ} 𝔓′_{ρ} where 𝔓′_{ρ} denotes the maximal invertible two-sided ideal Ker(A_{O′} → A′_{ρ}/ Rad(A′_{ρ})) of A_{O′}. For the distinguished element ι: = incl : k ↪ k′ in Hom_{𝔽_{q}}(k, k′) we put 𝔭′ = 𝔭′_{ρ} and 𝔓′: = 𝔓′_{ι}. Let M be a free right A_{O′}-module of rank 1. For m ∈ ℤ the A_{O′}- module ^{σ}(M(𝔓′)^{m}) is also free of rank 1. Hence there exists an isomorphism

ϕ : ^{σ}(M(𝔓′)^{m}) ⟶ M.

If we set

B: = End_{A_{O′}} (M, ϕ) = {f ∈ End_{A_{O′}} (M) | ϕ ∘ f = ∘ ϕ}. then one can deduce easily from Lemma 2.19 that B is a principal O-order of index e in the central simple K-algebra B = B_{K} and that equation (7) holds. Conversely given such a principal O-order B of index e and m ∈ ℤ such that (7) holds there exists a pair (M, ϕ) as above with B = End_{A_{O′}} (M, ϕ). To see this let M be any invertible B_{O′}-A_{O′}-bimodule. Since ^{σ}M is invertible as well we have

σ(M ∏_{ρ}(𝔓′_{ρ})^{m_{ρ}}) ≅ M  (8)

for certain m_{ρ} ∈ ℤ. Since ^{σ}(𝔓′_{ρ}) ≅ 𝔓′_{Frob_{q} ∘ρ} we may assume – after replacing M by M𝔄 for a suitable invertible two-sided A_{O′}-ideal 𝔄 – that m_{ρ} = 0 for all ρ ∈ Hom_{𝔽_{q}}(k, k′) except ρ = ι. As in the proof of Lemma 2.20 we deduce

inv(B) = inv(A) + \frac{m_{ι}}{e} mod ℤ

hence m_{ι} ≅ m mod e and therefore (𝔓′)^{m_{ι}} ≅ (𝔓′)^{m}. Hence there also exists an isomorphism ^{σ}(M(𝔓′)^{m}) ≅ M.

Definition 2.21. A pair (M, ϕ) consisting of an invertible B_{O′}-A_{O′}-bimodule M and an isomorphism ϕ : ^{σ}(M(𝔓′)^{m}) → M is called an invertible ϕ-A-B-bimodule of slope −\frac{m}{e} over O′.

We have seen that an invertible B_{O′}-A_{O′}-bimodule of a given slope r ∈ ℚ exists and only if r = inv(A) − inv(B) mod ℤ. It is also easy to see that any two invertible B_{O′}-A_{O′}-bimodules of the same slope differ (up to isomorphism) by a fractional A-ideal. This implies that if k′′ is an algebraic extension of k′ and O′′ = O ⊗_{𝔽_{q}} k′′ then any ϕ-A-B-bimodule over O′′ is obtained by base change from an ϕ-A-B-bimodule over O′.

Remark 2.22. Assume that [k′ : k] = e and let n = [k′ : 𝔽_{q}]. Let (M, ϕ) be an invertible B_{O′}-A_{O′}-bimodule of slope −\frac{m}{e}. For r ∈ ℤ/nℤ we put 𝔓′_{r} = σ^{r}𝔓′. We have ∏_{r∈ℤ/nℤ} 𝔓′_{r} = 𝔓_{O′}^{e} = 𝔭A_{O′}. For each two-sided invertible ideal 𝔄′ of A_{O′} and r ∈ ℤ/nℤ, the map ϕ induces isomorphisms (^{σ^{r}}M)𝔄′ → (^{σ^{r−1}}M)𝔄′𝔓′_{r}^{m} which will be also denoted by ϕ. Consider the map

ϕ^{n} : M = (^{σ^{n}}M) ⟶^{ϕ} (^{σ^{n−1}}M)𝔓′_{n}^{m} ⟶^{ϕ} … ⟶ M ∏_{r∈ℤ/nℤ} 𝔓′_{r}^{m} = M𝔭^{m} (9)

Since (9) is B_{O′}-A_{O′}-bilinear and commutes with ϕ there exists an element x ∈ K with v_{K}(x) = m such that (9) is given by multiplication with x. This fact will be used later when we discuss level structure at the pole of A-elliptic sheaves.

3 Global theory of hereditary orders

In this section we study hereditary orders in a central simple algebras over a function field of one variable (though most results hold also for number fields).

We shall show that two hereditary orders are Morita equivalent if their generic fibers are equivalent and all their local indices are the same. Furthermore any such hereditary order is Morita equivalent to a locally principal one. We will then study the Picard group of a locally principal order A and introduce the notion of A-degree of a locally free A-module of finite rank. In the final part we will introduce the notion of a special A-module. In this chapter k denotes a fixed perfect field of cohomological dimension ≤ 1 and X a smooth projective geometrically connected curve over k with function field F. For x ∈ |X| we denote by O_{x} the completion of O_{X,x} and by F_{x} the quotient field of O_{x}. The maximal ideal of O_{x} will be denoted by 𝔭_{x}. If V is a coherent O_{X}-module then we set V_{x} = V⊗_{O_{X}} O_{x} and if V is a finite-dimensional F -vector space we put V_{x} = V ⊗_{F} F_{x}.

3.1 Morita equivalence.

Let V be a finite-dimensional F -vector space. The set of locally free coherent O_{X}-modules V with generic fiber V_{η} = V is in one-to-one correspondence with the set of O_{x}-lattices V_{x} in V_{x} for all x ∈ |X| such that there exists an F -basis B of V with V_{x} = ∑_{b∈B} O_{x}b for almost all x. Consequently if U⊆X is an open subscheme then there is a one-to-one correspondence between coherent and locally free O_{X}-modules V and coherent and locally free O_{U}-module V_{U} and together with an O_{x}-lattice V_{x} in V_{U} ⊗ F_{x} for all x ∈ X − U. Let A be a central simple F -algebra and A a hereditary O_{X}-order in A. We put e_{x}(A): = e(A_{x}). There are only finitely many points x ∈ |X| with e_{x}(A) > 1. Define the divisor \underline{Disc}(A) as \underline{Disc}(A): = ∑_{x∈|X|} (e_{x}(A) − 1)x. If k is finite and x ∈ |X| then inv_{x}(A) denotes the image of the class of A_{x} under the canonical isomorphism of class field theory Br(F_{v}) → ℚ/ℤ.

Proposition 3.1. Let A_{1}, A_{2} be central simple algebras over F and let A_{1} and A_{2} be hereditary O_{X}-orders in A_{1} and A_{2} respectively. The following conditions are equivalent.

(i) A_{1} and A_{2} are equivalent. (ii) A_{1} and A_{2} are equivalent and (A_{1})_{x} and (A_{2})_{x} are equivalent for all x ∈ |X|. (iii) A_{1} and A_{2} are equivalent and \underline{Disc}(A_{1}) = \underline{Disc}(A_{2}). Moreover if k is a finite field then the above conditions are also equivalent to: (iv) inv_{x}(A_{1}) = inv_{x}(A_{2}) for all x ∈ |X| and \underline{Disc}(A_{1}) = \underline{Disc}(A_{2}).

Proof. (i) ⇒ (ii) is clear. (ii) ⇔ (iii) follows from Proposition 2.9 and (iii) ⇔ (iv) from the Theorem of Brauer–Hasse–Noether. It remains to show that (ii) implies (i). Let U be an affine open subscheme of X contained in the complement of \underline{Disc}(A_{1}) = \underline{Disc}(A_{2}) in X. By ([Re], 21.7) A_{1}|_{U} and A_{2}|_{U} are Morita equivalent. Let I_{U} be an invertible A_{1}|_{U}-A_{2}|_{U}-bimodule and let I_{x} be an invertible (A_{1})_{x}-(A_{2})_{x}-bimodule for each x ∈ X − U. Since there is only one invertible (A_{1})_{x}-(A_{2})_{x}-bimodule up to isomorphism we may assume that I_{x} ⊗ F_{x} = I_{U} ⊗ F_{x} i.e. that I_{x} is a lattice in I_{U} ⊗ F_{x}. It is easy to see that the locally free O_{X}-module I corresponding to I_{U} and the I_{x}, x ∈ X − U is then an invertible A_{1}-A_{2}-bimodule. □

A locally principal O_{X}-order A is a hereditary O_{X}-order in a central simple F -algebra A such that A_{x} is principal for all x ∈ |X|. The rank of A is its rank as an O_{X}-module, hence = dim_{F} (A). If A is a hereditary O_{X}-order in A then it is locally principal if for example A_{x} is either maximal or e_{x}(A) = d for all x ∈ | \underline{Disc}(A)|. Suppose that A is a locally principal O_{X}-order of rank d^{2}. We define two positive integers e(A), δ(A) by

e(A): = lcm{e_{x}(A) | x ∈ |X|}  (10)
δ(A): = lcm\{ numerator of \frac{e_{x}(A)}{deg(x)} | x ∈ |X|\}

According to Lemma 2.2 we have δ(A) | e(A) | d. If A is locally principal then one can easily see that

deg(A) = −\frac{d^{2}}{2} ∑_{x∈|X|} (1 − \frac{1}{e_{x}(A)}) deg(x).

In particular if B is a second locally principal O_{X}-order of rank d^{2} with \underline{Disc}(A) = \underline{Disc}(B) then

deg(A) = deg(B).  (11)

Corollary 3.2. Let A be a hereditary O_{X}-order in a central simple F -algebra A. Then there exists a locally principal O_{X}-order D which is Morita equivalent to A. In fact D can be chosen such that rank_{O_{X}} (D) = e(A)^{2}.

Proof. That A is equivalent to a locally principal O_{X}-order follows easily from the corresponding local statement 2.13. In fact if B: = M_{e}(A) then for all x ∈ | \underline{Disc}(A)| we can pick a principal O_{x}-order B_{x} in B_{x} equivalent to A_{x}. If U: = X − | \underline{Disc}(A)| and B_{U} is a maximal O_{U}-order in B then there exists a uniquely determined hereditary O_{X}-order B in B with B ⊗_{O_{X}} O_{x} = B_{x} for all x ∈ | \underline{Disc}(A)| and B|_{U} = B_{U}. The order B is locally principal and equivalent to A by 3.1. Thus to prove the second statement we may assume that A is locally principal. Let I be a locally stably free A-module which is of rank de as an O_{X}-module. By Lemma 2.10 and 3.1 above it follows that D: = \underline{End}_{A}(I) is a locally principal O_{X}-order in End_{A}(I_{η}). Moreover D is equivalent to A and rank_{O_{X}} (D) = e(A)^{2}. □

3.2 Locally free A-modules

The Picard group of a locally principal order. In this section A denotes a locally principal O_{X}-order of rank d^{2}. We are going to compute the Picard group of A. We define

Div(A): = { ∑_{x∈|X|} n_{x}x ∈ Div(X) ⊗ ℚ | e_{x}(A)n_{x} ∈ ℤ ∀ x ∈ |X|}.

Note that deg(Div(A)) = \frac{1}{δ(A)}ℤ. For a divisor D = ∑_{x∈|X|} n_{x}x ∈ Div(A)) we denote by A(D) the invertible A-A-bimodule given by A(D)|_{X−|D|} = A|_{X−|D|} and A(D)_{x} = 𝔓_{A_{x}}^{−n_{x}e_{x}(A)} for all x ∈ |X|. If D ∈ Div(X) then A(D) = A ⊗_{O_{X}} O_{X}(D).

Proposition 3.3. The sequence

0 ⟶ F^{∗}/k^{∗} ⟶^{div} Div(A) ⟶^{D↦A(D)} Pic(A) ⟶ 0

is exact.

Proof. This follows from ([Re], 40.9). □

We also need to consider the group of isomorphism classes of invertible A-A-bimodules with level structure and give a description of it as an idele class group. Let I = ∑_{x} n_{x}x be an effective divisor on X. The corresponding finite closed subscheme of X will be also denoted by I. A level-I-structure on an invertible A-A-bimodule L is an isomorphism β : A_{I} → L_{I} of right A_{I}- modules. We denote by Pic_{I}(A) the set of isomorphism classes of invertible A-A-bimodules with level-I-structure. If (L_{1}, β_{1}), (L_{2}, β_{2}) are invertible A-A-bimodules with level-I-structures we define the level-I-structure β_{1}β_{2} on L_{1} ⊗_{A} L_{2} as the composite

β_{1}β_{2} : A_{I} ⟶^{β_{2}} (L_{2})_{I} = A_{I} ⊗_{A_{I}} (L_{2})_{I} ⟶^{β_{1}⊗id} (L_{1} ⊗_{A} L_{2})_{I}  (12)

thus defining a group structure on Pic_{I}(A). Note that unlike Pic(A), Pic_{I}(A) is in general not abelian. In fact we have a short exact sequence

0 ⟶ Γ(I, A_{I})^{∗}/k^{∗} ⟶ Pic_{I}(A) ⟶ Pic(A) ⟶ 0  (13)

where the first map is given by a ∈ Γ(I, A_{I})^{∗} ↦ (A, l_{a} : A_{I} ⟶^{a·} A_{I}) . Let U_{I}(A): = Ker(∏_{x∈|X|} A_{x}^{∗} → ∏_{x∈|X|} (A_{x}/𝔭_{x}^{n_{x}}A_{x})^{∗} = Γ(I, A_{I})^{∗}) and let

C_{I}(A): = (∏′_{x∈|X|} N(A_{x}))/U_{I}(A)F^{∗}

where ∏′_{x∈|X|} N(A_{x})) denotes the restricted direct product of the groups {N(A_{x}))}_{x∈|X|} with respect to {A_{x}^{∗}}_{x∈|X|}. Given a = {a_{x}}_{x} ∈ ∏′_{x∈|X|} N(A_{x}) we put div(a) = ∑_{x∈|X|} v_{A_{x}}(a_{x})x. Left multiplication by a induces a level-I-structure β_{a} : A_{I} → A(div(a))_{I}.

Corollary 3.4. The assignement a ↦ (A(div(a)), β_{a}) induces an isomorphism C_{I}(A) ≅ Pic_{I}(A).

Relative divisors and invertible bimodules. Let S be a k-scheme and let π : X × S → S be the projection. We need to define the bimodule A(D) also for elements of a certain group of relative divisors Div(A ⊠ O_{S}). For the latter we use the following ad hoc definition. Assume first that S is of finite type over k. Let S be the collection of all connected components of x×S where x runs through all closed points of X. Thus if S′ ∈ S there exists a unique closed point x: = π(S′) with S′ ⊆ x × S. We set

Div(A ⊠ O_{S}): = ⨁_{S′∈S} \frac{1}{e_{π(S′)}(A)}ℤ.

Let R be the integral closure of k in Γ(S, O_{S}). Note that for x ∈ |X| the set of open and closed subschemes of x × S corresponds to the set of idempotents in k(x) ⊗_{k} Γ(S, O_{S}). If f : S_{1} → S_{2} is a morphism of k-schemes there is an obvious notion of a pull-back f^{∗} : Div(A ⊠ O_{S_{2}}) → Div(A ⊠ O_{S_{1}}). For an arbitrary k-scheme we define Div(A ⊠ O_{S}) as the direct limit of Div(A ⊠ O_{S′}) over the category of pairs (S′, g) consisting of a k-scheme S′ of finite type and a morphism g : S → S′ in Sch /k. Let S ∈ Sch /k. A k-morphism x_{S} : S → X which factors as S → Spec k(x) → X for some x ∈ |X| yields an element – denoted by x_{S} as well – of the group Div(A ⊠ O_{S}). For that we can assume that S is of finite type. Since the graph Γ_{x_{S}} = (x_{S}, id_{S}) : S ⟶ X × S is an open and closed subscheme of x × S it is a disjoint union of connected components and we define x_{S} ∈ Div(A ⊠ O_{S}) to be the sum of these components. There exists a unique homomorphism

Div(A ⊠ O_{S}) → Pic(A ⊠ O_{S}), D ↦ (A ⊠ O_{S})(D)  (14)

compatible with pull-backs which agrees with the previously defined map in case S = Spec k′ for a finite extension k′/k. It suffices to define (14) for \frac{1}{e_{x}(A)}D, where D is a connected component of x × S for some x ∈ |X|. It is also enough to consider the case where S is connected and of finite type over k. Let R be the integral closure of k in Γ(S, O_{S}). Then Spec R is connected and finite over Spec k, i.e. R is an artinian finite local k-algebra. Let k′ denotes the residue field of R. Since k is perfect the canonical projection R → k′ has a unique section. Therefore the structural morphism S → Spec k factors as S → Spec k′ → Spec k. Thus by replacing k, X and A by k′ and X_{k′} and A ⊠ k′ respectively we can assume that the residue field of R is k. However, in this case, x × S is connected for all x ∈ |X|, hence D = x × S with x = π(D). S o we are forced to define (A ⊠ O_{S})(\frac{1}{e_{x}(A)}D): = π^{∗}(A(\frac{1}{e_{x}(A)}x)).

A-rank and A-degree. Let f : S → X be a morphism. For E in _{f^{∗}(A)} Mod and F in Mod_{f^{∗}(A)} we put E ⊗_{A} F: = E ⊗_{f^{∗}(A)} F. If D = ∑_{x∈|X|} n_{x}x ∈ Div(A) we set E(D): = E ⊗_{A} f^{∗}(A(D)) and F(D): = f^{∗}(A(D) ⊠ O_{S}) ⊗_{A} F. Let S be a k-scheme. We denote by_{A}Vect(S) (resp. Vect_{A}(S)) the category coherent and locally free left (resp. right) A⊠O_{S}-modules. For F in _{A}Vect(S) or Vect_{A}(S) let rank_{A} F be the locally constant function s ↦ rank_{A⊠k(s)}(F|_{X×s}) on S (hence rank_{A} F can be viewed as an element of ℤ^{π_{0}(S)}). For a positive integer r we denote by _{A}Vect^{r}(S) (resp. Vect_{A}^{r}(S)) the subcategory of F ∈ _{A}Vect(S) (resp. F ∈ Vect_{A}(S)) with rank_{A} F = r. Let F be a locally free A⊠O_{S}-module of rank r. Define det_{A} F as the image of the isomorphism class of F (viewed as an element of H^{1}(X × S, GL_{r}(A ⊠ O_{S})) under the map

H_{zar}^{1}(X × S, GL_{r}(A ⊠ O_{S}) ⟶ H_{zar}^{1}(X × S, O^{∗}) = Pic(X × S)

induced by the reduced norm Nrd : M_{r}(A) → F. We obtain a locally constant function

deg_{A}(F) : S → \frac{1}{d}ℤ, s ↦ deg((det_{A} F)|_{X×s})

It is easy to see that

deg_{A}(F) = \frac{1}{d^{2}} (deg(F) − rank_{A}(F) deg(A)).

In particular since deg(A(D)) = deg(A) + d^{2} deg(D) we have

deg_{A}(A(D)) = deg(D)

for D ∈ Div(A).

Lemma 3.5. (a) Let 0 → F_{1} → F_{2} → F_{3} → 0 be a short exact sequence of coherent and locally free A ⊠ O_{S}-modules. Then

deg_{A}(F_{2}) = deg_{A}(F_{1}) + deg_{A}(F_{3}).

(b) Let E be an object of Vect_{A}^{r}(S) and F be an object of _{A}Vect^{s}(S). Then

\frac{1}{d^{2}} (deg(E ⊗_{A} F) − rs deg(A)) = r deg_{A}(F) + s deg_{A}(E).

(c) Let B be a second locally principal O_{X}-order of rank d^{2} equivalent to A. Let E be an object of Vect_{A}^{r}(S) and let I be an invertible A-B-bimodule. Then

deg_{B}(E ⊗_{A} I) = deg_{A}(E) + r deg_{A}(I).

(d) Let E be an object of _{A}Vect^{r}(S) and D ∈ Div(A). Then

deg_{A}(E(D)) = deg_{A}(E) + r deg(D)

Proof. (a) is obvious, (c) follows from (b) and (11) and (d) is a special case of (c). Note that by 2.12 the bimodule L in (c) is a locally-free left A- and right B-module of rank 1.

For (b) it is enough to consider the case when S is a connected k-scheme of finite type and therefore – by chosing a fixed closed point s ∈ S and taking the base change Spec k(s) → Spec k – to consider the case S = Spec k. If

\xymatrix{ & E \\ E′′ \ar@{-->}[ur] \ar[dr] \\  & E′}  (15)

is a diagram of locally free A ⊠ O_{S}-modules of the same rank r and injective A ⊠ O_{S}-linear homomorphisms then it is easy to see that (b) holds for E if and only if it holds for E′. Since E|_{U} ≅ A^{r}|_{U} for some non-empty open subscheme U⊆X there exists a diagram (15) with E′ = A^{r}. The assertion follows. □

It follows from 3.3 or 3.5 (b) that deg_{A} : Pic(A) → ℚ is a homomorphism. We denote its kernel by Pic_{0}(A). Also if I ∈ Div(X) we let Pic_{I,0}(A) be the subgroup of (L, β) ∈ Pic_{I}(A) with deg_{A}(L) = 0. The image deg_{A}(Pic(A)) is equal to \frac{1}{δ(A)}ℤ.

Remark 3.6. Let A, B be locally principal O_{X}-order of rank d^{2} and suppose that A and B are equivalent. The set of isomorphism classes of invertible A-B-bimodule has a simple transitive left Pic(A)-action. Hence for any two invertible A-B-bimodule I, J the degrees deg_{A}(J) and deg_{A}(I) differ by a multiple of \frac{1}{δ(A)}. Call A and B strongly Morita equivalent if there exists an invertible A-B-bimodule I with deg_{A}(I) = 0. It is easy to see that a given equivalence class of locally principal O_{X}-orders of rank d^{2} decomposes into \frac{e}{δ} strong equivalence classes (where e and δ are defined in (10)).

3.3 Special A-modules

Let A be a locally principal O_{X}-order of rank d^{2}. If g : U → X is an étale morphism then a maximal torus in A_{U}: = g^{∗}(A) is a maximal commutative étale O_{U}-subalgebra of A_{U}.

Definition 3.7. A right A ⊠ O_{S}-module K is called special of rank r if the following holds:

(i) K is coherent as an O_{X×S}-module and the map Supp(K) ↪ X ×S → S is an isomorphism. Hence Supp(K) is the image of the graph of a morphism N = N(K) : S → X and K is the direct image of a N^{∗}(A)-module – also denoted by K – by the graph Γ_{N} = (N, id_{S}) : S → X × S.

(ii) Consider K as a sheaf on S as in (i). For any étale morphism g : U → X and maximal torus T of A_{U}, (g_{S})^{∗}(K) is a locally free (N_{U})^{∗}(T )-module of rank r. Here g_{S} (resp. N_{U}) denote the base change of g (resp. N) with respect to N (resp. g).

We denote by Coh_{A,sp}^{r} the stack over k such that for each S ∈ Sch /k, Coh_{A,sp}^{r}(S) is the groupoid of special A ⊠ O_{S}-modules of rank r. The morphism K ↦ N(K) will be denoted by N : Coh_{A,sp}^{r} → X.

Remarks 3.8. (a) By Lemma 2.16 it suffices to check condition (ii) for a fixed étale covering {U_{i} → U} and maximal tori T_{i} of A_{U_{i}}.

(b) Let K be as in 3.7 satisfying (i) and assume that N(K) : S → X factors through X − | \underline{Disc}(A)|. Then K is special of rank r if and only if K is a locally free of rank rd as an O_{S}-module.

(c) Let A′ be another locally principal O_{X}-order of rank d^{2} equivalent to A and let I be an invertible A-A′-bimodule. Tensoring with I maps Coh_{A,sp}^{r} isomorphically to Coh_{A′,sp}^{r}. This follows easily from the fact that, locally on X, A and A′ are isomorphic. More generally if A are equivalent on some open subscheme U ⊆ X and I is a A-A′-bimodule which is invertible on U then tensoring with I yields an isomorphism · ⊗_{A} I : Coh_{A,sp}^{r} ×_{X}U → Coh_{A′,sp}^{r} ×_{X}U.

Except in the appendix, we need to consider only the case r = 1. In the following we investigate the geometric properties of Coh_{A,sp} : = Coh_{A,sp}^{r}. Recall that a morphism f : Y → X is said to be semistable if its generic fiber is smooth and for any y ∈ Y there exists an étale neighbourhood Y′ of y, an open affine neighbourhood Spec R of x = f(y) and a smooth X-morphism Y′ ⟶^{g} Spec R[T_{1}, …, T_{r}]/(T_{1} ⋯ T_{r} − ϖ) for some r ≥ 1, where ϖ is a local parameter at x. Equivalently, Y is a smooth k-scheme, the generic fiber Y_{η} is smooth over F and the closed fiber Y_{x} is a reduced divisor with normal crossings for all x ∈ |X|. Therefore if f is semistable it is flat. We have the following simple Lemma whose proof will be left to the reader:

Lemma 3.9. Let Y_{1} ⟶^{f} Y_{2} ⟶^{g} X be morphism of schemes such that f is smooth and surjective. Then g is semistable if and only if g ∘ f is semistable.

Let Y be an algebraic stack over k. We will call a morphism f : Y → X semistable if there exists a scheme Y and a presentation P : Y → Y (i.e. P is smooth and surjective) such that f ∘ P : Y → X is semistable. It follows from 3.9 that if this holds then any presentation P′ : Y′ → Y (with Y′ a scheme) has this property. In particular if Y is a scheme the two notions of semistability agree. Our aim in this section is to prove the following result.

Proposition 3.10. Coh_{A,sp} is an algebraic stack over 𝔽_{q}. The morphism N : Coh_{A,sp} → X is semistable of relative dimension −1. Its restriction to the open subset X − \underline{Disc}(A) is smooth. Consequently Coh_{A,sp} is locally of finite type and smooth over 𝔽_{q}.

Proof. The last assertion follows from ([Lau], 3.2.1). Since the assertion is étale local on X we may assume that X = Spec R is affine with R a principal ideal domain, | \underline{Disc}(A)| = \{𝔭\} and the generic fiber of A is ≅ M_{d}(F). By 3.2 we may also assume that e_{𝔭}(A) = d. Let ϖ be a generator of 𝔭. Then Γ(Spec R, A) is isomorphic to the R-subalgebra of M_{d}(R) of matrices which are upper triangular modulo 𝔭. Hence Γ(Spec R, A) can be identified with the R-algebra R^{d}{Π} defined by the relations

Π(x_{1}, …, x_{d}) = (x_{2}, …, x_{d}, x_{1})Π, Π^{d} = ϖ.

Let Coh_{A,sp}^{□}(S) denote the groupoid of pairs (K, α) where K ∈ Coh_{A,sp}(S) and α : O_{S}^{n} → N^{∗}(K) is an isomorphism. The action of Π on K yields – by transport of structure via α – a map O_{S}^{n} → O_{S}^{n} of the form (x_{1}, …, x_{d}) ↦ (x_{2}a_{1}, …, x_{d}a_{d−1}, x_{1}a_{d}) for some (a_{1}, …, a_{d}) ∈ Γ(S, O_{S}) such that a_{1} ⋯ a_{d} = N^{∗}(ϖ). Thus the assignement (K, α) ↦ (N, a_{1}, …, a_{d}) defines an isomorphism

Coh_{A,sp}^{□} ≅ Spec R[T_{1}, …, T_{d}]/(T_{1} ⋯ T_{d} − ϖ)

Finally the forgetful morphism Coh_{A,sp}^{□} → Coh_{A,sp} is a presentation. In fact it induces an isomorphism 𝔾_{m}^{d}\ Coh_{A,sp}^{□} ≅ Coh_{A,sp}. Here the 𝔾_{m}^{d} action on Coh_{A,sp}^{□} is defined by the natural 𝔾_{m}^{d}(S)-action on the set of isomorphisms α : O_{S}^{n} → N^{∗}(K). □

We finish this section with the following criterion for an A ⊠ O_{S}-module E to be a locally free.

Lemma 3.11. Let U ⊆ X be a non-empty open subscheme such that E|_{U×S} is a locally free A_{U} ⊠ O_{S}-module. The following conditions are equivalent.

(i) E is a locally free A ⊠ O_{S}-module of rank r. (ii) For x ∈ |X − U| and any pair of k-morphism g : S′ → S and x_{S′} : S′ → Spec k(x) → X the quotient g^{∗}(E)/g^{∗}(E)(−\frac{1}{e_{x}(A)}x_{S′}) is a special A-module of rank r. (iii) For x ∈ | \underline{Disc}(A)| − U and any pair of k-morphism g : S′ → S and x_{S′} : S′ → Spec k(x) → X the quotient g^{∗}(E)/g^{∗}(E)(−\frac{1}{e_{x}(A)}x_{S′}) is a special A-module of rank r.

Proof. That (i) implies (ii) follows from Lemma 2.17 and the equivalence of (ii) and (iii) from 3.8 (b) above. (ii) ⇒ (i) We may assume that S is affine, hence that S and of finite type over k. For y ∈ |X × S| we have to show that E ⊗ O_{(X×S),y} is a free (A ⊗_{O_{X}} O_{(X×S),y}- module. It follows from ([Laf], I.2, lemme 4) that we may even replace S by the image s of y → X × S → S. It follows from ([Laf], I.2, lemme 4). Thus it is enough to prove (i) if S = Spec \overline{k} is the algebraic closure of k. However in this case the assertion follows from 2.11 and 2.17. □

We have the following generalization of ([Lau], Lemma 1.2.6).

Lemma 3.12. Let 0 → E′ → E → K → 0 be a short exact sequence of right A ⊠ O_{S}-modules. We assume K is coherent as an O_{X×S}-module, the map Supp(K) ↪ X × S → S is an isomorphism and K is as an O_{S}-module locally free of rank rd. We also assume that E is a locally free A ⊠ O_{S}-module of rank r. Then the following conditions are equivalent.

(i) E′ is a locally free A ⊠ O_{S}-module of rank r. (ii) K is special of rank r.

Proof. Again by using Lafforgues Lemma ([Laf], I.2.4) (applied to A and maximal tori in A) it suffices to consider the case where k is algebraically closed and S = Spec k. The assertion follows then from Lemma 2.18. □

4 The moduli space of A-elliptic sheaves

4.1 A-elliptic sheaves

In this chapter X denotes a smooth projective geometrically connected curve over the finite field 𝔽_{q} of characteristic p, F the function field of X. We also fix a closed point ∞ ∈ |X|. Let A be a locally principal O_{X}-order of rank d^{2} and let A be its generic fiber. We make the following

Assumption 4.1. e_{∞}(A) = d.

Definition 4.2. Let S be an 𝔽_{q}-scheme. An A-elliptic sheaf over S with pole ∞ is a triple E = (E, ∞_{S}, t), where E is a locally free right A ⊠ O_{S}-module of rank 1, where ∞_{S} : S → X is an 𝔽_{q}-morphism with ∞_{S}(S) = {∞} and where

t : ^{τ} (E(−\frac{1}{d}∞_{S})) ⟶ E

is an injective A ⊠ O_{S}-linear homomorphism such that the following condition holds: (*) The map Supp(Coker(t)) ↪ X × S → S is an isomorphism. Considered as a sheaf on S, K is a locally free O_{S}-module of rank d. Hence Supp(Coker(t)) is the image of the graph of a 𝔽_{q}-morphism ι_{0} : S → X called the zero (or characteristic) of E. We denote by Eℓℓ_{A}^{∞} the stack over 𝔽_{q} such that for each S ∈ Sch /k, Eℓℓ_{A}^{∞}(S) is the category whose objects are A-elliptic sheaves over S and whose morphisms are isomorphisms between A-elliptic sheaves.

For n ∈ \frac{1}{d}ℤ we define Eℓℓ_{A,n}^{∞} to be the open and closed substack of A-elliptic sheaves E = (E, ∞_{S}, t) with fixed degree deg_{A}(E) = n. The functor which maps an A-elliptic sheaf over E = (E, ∞_{S}, t) over S to its zero ι_{0} : S → X defines a morphism char : Eℓℓ_{A}^{∞} → X (called the characteristic morphism). Similarly E = (E, ∞_{S}, t) ↦ ∞_{S} defines a morphism pole : Eℓℓ_{A}^{∞} → Spec k(∞). By Lemma 3.12, Coker(t) is a special A-module of rank 1. This fact allows us to compare the above condition (*) with the condition spéciale in ([Hau], section 3) (see also 5.11 (b) below). It follows that the characteristic morphism factors as

char : Eℓℓ_{A}^{∞} ⟶ Coh_{A,sp} ⟶^{N} X  (16)

We will see in the proof of Theorem 4.11 below that the first arrow is smooth.

Remarks 4.3. (a) The concept of an A-elliptic sheaf is due to Laumon, Rapoport and Stuhler ([LRS], section 2). The definition given above is different but, as will be explained in the appendix, equivalent to the one given in ([LRS], section 2). In fact our Definition 4.2 is slightly more general. Their notion corresponds to an A-elliptic sheaf where (i) A is a division algebra which is unramified at ∞, (ii) A|_{X−{∞}} is a maximal order in A and (iii) the zero ι_{0} is disjoint from | \underline{Disc}(A)|, i.e. ι_{0} factors through X − | \underline{Disc}(A)| ↪ X (the latter condition was weakened in [BS] and [Hau] to require only that ι_{0} factors through (X − | \underline{Disc}(A)|) ∪ \{∞\} ∪ \{x ∈ |X| | inv_{x}(A) = \frac{1}{d}\}).

(b) Let A be the subsheaf of M_{d}(O_{X}) of matrices which are upper triangular modulo ∞. In this case Eℓℓ_{A}^{∞} is isomorphic to the stack Eℓℓ_{X}^{(d)} of elliptic sheaves of rank d (hence above X − {∞} it is isomorphic to the stack of Drinfeld modules of rank d; compare ([BS], section 3)). In fact by Proposition 5.10 of the appendix we have Eℓℓ_{A}^{∞} ≅ PEℓℓ_{M_{d}(O_{X})}^{∞} and the latter is isomorphic to the

stack of Eℓℓ_{X}^{(d)} by Morita equivalence.

(c) If A is a division algebra then A-elliptic sheaves are special cases of right A-shtukas of rank 1 ([Laf], 1.1). Recall that an A-shtuka of rank 1 is a diagram

\xymatrix{E \ar[dr]^{j} \\  & E′ \\ ^{τ} E \ar[ur]^{t}}

where E, E′ are locally free right A ⊠ O_{S}-modules of rank 1 and where j and t are injective A ⊠ O_{S}-linear homomorphism such that the cokernels of j and t and of the dual morphisms j^{∨} and t^{∨} satisfy condition (*) above (actually, it follows from Lemma 3.12 (compare also the proof of 4.14 (b) below) that it is enough to require that the cokernels of j and t satisfies (*)). Hence we have Coker(j), Coker(t) ∈ Coh_{A,sp}(S). In fact if E = (E, ∞_{S}, t) ∈ Eℓℓ_{A}^{∞}(S) is an A-elliptic sheaf with zero ι_{0} : S → X then the diagram

\xymatrix{E(−\frac{1}{d}∞_{S}) \ar@{-->}[dr]^{j} \\  & E \\ ^{τ} (E(−\frac{1}{d}∞_{S})) \ar@{-->}[ur]^{t}}  (17)

is an A-shtuka with pole ∞_{S} and zero ι_{0}. Therefore we have a 2-cartesian square

Eℓℓ_{A}^{∞} ⟶ Sht_{A}^{1}
↓^{pole}  ↓
Spec k(∞) ⟶ Coh_{A,sp}

(18)

Here the second vertical arrow is given by mapping an A-shtuka (E, E′, j, t) to Coker(j). The lower horizontal arrow is defined by A/A(−\frac{1}{d}∞_{Spec k(∞)}) ∈ Coh_{A,sp}(Spec k(∞)). It is easy to see that it is representable and a closed immersion. Hence the morphism Eℓℓ_{A}^{∞} → Sht_{A}^{1} given by (17) is a closed immersion.

(d) One could consider A-elliptic sheaves more generally for a hereditary O_{X}- order A. However since any hereditary O_{X}-order is Morita equivalent to a locally principal O_{X}-order we do not obtain new moduli spaces in this way.

(e) If we consider Eℓℓ_{A}^{∞} as a k(∞)- rather than a 𝔽_{q}-stack we can (and will) drop ∞_{S} from the definition. More precisely for S ∈ Sch /k(∞) the objects of Eℓℓ_{A}^{∞}(S) are just pairs E = (E, t) such that (E, ∞_{S}, t) is an A-elliptic sheaf as in 4.2 where ∞_{S} is composite S → Spec k(∞) ↪ X.

(f) Define an automorphism of stacks θ : Eℓℓ_{A}^{∞} → Eℓℓ_{A}^{∞} by

θ(E, ∞_{S}, t) = (E(\frac{1}{d}^{τ} ∞_{S}),^{τ} ∞_{S}, t(\frac{1}{d}^{τ} ∞_{S}))  (19)

where ^{τ} ∞_{S} = ∞_{S} ∘ Frob_{S}. We have θ(Eℓℓ_{A,n}^{∞}) = Eℓℓ_{A,n+ \frac{1}{d}}^{∞} for all n ∈ \frac{1}{d}ℤ and θ^{deg(∞)}(E) = E ⊗_{A} A(\frac{1}{d}∞) for all A-elliptic sheaves E.

(g) Let A′ be a locally principal O_{X}-order which is Morita equivalent to A and let L be an invertible A-A′-bimodule. Then

E = (E, ∞_{S}, t) ↦ E ⊗_{A} L: = (E ⊗_{A} L, ∞_{S}, t ⊗_{A} id_{L})

defines an isomorphism between Eℓℓ_{A}^{∞} and Eℓℓ_{A′}^{∞}. If m = deg_{A}(L) then it maps the substack Eℓℓ_{A,n}^{∞} isomorphically onto the substack Eℓℓ_{A,m+n}^{∞}. In particular E ↦ E ⊗_{A} L defines an action of the abelian group Pic(A) on Eℓℓ_{A}^{∞}.

We define Pic(A)[θ] to be the group generated by its subgroup Pic(A) and the element θ which satisfies the relations θ^{deg(∞)} = A(\frac{1}{d}∞) and θL = Lθ for all L ∈ Pic(A). Thus Pic(A)[θ] acts on Eℓℓ_{A}^{∞}. The group Pic(A)[θ] is an extension of ℤ/ deg(∞)ℤ ≅ G(k(∞)/𝔽_{q}) by Pic(A). The map deg_{A} : Pic(A) → \frac{1}{d}ℤ extends to a homomorphism deg_{A} : Pic(A)[θ] → \frac{1}{d}ℤ by defining deg_{A}(θ) = \frac{1}{d}.

Definition 4.4. The group of modular automorphisms W(A, ∞) is defined as the kernel of deg_{A} : Pic(A)[θ] → \frac{1}{d}ℤ.

W(A, ∞) stabilizes the substack Eℓℓ_{A,n}^{∞} for all n ∈ \frac{1}{d}ℤ. There exists a canonical homomorphism

W(A, ∞) → G(k(∞)/𝔽_{q})  (20)

so that pole : Eℓℓ_{A}^{∞} → Spec k(∞) is W(A, ∞)-equivariant. The kernel of (20) is Pic_{0}(A) and the image is of order \frac{δ(A) deg(∞)}{d} (thus (20) is surjective if and only if deg_{A} : Pic(A) → \frac{1}{d}ℤ is surjective).

4.2 Level structure

We reformulate now the notion of a level structure on an A-elliptic sheaf given in ([LRS], 2.7 and 8.4) in our framework. Let I = ∑_{x} n_{x}x be an effective divisor on X. We recall first from ([LRS], 2.7; see also [Dr3]) the notion of a level-I-structure when ∞ does not divide I, i.e. n_{∞} = 0.

Definition 4.5. Suppose that ∞ ∉ |I|. Let E = (E, ∞_{S}, t) be an A-elliptic sheaf over an 𝔽_{q}-scheme S with zero ι_{0} : S → X disjoint from I i.e. ι_{0}(S)∩I = ∅. A level-I-structure on E is an A_{I} ⊠ O_{S}-linear isomorphism

α : A_{I} ⊠ O_{S} ⟶ E|_{I×S} ⊗_{A} A(\frac{1}{d}∞)

compatible with t, i.e. the diagram

\xymatrix{^{τ} E|_{I×S} \ar[rr]^{t|_{I×S}} &  & E|_{I×S} \\  & A_{I} ⊠ O_{S} \ar[ul]^{^{τ} α} \ar[ur]_{α}}

commutes.

We denote by Eℓℓ_{A,I}^{∞} the stack of A-elliptic sheaves with level I-structure and for n ∈ \frac{1}{d}ℤ by Eℓℓ_{A,I,n}^{∞} the open and closed substack of A-elliptic sheaves with level I-structure with fixed degree deg_{A} = n. Again we obtain morphisms char : Eℓℓ_{A,I}^{∞} → X −I and pole : Eℓℓ_{A,I}^{∞} → Spec k(∞). The automorphism (19) of Remark 4.3 (f) extends canonically to an automorphism θ : Eℓℓ_{A,I}^{∞} → Eℓℓ_{A,I}^{∞}. The right action of Pic(A) on Eℓℓ_{A}^{∞} lifts to a right action of Pic_{I}(A) on Eℓℓ_{A,I}^{∞} as follows. If (L, β) is an invertible A-A-bimodule with level-I-structure and (E, α) an A-elliptic sheaf with level-I-structure (E, α) over S then we define (E, α) ⊗ (L, β): = (E ⊗ L, α • β) with

α •β : A_{I} ⊠O_{S} ⟶^{β⊠id} L_{I} ⊠O_{S} = (A_{I} ⊠O_{S})⊗_{A} L⊠O_{S} ⟶^{α⊠id} (E ⊗_{A} L)|_{I×S}. (21)

As before we have θ^{deg(∞)}(E, α) = (E, α) ⊗ (A(\frac{1}{d}∞), id). Suppose now that |I| = {∞}, i.e. I = n∞ with n > 0. Let k(∞)_{d} be a fixed extension of degree d of k(∞). According to section 2.6 there exists a pair (M_{∞}, ϕ_{∞}) consisting of a free right A_{∞} ⊗_{𝔽_{q}} k(∞)_{d}-module M_{∞} of rank 1 and an isomorphism

ϕ_{∞} : ^{σ}(M_{∞}𝔓) ⟶ M_{∞}

where 𝔓 denotes the maximal invertible two-sided ideal of A_{∞} ⊗_{𝔽_{q}} k(∞)_{d} corresponding to the inclusion k(∞) ↪ k(∞)_{d}. Let M_{I} denote the sheaf of A_{I} ⊗_{𝔽_{q}} k(∞)_{d}-modules associated to the M_{∞}/M_{∞}𝔭_{∞}^{n}. The map ϕ_{∞} induces an isomorphism

ϕ_{I} : ^{τ} (M_{I}(−\frac{1}{d}∞_{d})) ⟶ M_{I}

where ∞_{d} denotes the morphism ∞_{k(∞)_{d}} : Spec k(∞)_{d} → Spec k(∞) ↪ X.

Definition 4.6. Let E = (E, ∞_{S}, t) be an A-elliptic sheaf over an 𝔽_{q}-scheme S with zero ι_{0} : S → X disjoint from I.

(a) Suppose that I = n∞ with n > 0. Let E = (E, ∞_{S}, t) be an A-elliptic sheaf over an 𝔽_{q}-scheme S with zero ι_{0} : S → X disjoint from I. A level-I-structure on E consist of a pair (λ, α) where λ : S → Spec k(∞)_{d} is an 𝔽_{q}-morphism of schemes which lifts the pole ∞_{S} and where α is an A_{I_{∞}} ⊠ O_{S}- linear isomorphism

α : (id_{I} ×λ)^{∗}(M_{I}) ⟶ E|_{I×S}

such that the diagram

\xymatrix{(^{τ} (E(−\frac{1}{d}∞_{S}))|_{I×S} \ar[r]^{t|_{I×S}} & E|_{I×S} \\ (id_{I} ×λ)^{∗}(^{τ} (M_{I}(−\frac{1}{d}∞_{d}))) \ar[r]^{(id_{I} ×λ)^{∗}(ϕ_{I})} \ar[u]_{^{τ} (α(− \frac{1}{d} ∞_{S}))} & (id_{I} ×λ)^{∗}(M_{I}) \ar[u]_{α}}

commutes.

(b) Suppose that I is an arbitrary effective divisor on X with ∞ ∈ |I| and write I = n∞ + I^{∞} = I_{∞} + I^{∞} with n > 0 such that ∞ does not divide I^{∞}. A level-I-structure on E is a triple (α_{f}, λ, α_{∞}) consisting of a level-I^{∞}-structure α_{f} and a level-I_{∞}-structure (λ, α_{∞}).

Let I be an effective divisor on X with ∞ ∈ |I|. Again we define Eℓℓ_{A,I}^{∞} as the stack of A-elliptic sheaves with level-I-structure (E, t, α_{f}, λ, α_{∞}) and denote for n ∈ \frac{1}{d}ℤ by Eℓℓ_{A,I,n}^{∞} the substack where deg_{A}(E) = n. There are canonical morphisms

char : Eℓℓ_{A,I}^{∞} → X − I, pole : Eℓℓ_{A,I}^{∞} → Spec k(∞)_{d}

(the latter is given by (E, α_{f}, λ, α_{∞}) ↦ λ; it lifts the morphism pole : Eℓℓ_{A}^{∞} → Spec k(∞)).

Modular automorphisms. Next we are going to extend the definition of the automorphisms (19) and define a natural right action of a certain idele class group on Eℓℓ_{A,I}^{∞} (thus lifting the action of Pic_{I}(A) when ∞ ∉ |I|). Define θ : Eℓℓ_{A,I}^{∞} → Eℓℓ_{A,I}^{∞} by

θ(E, ∞_{S}, t, α_{f}, λ, α_{∞}) = (E(\frac{1}{d}^{τ} ∞_{S}),^{τ} ∞_{S}, t(\frac{1}{d}^{τ} ∞_{S}), α_{f},^{τ} λ, α_{∞}^{♯})  (22)

where α_{∞}^{♯} is the composite

(id_{I} ×^{τ} λ)^{∗}(M_{I}) ⟶^{ϕ_{I}( \frac{1}{d}^{τ} ∞_{d})} (id_{I} ×λ)^{∗}(M_{I}(\frac{1}{d}^{τ} ∞_{d})) ⟶^{α_{∞}( \frac{1}{d}^{τ} ∞_{S})} E(\frac{1}{d}^{τ} ∞_{S})|_{I×S}.

Write I = n∞ + I^{∞} = I_{∞} + I^{∞} with n > 0 and ∞ ∉ |I^{∞}|. Let D_{∞} be a principal order in a central F -algebra D_{∞} of dimension d^{2} such that

e(D_{∞}) = e(A_{∞}) and inv(D_{∞}) = inv(A_{∞}) + \frac{1}{d}.

We have seen in section 2.6 that

D_{∞} ≅ End_{A_{∞}⊗_{𝔽q} k(∞)_{d}}(M_{∞}, ϕ_{∞}).

We choose an isomorphism (thus making (M_{∞}, ϕ_{∞}) into an invertible ϕ-D_{∞}- A_{∞}-bimodule of slope −\frac{1}{d}). Let

U_{I}(A^{∞} × D_{∞}): = Ker( ∏_{x∈|X|−{∞}} A_{x}^{∗} × D_{∞}^{∗} → Γ(I^{∞}, A_{I^{∞}})^{∗} × (D_{∞}/𝔭_{∞}^{n}D_{∞})^{∗})

and define

C_{I}(A^{∞} × D_{∞}): = (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞})/U_{I}(A^{∞} × D_{∞})F^{∗}

For a = (a_{f}, a_{∞}) = ({a_{x}}_{x≠∞}, a_{∞}) ∈ (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞}) let div(a) = ∑_{x∈|X|−{∞}} v_{A_{x}}(a_{x})x + v_{D_{∞}}(a_{∞})∞ ∈ Div(A). Let (E, α_{f}, λ, α_{∞}) ∈ Eℓℓ_{A,I}^{∞}. Left multiplication by a_{f} on ∏′_{x∈|X|−{∞}} N(A_{x}) induces a level-I^{∞}-structure α_{f} · a_{f} on E(div(a)). Similarly left multiplication by a_{∞} on M_{∞} yields a level-I_{∞}-structure α_{∞} · a_{∞} on E ⊗ A(div(a)). One easily verifies that

(E, α_{f}, λ, α_{∞}) · a: = (E ⊗ A(div(a)), α_{f} · a_{f}, λ, α_{∞} · a_{∞})

yields a right (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞})-action on Eℓℓ_{A,I}^{∞} and that it factors through C_{I}(A^{∞} × D_{∞}). The canonical projection C_{I}(A^{∞} × D_{∞}) → C_{I^{∞}}(A) (given on the ∞-factor by N(D_{∞}) ⟶^{v_{D∞}} \frac{1}{d}ℤ ≅ N(A_{∞})/A_{∞}^{∗}) followed by the isomorphism C_{I^{∞}}(A) → Pic_{I^{∞}}(A) from 3.4 yields also a C_{I}(A^{∞}×D_{∞})-action on Eℓℓ_{A,I^{∞}}^{∞} and one checks that the forgetful morphism of stacks Eℓℓ_{A,I}^{∞} ⟶ Eℓℓ_{A,I^{∞}}^{∞} commutes with the C_{I}(A^{∞} × D_{∞})-actions. By Remark 2.22, there exists a prime element ϖ_{∞} ∈ O_{∞} such that the class ξ ∈ C_{I}(A^{∞} × D_{∞}) of the idele ({1}_{x≠∞}, ϖ_{∞}) satisfies θ^{d deg(∞)}(E) = E · ξ for all E ∈ Eℓℓ_{A,I}^{∞}(S). If ∞ does not divide the level I we define the group Pic_{I}(A)[θ] similar to Pic(A)[θ] in the last section. Pic_{I}(A)[θ] contains Pic_{I}(A) as a subgroup and the element θ lies in the center and satisfies the relation θ^{deg(∞)} = (A(\frac{1}{d}∞), id). Let deg_{A} : Pic_{I}(A)[θ] → \frac{1}{d}ℤ be given by (L, β) ↦ deg_{A}(L) on Pic_{I}(A) and deg_{A}(θ) = \frac{1}{d}. Assume that ∞ divides I and write I = n∞ + I^{∞} = I_{∞} + I^{∞} as above. Define C_{I}(A^{∞} ×D_{∞})[θ] as the group generated by C_{I}(A^{∞} ×D_{∞}) and a central element θ satisfying the relation θ^{d deg(∞)} = ξ. The homomorphism C_{I}(A^{∞} × D_{∞}) → C_{I^{∞}}(A) ≅ Pic_{I^{∞}}(A) ⟶^{deg_{A}} \frac{1}{d}ℤ extends to a homomorphism deg_{A} : C_{I}(A^{∞} × D_{∞})[θ] → \frac{1}{d}ℤ by setting deg_{A}(θ) = \frac{1}{d}ℤ.

Definition 4.7. Let I be an effective divisor on X. The group of modular automorphisms W(A, I, ∞) of Eℓℓ_{A,I}^{∞} is defined as follows:

W(A, I, ∞): = \{ \begin{matrix}Ker(deg_{A} : Pic_{I}(A)[θ] → \frac{1}{d}ℤ) & if ∞ ∉ |I|, \\ Ker(deg_{A} : C_{I}(A^{∞} × D_{∞})[θ] → \frac{1}{d}ℤ) & if ∞ ∈ |I|. \end{matrix}

Remarks 4.8. (a) If ∞ ∉ |I| (resp. ∞ ∈ |I|) there exists a canonical homomorphism W(A, I, ∞) → G(k(∞)/𝔽_{q}) (resp. W(A, I, ∞) → G(k(∞)_{d}/𝔽_{q})) with kernel Pic_{I,0}(A) (resp. C_{I}(A^{∞} × D_{∞})_{0}) such that Eℓℓ_{A,I}^{∞} → Spec k(∞) (resp. Eℓℓ_{A,I}^{∞} → Spec k(∞)_{d}) is W(A, I, ∞)-equivariant.

(b) Let I < J be effective divisors on X there exists a canonical projection W(A, J, ∞) → W(A, I, ∞) such that the forgetful morphism Eℓℓ_{A,J}^{∞} ⟶ Eℓℓ_{A,I}^{∞} is W(A, J, ∞)-equivariant.

(c) The map x ↦ xθ^{−d deg_{A}(x)} induces an isomorphism

C_{I}(A^{∞} × D_{∞})/ξ_{∞}^{ℤ} ≅ W(A, I, ∞).

This fact will be used in section 4.6.

We have (compare ([LRS], 8.10) and ([Laf], I.3.5))

Lemma 4.9. Let I < J be effective divisors on X. Over X − J the forgetful morphism

Eℓℓ_{A,J}^{∞} ⟶ Eℓℓ_{A,I}^{∞}

is representable and is a finite, étale Galois covering. Its Galois group is ≅ Ker(W(A, J, ∞) → W(A, I, ∞)). If ∞ ∉ |J| − |I| it is ≅ Ker(A_{J}^{∗} → A_{I}^{∗}).

Corollary 4.10. Let A′ be a locally principal O_{X}-suborder of A with the same generic fiber A and denote by ι : Y ↪ X the reduced closed subscheme with |Y | = {x ∈ |X| | e_{x}(A′) > e_{x}(A′)}. Note that ∞ ∉ Y. Let I be an effective divisors disjoint from Y and put J: = I + Y. Then over X − J the forgetful morphism factors canonically as

Eℓℓ_{A,J}^{∞} ⟶ Eℓℓ_{A′,I}^{∞} ⟶ Eℓℓ_{A,I}^{∞}.  (23)

Both maps are representable and finite and étale. Moreover the first arrow is Galois.

Proof. Let P: = Im(A′|_{Y} → A|_{Y}). Then the diagram

A′ ⟶ ι_{∗}(P)
↓  ↓
A ⟶ ι_{∗}(A_{Y})

is cartesian. Here we view J as a closed subscheme of X and denote by ι : J → X the inclusion. For E = (E, t, ∞_{S}, α) in Eℓℓ_{A,J}^{∞}(S) we decompose α into a level-I-structure α_{I} and a level-Y -structure α_{Y}. Define E′ by the cartesian square

\xymatrix{E′ \ar[rr] \ar[d] &  & ι_{∗}(P) \ar[d] \\ E \ar[r] & ι_{∗}(E|_{Y ×S}) \ar[r]^{_{α_{Y} _{−1}}} & ι_{∗}(A_{Y})}

Then E′ is a locally free A′ ⊠ O_{S}-module of rank 1. The first morphism in (23) is induced by E ↦ E′ whereas the second by E′ ↦ E′ ⊗_{A′} A. The proof of the remaining assertions is left to the reader. □

4.3 The coarse moduli scheme

Our aim now is to prove the following theorem.

Theorem 4.11. (a) Eℓℓ_{A,I}^{∞} is a Deligne-Mumford stack over 𝔽_{q}. It is locally of finite type over X. The morphism char : Eℓℓ_{A,I}^{∞} → X − I is semistable of relative dimension d − 1.

(b) The open and closed substack Eℓℓ_{A,I,0}^{∞} of Eℓℓ_{A,I}^{∞} is of finite type over X′: = X − (\underline{Disc}(A) ∪ I). It admits a coarse moduli scheme which will be denote by Ell_{A,I}^{∞}. The structural map Eℓℓ_{A,I,0}^{∞}|_{X′} → Ell_{A,I}^{∞} is an isomorphism if I ≠ 0.

(c) The morphism char : Ell_{A,I}^{∞} → X′ is quasiprojective and smooth of relative dimension d − 1. In particular Ell_{A,I}^{∞} is a smooth, quasiprojective 𝔽_{q}-scheme.

Remark 4.12. This is known if A is a division algebra or A = M_{d}(F) and if we restrict Eℓℓ_{A,I}^{∞} to the open subset X′ ([LRS], Theorem 4.1 and [Dr1]). In fact if we assume that A is a division algebra and let S denote the subset of \underline{Disc}(A) consisting of all points 𝔭 ∈ \underline{Disc}(A) − \{∞\} with inv_{𝔭} A = 1 and of ∞ if inv_{∞} A = 0 then Eℓℓ_{A,I,0}^{∞} admits a coarse moduli scheme Ell_{A,I}^{∞} over X ∪ S which is projective and semistable of relative dimension d − 1 (at the pole ∞ this is proved in [BS]; at 𝔭 ∈ S − {∞} it is proved in certain cases by [Hau] and can be deduce in general from the first case using the main result of section 4.5).

The proof of 4.11 consists essentially of two parts. In the first part one shows that Eℓℓ_{A}^{∞} → X is a Deligne-Mumford stack and semistable. In the second part one proves that for I ≠ 0, Eℓℓ_{A,I,n}^{∞} is a quasiprojective scheme over X′ by showing that for a large m the map Eℓℓ_{A,mI,n}^{∞,stab} → Eℓℓ_{A,I,n}^{∞} is surjective. Here Eℓℓ_{A,I}^{∞,stab} denotes the substack of A-elliptic sheaves whose underlying vector bundle is I-stable. It is a consequence of a theorem of Seshadri that Eℓℓ_{A,I}^{∞,stab} is a quasiprojective scheme. For the surjectivity one can follow the arguments in ([LRS], section 5) so we will omit the proof. The proof of the first part is also mainly a reproduction of the corresponding arguments in ([LRS], section 4; compare also ([Laf] Chapitre I), [La] and ([Lau], 1.3 and 1.4)) so we will be rather brief and elaborate only on those steps were essential modification have to be made. We follow ([Lau], 1.2) and work with the factorization (16), i.e. we consider Eℓℓ_{A,I}^{∞} mostly over Coh_{A,sp} rather than over X. Let Inj_{A,sp} be the stack over k(∞) such that for each S ∈ Sch /k(∞), Inj_{A,sp}(S) is the groupoid of injective morphisms j : E′ → E locally free right A ⊠ O_{S}-modules of rank 1 with Coker(j) ∈ Coh_{A,sp}(S).

Lemma 4.13. (a) The two morphism

Inj_{A,sp} ⟶ Vect_{A,0}^{1} × Coh_{A,sp}

given by (j : E′ → E) ↦ (E, Coker(j)) and (j : E′ → E) ↦ (E′, Coker(j)) are representable and quasiaffine of finite type and smooth of relative dimension d. Consequently Inj_{A,sp} is algebraic, smooth and of finite type over 𝔽_{q}.

(b) The two morphism

Inj_{A,sp} ⟶ Vect_{X}^{d^{2}}

given by E and E′ are representable and quasiprojective and in particular of finite type.

The proof of (a) for the first morphism is literally the same as ([Lau], 1.3.2). The statement for second morphism can be deduce from that for the first as in ([Lau], 1.3.2). We need to remark only that for a short exact sequence 0 → E′ → E → K → 0 of right A ⊗ O_{S}-modules with E′, E ∈ Vect_{A}^{1}(S) and K ∈ Coh_{A,sp}(S) the third term of the dual sequence of A^{opp} ⊗O_{S}-modules 0 → E^{∨} → E′^{∨} → \underline{Ext}_{A⊗O_{S}}^{1}(K, A ⊠ O_{S}) → 0 lies, by Lemma 3.12, in Coh_{A^{opp},sp}(S).

(b) follows from ([Laf], I.2.2 and I.2.8). □

Consider now the following obvious diagram of stacks

Eℓℓ_{A}^{∞} ⟶  Vect_{A}^{1} ⊗_{𝔽_{q}}k(∞)
↓  ↓
Inj_{A,sp} ⟶ (Vect_{A}^{1} × Vect_{A}^{1}) ⊗_{𝔽_{q}} k(∞)
↓
Coh_{A,sp}

(24)

where the right vertical arrow in the (2-cartesian) square is the graph of the endomorphism Frob ∘θ^{−1} : Vect_{A}^{1} ⊗_{𝔽_{q}}k(∞) → Vect_{A}^{1} ⊗_{𝔽_{q}}k(∞) (for the definition of θ^{−1} compare 4.3 (f); if deg(∞) = 1 it is given by E ↦ E(−\frac{1}{d}∞)). By ([Laf], I.2.5) the stack Vect_{A}^{1} is algebraic, locally of finite type and smooth over 𝔽_{q}. Together with Proposition 3.10, Lemma 4.9 and Lemma 4.13 above the same argument as in ([LRS], section 4; see also ([Laf], I.2.5) and ([Lau], I.3.5)) imply part (a) of Lemma 4.14. (a) Let I be an effective divisor on X. The morphism Eℓℓ_{A,I}^{∞} → Coh_{A,sp} is algebraic, locally of finite type and smooth of relative dimension d. The morphism char : Eℓℓ_{A,I}^{∞} → X is semistable of relative dimension d − 1.

(b) Eℓℓ_{A}^{∞} is a Deligne-Mumford stack, locally of finite type and smooth over 𝔽_{q}. Moreover if I ≠ 0 then Eℓℓ_{A,I}^{∞}|_{X′} is isomorphic to an algebraic space.

Proof of (b). Everything is clear if we replace “Deligne-Mumford” by “algebraic”. To prove that Eℓℓ_{A}^{∞} is indeed a Deligne-Mumford stack we use ([LM], 8.1). If we replace the lower vertical map in (24) by Inj_{A,sp} → Spec k(∞) then, by Lemma 4.13 and ([La], Lemma on p. 60), the diagonal morphism Eℓℓ_{A}^{∞} → Eℓℓ_{A}^{∞} ×_{𝔽_{q}} Eℓℓ_{A}^{∞} is unramified. Note that for E ∈ Eℓℓ_{A,I}^{∞}(S) with zero S → X′ we have Aut(E) = 𝔽_{q}^{∗} if I = 0 or Aut(E) = 1 otherwise. Hence the last assertion follows from ([LM], 8.1.1). □

Remarks 4.15. (a) Note that by 4.3 (f) we could have defined Ell_{A}^{∞} also as the coarse moduli scheme of the quotient Eℓℓ_{A}^{∞}/θ^{ℤ} or of Eℓℓ_{A,n}^{∞} for any n ∈ \frac{1}{d}ℤ.

(b) Let I ↪ X be a reduced closed subscheme with ∞ ∉ I and let A be the subsheaf of M_{2}(O_{X}) of matrices which are upper triangular modulo I. Then by using 4.10 and 4.3 (b) it is easy to see that the Ell_{A}^{∞} is isomorphic to the (open) Drinfeld modular curve Y_{0}(I) = Y_{0}^{∞}(I).

(c) If A is a central division algebra which is unramified at ∞ and A|_{X−{∞}} is a maximal order in A then char : Ell_{A}^{∞} → X is proper (see [LRS], Theorem 6.1 and [Hau], 6.4). In the general case this is not true anymore even if A is a division algebra. In fact if d = 2 and A is ramified only at ∞ and at 𝔭 ∈ |X| and if A is a maximal O_{X}-order in A then we will show in section 4.5 that Ell_{A}^{𝔭} is a twist of the affine curve Y_{0}^{∞}(𝔭) → X.

4.4 Invertible Frobenius bimodules

We consider now two locally principal O_{X}-orders A and B, both of rank d^{2} with \underline{Disc}(A) = \underline{Disc}(B) and assume that e(A) = d = e(B). We denote by A and B the generic fibers of A and B respectively. Let D = ∑_{x∈|X|} m_{x}x ∈ Div(A) be a divisor such that ∑_{x∈|X|} m_{x} = 0. We consider the following moduli problem associated to A, B, D.

Definition 4.16. Let S be an 𝔽_{q}-scheme. An invertible Frobenius A-B-bimodule (or Φ-A-B-bimodule for short) over S of slope D is a tuple L = (L, (x_{S})_{x∈|D|}, Φ) where L is an invertible A ⊠ O_{S}-B ⊠ O_{S}-bimodule which is locally free of rank 1 as a left A ⊠ O_{S}- and right B ⊠ O_{S}-module, where for x ∈ |D|, x_{S} : S → X is a morphism in Sch /𝔽_{q} which factors through x → X and where Φ is a bimodule isomorphism

Φ : ^{τ} (L(D_{S})) ⟶ L.

with D_{S}: = ∑_{x∈|D|} m_{x}x_{S}. The morphisms x_{S} are called the poles of L.

Note we have deg_{A}(L) = deg_{B}(L). Note also that (A ⊠ O_{S})(D_{S}) ⊗_{A} L = L ⊗_{B} (B ⊠ O_{S})(D_{S}). Thus the notion L(D_{S}) is unambiguous. Obviously the concept of invertible Φ-A-B-bimodules of slope D defines a stack which we denote by SE_{A,B}^{D}. It is equipped with canonical morphisms SE_{A,B}^{D} → Spec k(x) for all x ∈ |D|. For n ∈ \frac{1}{d}ℤ let SE_{A,B,n}^{D} be the substack of (L, (x_{S})_{x∈T} , Φ) ∈ SE_{A,B}^{D}(S) with deg_{A}(L) = n. There is a canonical left Pic(A)- and right Pic(B)-action on SE_{A,B}^{D} compatible with deg_{A} (in fact the left and right action are the same if we identify the two groups under the canonical isomorphism Pic(A) ≅ Div(A) /F^{∗} ≅ Pic(B)). For x ∈ |D| we define an automorphism θ_{x} : SE_{A,B}^{D} → SE_{A,B}^{D} by

θ_{x}(L, (x′_{S})_{x′∈|D|}, Φ) = (L(−m_{x}^{τ} x_{S}),^{τ} x_{S}, (x′_{S})_{x′∈|D|,x′≠x}, Φ(−m_{x}^{τ} x)). (25)

The automorphisms θ_{x} for different x ∈ |D| commute with each other and with the Pic(A)- and Pic(B)-action. We have θ_{x}(SE_{A,B,n}^{D}) = SE_{A,B,n−m_{x}}^{D} for all n ∈ \frac{1}{d}ℤ and θ_{x}^{deg(x)}(L) = A(−m_{x}x) ⊗ L = L ⊗ B(−m_{x}x) for all L ∈ SE_{A,B}^{D}(S). Moreover if |D| = {x_{1}, …, x_{m}} and if we put Θ_{D}: = θ_{x_{1}} ∘ … ∘ θ_{x_{m}} then Θ_{D}(L) = Frob_{S}^{∗}(L) for all S ∈ Sch /𝔽_{q} and L ∈ SE_{A,B}^{D}(S). Hence Θ_{D} = Frob_{SE}_{A,B}^{D}.

Level structure. Let L = (L, (x_{S})_{x∈|D|}, Φ) ∈ SE_{A,B}^{D}(S). We view L as a right B-module only and proceed as in section 4.2. Let I be an effective divisor on X. Assume first that |I| ∩ |D| = ∅. Then a level-I-structure on L is an isomorphism of right B_{I} ⊠ O_{S}-modules β : B_{I} ⊠ O_{S} ⟶ L|_{I×S} such that the diagram

\xymatrix{^{τ} L|_{I×S} \ar[rr]^{Φ|_{I×S}} &  & L|_{I×S} \\  & A_{I} ⊠ O_{S} \ar[ul]^{^{τ} β} \ar[ur]_{α}}

commutes. Next assume that I = nx with n > 0 for some x ∈ |D|. Put e = e_{x}(A), m = m_{x} and let k(x)_{e} be an extension of degree e of k(x). If m > 0 we denote by M = (M_{x}, ϕ_{x}) a fixed invertible ϕ-A_{x}-B_{x}-bimodule of slope −m over O_{x} ⊗_{𝔽_{q}} k(x)_{e}. In case m < 0, M = (M_{x}, ϕ_{x}) denotes a ϕ-B_{x}-A_{x}-bimodule of slope m over O_{x} ⊗_{𝔽_{q}} k(x)_{e}. Thus if m > 0 (resp. m < 0) then ϕ_{x} is an isomorphism

ϕ_{x} : ^{σ}(M_{x}𝔓^{me}) ⟶ M_{x} (resp. ϕ_{x} : ^{σ}(𝔓^{−me}M_{x}) ⟶ M_{x})

where 𝔓 denotes the maximal ideal of B_{x} ⊗_{𝔽_{q}} k(x)_{e} corresponding to the inclusion k(x) ↪ k(x)_{e}. As in 4.2 the pair (M_{x}, ϕ_{x}) induces a pair (M_{I}, ϕ_{I}) consisting of A_{I} ⊗_{𝔽_{q}} k(x)_{e}-B_{I} ⊗_{𝔽_{q}} k(x)_{e}-bimodule and an isomorphism ϕ_{I} : τ (M_{I}(mx_{e})) ⟶ M_{I} where x_{e} is the map Spec k(x)_{e} → x ↪ X. A level-I-structure on L consists of a pair (μ, β) where μ : S → Spec k(x)_{e} is an 𝔽_{q}-morphism which lifts x_{S} and an isomorphism of right B_{I} ⊠ O_{S}-modules β. If m < 0, then β is a map

β : (id_{I} ×μ)^{∗}(M_{I}) ⟶ L|_{I×S}  (26)

such that

\xymatrix{^{τ} (L(D))|_{I×S} \ar[r] & L|_{I×S} \\ (id_{I} ×μ)^{∗}(^{τ} (M_{I}(mx_{e})) \ar[u] \ar[r] & (id_{I} ×μ)^{∗}(M_{I}) \ar[u]}

commutes. If m > 0 then

β : B_{I} ⊠ O_{S} ⟶ (id_{I} ×μ)^{∗}(M_{I}) ⊗_{A} L|_{I×S}  (27)

and

\xymatrix{^{τ} ((id_{I} ×μ)^{∗}(M_{I}) ⊗_{A} L|_{I×S}) \ar[rr]^{ϕ_{I}⊗Φ|_{I×S}} &  & (id_{I} ×μ)^{∗}(M_{I}) ⊗_{A} L|_{I×S} \\  & B_{I} ⊠ O_{S} \ar[ul]^{^{τ} β} \ar[ur]_{β}}

should commute. For an arbitrary effective divisor I on X we write I = I_{0} + ∑_{x∈|I|∩|D|} n_{x}x = I_{0} + ∑_{x∈|I|∩|D|} I_{x} with |I_{0}| ∩ |D| = ∅ and n_{x} > 0 for x ∈ |I| ∩ |D|. Then a level-I-structure on L is a tuple (β_{0}, (μ_{x}, β_{x})_{x∈|I|∩|D|}) consisting of a level-I_{0}- structure β_{0} and level-I_{x}-structures (μ_{x}, β_{x}) for all x ∈ |I| ∩ |D|. This yields stacks SE_{A,B,I}^{D}, SE_{A,B,I,n}^{D} equipped with forgetful morphisms

SE_{A,B,I}^{D} → SE_{A,B}^{D}, SE_{A,B,I}^{D} → Spec k(x)_{e_{x}(A)} for all x ∈ |I| ∩ |D|

(the latter lifts the morphism SE_{A,B}^{D} → Spec k(x)).

Modular automorphisms. Let T: = {x ∈ |D| | m_{x} > 0}. If |I| ∩ T = ∅ then there is a canonical left Pic_{I}(A)-action on SE_{A,B,I}^{D} lifting the Pic(A)- action on SE_{A,B}^{D}. We want to extend this to a natural left action of an idele class group C_{I}(A^{T} ×B_{T}) on SE_{A,B,I}^{D} for arbitrary I (similarly to the right action of C_{I}(A^{∞} × D_{∞})-action on Eℓℓ_{A,I^{∞}}^{∞} defined in 4.2). Write I = I^{T} + I_{T} with |I^{T} | ∩ T = ∅ and |I_{T} | ⊆ T. Put

U_{I}(A^{T} × B_{T}): = Ker( ∏_{x∈|X|−T} A_{x}^{∗} × ∏_{x∈T} B_{x}^{∗} → Γ(I^{T} , A_{I^{T}})^{∗} × Γ(I_{T} , B_{I_{T}})^{∗})

and define

C_{I}(A^{T} × B_{T}): = ∏′_{x∈|X|−T} N(A_{x}) × ∏_{x∈T} N(B_{x})/U_{I}(A^{T} × B_{T})F^{∗}.

There is a canonical epimorphism C_{I}(A^{T} × B_{T}) ⟶ Pic(A) given on the class [g] represented by g = ({a_{x}}_{x∈|X|−T} , {b_{x}}_{x∈T}) ∈ ∏′_{x∈|X|−T} N(A_{x})) × ∏_{x∈T} N(B_{x}) by A(div(g)) where

div(g) = ∑_{x∈|X|−T} v_{A_{x}}(a_{x})x + ∑_{x∈T} v_{B_{x}}(a_{x})x

The kernel of the composition C_{I}(A^{T} × B_{T}) → Pic(A) ⟶^{deg_{A}} ℚ will be denoted by C_{I}(A^{T} × B_{T})_{0}. Let g = (a^{T} , b_{T}) = ({a_{x}}_{x∉T} , {b_{x}}_{x∈T}) ∈ ∏′_{x∈|X|−T} N(A_{x})) × ∏_{x∈T} N(B_{x}) and L = (L, β_{0}, (μ_{x}, β_{x})_{x∈|I|∩|D|}) ∈ SE_{A,B,I}^{D}(S). Left multiplication by a^{T} on the target of β_{0} and β_{x} for x ∈ |I^{T} | (respectively by b_{T} on the target of β_{x} for x ∈ |I_{T} |) yields a level-I-structure on A(div(g)) ⊗ L. This defines a left action of ∏′_{x∈|X|−T} N(A_{x})) × ∏_{x∈T} N(B_{x}) on SE_{A,B,I}^{D} which factors through C_{I}(A^{T} × B_{T}).

4.17. Similar to (22), for x ∈ |D| there exists a canonical lift of (25) to an automorphism θ_{x} : SE_{A,B,I}^{D} → SE_{A,B,I}^{D} having the following properties:

(i) The following diagram commutes

SE_{A,B,I}^{D} ⟶^{θ_{x}} SE_{A,B,I}^{D}
↓  ↓
Spec k(x)_{∗} ⟶^{Frob_{q}} Spec k(x)_{∗}

where ∗ = 1 or ∗ = e_{x}(A) depending on whether x ∉ |I| or x ∈ |I|.

(ii) For n ∈ \frac{1}{d}ℤ we have θ_{x}(SE_{A,B,I,n}^{D}) = SE_{A,B,I,n−m_{x}}^{D}.

(iii) The automorphisms θ_{x} for different x ∈ |D| commute with each other and with the C_{I}(A^{T} × B_{T})-action.

(iv) For x ∈ |D| there exists ξ_{x} ∈ C_{I}(A^{T} × B_{T}) such that θ_{x}^{deg(x)}(L) = ξ_{x}L (resp. θ_{x}^{e_{x}(A) deg(x)}(L) = ξ_{x}L) for all L ∈ SE_{A,B,I}^{D}(S).

(v) If |D| = {x_{1}, …, x_{m}} put Θ_{D}: = θ_{x_{1}} ∘…∘θ_{x_{m}}. Then Θ_{D} = Frob_{SE}_{A,B,I}^{D}.

Let G be the group of automorphism of SE_{A,B,I}^{D} generated by C_{I}(A^{T} × B_{T}) and the set \{θ_{x} | x ∈ |D|\}. For g ∈ G the degree m ∈ \frac{1}{d}ℤ of g is defined by gSE_{A,B,I,n}^{D} = SE_{A,B,I,n+m}^{D} for all n ∈ \frac{1}{d}ℤ. Let G_{0} be the subgroup of elements of degree 0. Since the degree of θ_{x}, x ∈ |D| is −m_{x} we have Θ_{D} ∈ G_{0}.

Definition 4.18. Suppose that SE_{A,B,I,0}^{D} ≠ ∅. We define W(A, B, I, D) to be the group of automorphisms of SE_{A,B,I,0}^{D} of the form g|_{SE}_{A,B,I,0}^{D} for g ∈ G_{0}.

Remark 4.19. Assume that SE_{A,B,I,0}^{D} ≠ ∅. For all x ∈ |D| there exists canonical homomorphisms W(A, B, I, D) → G(k(x)_{∗}/𝔽_{q}) where ∗ = ∅ or ∗ = e_{x}(A) depending on whether x ∉ |I| or x ∈ |I|. It is surjective since Θ_{D} is mapped to Frob_{k(x)_{∗}} by property (v) above. The kernel of the homomorphism

W(A, B, I, D) ⟶ ∏_{x∈|D|−|I|} G(k(x)/𝔽_{q}) × ∏_{x∈|D|∩|I|} G(k(x)_{e_{x}(A)})/𝔽_{q}) (28)

is C_{I}(A^{T} × B_{T})_{0}. It is easy to see that (28) is surjective provided that δ(A) = d (i.e. deg_{A} : Pic(A) → \frac{1}{d}ℤ is surjective). Under this condition W(A, B, I, D) can be defined in a similar way as 4.7, i.e. as the subgroup of deg_{A} = 0 elements in the abstract group C_{I}(A^{T} × B_{T})[θ_{x}, x ∈ |D|] generated by C_{I}(A^{T} × B_{T}) and a set of central element {θ_{x} | x ∈ |D|} with deg_{A}(θ_{x}) = −m_{x} and such that the relations (iv) above hold.

Tensor product and Inverse. There is also a notion of a tensor product of invertible Frobenius bimodules and of an inverse. These constructions are needed in the proof of Proposition 4.20 below. Let C be a third locally principal O_{X}-order of rank d^{2} with \underline{Disc}(C) = \underline{Disc}(A). Let D_{1} = ∑_{x∈|X|} m_{x}^{(1)}x, D_{2} =

∑_{x∈|X|} m_{x}^{(2)}x ∈ Div(A) with ∑_{x∈|X|} m_{x}^{(i)} = 0 for i = 1, 2. Let Y = Spec 𝔽_{q} if |D_{1}| ∩ |D_{2}| = ∅ or Y = Spec(⨂_{x∈|D_{1}|∩|D_{2}|} k(x)) otherwise. We view SE_{A,B}^{D_{1}} and SE_{B,C}^{D_{2}} as stacks over Y. Let S ∈ Sch /Y and let L = (L, (x_{S})_{x∈|D|}, Φ) ∈ SE_{A,B}^{D_{1}}(S), M = (M, (x_{S})_{x∈|D_{2}|}, Ψ) ∈ SE_{B,C}^{D_{2}}(S) (hence for x ∈ |D_{1}| ∩ |D_{2}|, the morphisms x_{S} for L and M agree and are equal to the canonical morphism S → Spec k(x) → X). Define

L ⊗ M = (L ⊗ M, (x_{S})_{x∈|D_{1}+D_{2}|}, Φ ⊗_{B} Ψ) ∈ SE_{A,C}^{D_{1}+D_{2}}(S).

Thus we get a morphism of stacks

⊗ : SE_{A,B}^{D_{1}} ×_{Y} SE_{B,C}^{D_{2}} ⟶ SE_{A,C}^{D_{1}+D_{2}}  (29)

which is compatible with degrees. The inverse L^{−1} of L = (L, (x_{S})_{x∈|D|}, Φ) ∈ SE_{A,B}^{D}(S) is defined as

L^{−1} = (L^{∨}, (x_{S})_{x∈|D|}, (Φ^{∨})^{−1}) ∈ SE_{B,A}^{−D}(S).  (30)

We leave it to the reader to extend the Definition (29) and (30) to invertible Frobenius bimodules with level-I-structure (see also the next section where the tensor product of an A-elliptic sheaf with level-I-structure with a Frobenius bimodule with level-I-structure is defined).

Moduli spaces. Let D = ∑_{x∈|X|} m_{x}x ∈ Div(A), D ≠ 0 be such that ∑_{x∈|X|} m_{x} = 0 and let I ∈ Div(X) with I ≥ 0. Our aim is to prove the following result.

Proposition 4.20. (a) SE_{A,B,I}^{D} ≠ ∅ if and only if

∑_{x∈|X|} inv_{x}(B)x = ( ∑_{x∈|X|} inv_{x}(A)x) + D mod Div(X).  (31)

(b) SE_{A,B,I}^{D} is a Deligne-Mumford stack which is étale over 𝔽_{q}. The open and closed substack SE_{A,B,I,n}^{D} is finite over 𝔽_{q} for all n ∈ \frac{1}{d}ℤ.

(c) SE_{A,B,I,0}^{D} admits a coarse moduli space SE_{A,B,I}^{D}. The structural morphism SE_{A,B,I,0}^{D} → SE_{A,B,I}^{D} is an isomorphims if I ≠ 0.

(d) Suppose that SE_{A,B,I,0}^{D} ≠ ∅. Then SE_{A,B,I}^{D} → Spec 𝔽_{q} is a W(A, B, I, D)- torsor. In particular SE_{A,B,I}^{D} is a finite, étale 𝔽_{q}-scheme.

We begin with the proof of (a). Since SE_{A,B,I}^{D} is locally of finite presentation it suffices to show that SE_{A,B,I}^{D}(Spec \overline{𝔽}_{q}) ≠ ∅ if and only if (32) holds. We write \overline{X}, \overline{A} etc. for X ⊗_{𝔽_{q}} \overline{𝔽}_{q}, A ⊗ \overline{𝔽}_{q} etc. Let σ: = id_{X} ⊗ Frob_{q} : \overline{X} → \overline{X} and let π : \overline{X} → X be the projection. Define div(π) : Div(\overline{X}) ⊗ ℚ → Div(X) ⊗ ℚ by div(π)(∑_{i} n_{i}x̄_{i}) = ∑_{i} n_{i}π(x̄_{i}) (Note that deg(div(π)(D)) ≠ deg(D) in general). Part (a) of Proposition 4.20 follows from the following slighty more general result.

Lemma 4.21. Let \overline{D} ∈ Div_{0}(\overline{A}). The following conditions are equivalent:

(i) There exists an invertible \overline{A}-\overline{B}-bimodule L such that ^{τ} (L(\overline{D})) ≅ L. (ii) We have

∑_{x∈|X|} inv_{x}(B)x = ( ∑_{x∈|X|} inv_{x}(A)x) + div(π)(\overline{D}) mod Div(X).  (32)

Proof. That (i) implies (ii) can be easily deduced from the corresponding local result. To show the converse we consider first the special case div(π)(\overline{D}) ∈ Div(X), i.e. A ≃ B. Then \overline{D} can be written as a sum of divisors of the form π^{∗}(D_{1}), D_{1} ∈ Div_{0}(X) and of the form \frac{1}{e_{x}(A)}(x − σ(x)) for x ∈ |X|. Hence we can assume that either \overline{D} = \frac{1}{e_{x}(A)}(x − σ(x)) or \overline{D} = pr^{∗}(D_{1}). In the first case the assertion is obvious. In the second case it follows from the fact that the homomorphism of abelian varieties

id − Frob : Jac_{X} → Jac_{X}

is an isogeny hence faithfully flat. Returning to the general case note that by 3.1 at least \overline{A} and \overline{B} are Morita equivalent. Let L be an arbitrary invertible \overline{A}-\overline{B}-bimodule. Then ^{τ} L is also invertible hence ^{τ} (L(\overline{D}′)) ≅ L for some \overline{D}′ ∈ Div_{0}(\overline{A}). It follows that the congruence (32) holds with \overline{D}′ instead of \overline{D} as well and therefore div(π)(\overline{D} − \overline{D}′) ∈ Div(X). Hence by what we have shown above we may alter L by some element of Pic(\overline{A}) so that ^{τ} (L(\overline{D})) ≅ L. □

To prove the other assertions of 4.20 we first note that for a connected S ∈ Sch /𝔽_{q} and L ∈ SE_{A,B,I,n}^{D}(S) the group of automorphisms Aut(L) of L is = 𝔽_{q}^{∗} if I = 0 or = 1 otherwise. Hence for I > 0 the presheaf SE_{A,B,I}^{D} defined by

SE_{A,B,I,n}^{D}(S): = isomorphism classes of objects of SE_{A,B,I,0}^{D}(S)

is a fppf sheaf and the canonical morphism SE_{A,B,I,n}^{D} ⟶ SE_{A,B,I,n}^{D} is an isomorphism. We put SE_{A,B,I}^{D} = SE_{A,B,I,0}^{D}. For I > 0, 4.20 (c), (d) follows from:

Lemma 4.22. Suppose that I ≠ 0 and SE_{A,B,I,0}^{D} ≠ ∅. Then SE_{A,B,I}^{D} is a W(A, B, I, D)-torsor.

Proof. Assume first that D = 0, A = B. It follows from ([Laf], I.3, Théorème 2) that the map

(f : S → Spec 𝔽_{q}) ↦ f^{∗} : Pic_{I}(A) → SE_{A,A,I}^{0}(S)

yields an isomorphism between SE_{A,A,I}^{0} and the trivial Pic_{I}(A)-torsor over 𝔽_{q}. In particular SE_{A,A,I}^{0} is isomorphic to the trivial Pic_{I,0}(A)-torsor. Now let D ≠ 0. To simplify the notation we assume |D| ∩ |I| = ∅ so that C_{I}(A^{T} × B_{T})_{0} ≅ Pic_{I,0}(A) (the proof in the general case is analogous). Let S ∈ Sch /𝔽_{q} be connected and let L_{1}, L_{2} ∈ SE_{A,B,I}^{D}(S). If L_{1} and L_{2} have the same poles then ξ = L_{2} ⊗ L_{1}^{−1} ∈ SE_{A,A,I}^{0}(S) ≅ Pic_{I,0}(A) by the remark above, hence ξL_{1} = L_{2}. In general there exists suitable r_{x} ∈ ℤ such that L_{2} and (∏_{x∈|D|} θ_{x}^{r_{x}})(L_{1}) have the same poles, hence ξ(∏_{x∈|D|} θ_{x}^{r_{x}})(L_{1}) = L_{2} for some ξ ∈ Pic_{I}(A). Thus wL_{1} = L_{2} for w = ξ(∏_{x∈|D|} θ_{x}^{r_{x}}) ∈ G_{0}. Let w ∈ W(A, B, I, D), L ∈ SE_{A,B,I}^{D}(S) such that wL = L. Write w = ξ ∏_{x∈|D|} θ_{x}^{r_{x}} with ξ ∈ Pic_{I}(A) and r_{x} ∈ ℤ. By 4.17 (ii), for x ∈ |D| and the pole x_{S} of wL = L we have x_{S} ∘ Frob_{S}^{r_{x}} = x_{S}, hence deg(x) | r_{x}. By 4.17 (iv) it follows that w ∈ Pic_{I,0}(A). However wL = L implies that w corresponds to (wL) ⊗ L^{−1} = L ⊗ L^{−1} ∈ SE_{A,A,I,0}^{0}(S) under the canonical bijection Pic_{I}(A) ≅ SE_{A,A,I,0}^{0}(S), i.e. w = 1. This proves that for a connected S ∈ Sch /𝔽_{q}, SE_{A,B,I}^{D}(S) is either empty or W(A, B, I, D) acts simply transitively on it. To finish the proof we have to show that SE_{A,B,I,0}^{D} ≠ ∅ implies that SE_{A,B,I}^{D}(Spec \overline{𝔽}_{q}) ≠ ∅. This is a consequence of the fact that SE_{A,B,I,0}^{D} is locally of finite presentation. □

Similarly one shows that SE_{A,B,I,n}^{D} is a Im(G_{0} → Aut(SE_{A,B,I,n}^{D}))-torsor for all n ∈ \frac{1}{d}ℤ. In particular each SE_{A,B,I,n}^{D} is a finite étale 𝔽_{q}-scheme. This proves (b) for I ≠ 0. It remains to consider the case I = 0. Choose an auxiliary level J ∈ Div(X) with J > 0 and |D| ∩ |J| = ∅. A similar argument as in 4.3 shows that

SE_{A,B,n}^{D} ≅ Γ(J, A_{J})^{∗}\ SE_{A,B,J,n}^{D}

Hence SE_{A,B}^{D} is a Deligne-Mumford stack. Moreover as in 4.3 one shows that the quotient SE_{A,B}^{D}: = (Γ(J, A_{J})^{∗}/𝔽_{q}^{∗})\ SE_{A,B,J}^{D} is a coarse moduli scheme of SE_{A,B,0}^{D} and that SE_{A,B,0}^{D} ≅ 𝔽_{q}^{∗}\ SE_{A,B}^{D}. Finally since W(A, B, D) ≅ W(A, B, J, D)/(Γ(J, A_{J})^{∗}/𝔽_{q}^{∗}) it follows from Lemma 4.22 that SE_{A,B}^{D} is a W(A, B, D)-torsor over 𝔽_{q}. This completes the proof of 4.20.

Remarks 4.23. (a) Let D = ∑_{x∈|X|} m_{x}x ∈ Div(A), D ≠ 0 be such that ∑_{x∈|X|} m_{x} = 0. Condition (31) is not sufficient for SE_{A,B,I,0}^{D} ≠ ∅ (compare Remark 3.6). However if additionally we have ∑_{x∈|X|} ℤm_{x} = \frac{1}{d}ℤ then by taking suitable products of θ_{x}’s we obtain elements g in the center of G of arbitrary degree m ∈ \frac{1}{d}ℤ. Thus SE_{A,B,I,m}^{D} ≠ ∅ implies SE_{A,B,I,0}^{D} ≠ ∅. We also see that an automorphism g ∈ G of degree zero is uniquely determined by its restriction to SE_{A,B,I,0}^{D}, i.e. we have W(A, B, I, D) = G_{0}.

(b) Suppose that SE_{A,B,I,0}^{D} ≠ ∅. One can describe the W(A, B, I, D)-torsor SE_{A,B,I}^{D} /𝔽_{q} explicitely as follows. For L ∈ SE_{A,B,I,0}^{D}(\overline{𝔽}_{q}) let

ψ_{L} : W(A, B, I, D) × Spec \overline{𝔽}_{q} = ∐_{w∈W(A,B,I,D)} Spec \overline{𝔽}_{q} → SE_{A,B,I}^{D}

be given on the w-component by the morphism corresponding to wL. By 4.17 (v) the diagram

W(A, B, I, D) × Spec \overline{𝔽}_{q} ⟶^{ψ_{L}×id} SE_{A,B,I}^{D} × Spec \overline{𝔽}_{q}
↓^{Θ_{D}^{−1}×Frob_{q}}  ↓^{id × Frob_{q}}
W(A, B, I, D) × Spec \overline{𝔽}_{q} ⟶^{ψ_{L}×id} SE_{A,B,I}^{D} × Spec \overline{𝔽}_{q}

commutes. Thus ψ_{L} induces an isomorphism

SE_{A,B,I}^{D} ≅ (W(A, B, I, D) × Spec \overline{𝔽}_{q})/ < Θ_{D}^{−1} × Frob_{q} >

4.5 Twists of moduli spaces of A-elliptic sheaves

In this section A denotes a locally principal O_{X}-order of rank d^{2} with generic fiber A such that e_{∞}(A) = d. We also assume that there is a second closed point 𝔭 ≠ ∞ such that e_{𝔭}(A) = d and we put D: = \frac{1}{d}∞ − \frac{1}{d}𝔭. Let B be a locally principal O_{X}-order of rank d^{2} with \underline{Disc}(B) = \underline{Disc}(A) and such that for the generic fiber B of B we have

∑_{x∈|X|} inv_{x}(B)x = ( ∑_{x∈|X|} inv_{x}(A)x) + D mod Div(X).

In order to show that the moduli spaces Ell_{A,I}^{∞} and Ell_{B,I}^{𝔭} are twists of each other we are going to define a canonical tensor product Eℓℓ_{A,I}^{∞} × SE_{A,B,I}^{D} → Eℓℓ_{B,I}^{𝔭}.

We introduce more notation. Recall that the notion of level structure at ∞ for objects of Eℓℓ_{A,I}^{∞} and SE_{A,B,I}^{D} and at 𝔭 for objects of Eℓℓ_{B,I}^{𝔭} and SE_{A,B,I}^{D} depend on the choice of certain local Frobenius bimodules. In order to define the tensor product (33) below these choices have to be compatibly matched. For ∞ let M = (M_{∞}, ϕ_{∞}) be an invertible ϕ-B_{∞}-A_{∞}-bimodule of slope −\frac{1}{d}. For 𝔭 we choose an invertible ϕ-A_{𝔭}-B_{𝔭}-bimodule N = (N_{𝔭}, ϕ_{𝔭}) also of slope −\frac{1}{d}. We use M to define level structure at ∞ and N to define level structure at 𝔭. By Remark 2.22 there exists prime elements ϖ_{∞} ∈ O_{∞} and ϖ_{𝔭} ∈ O_{𝔭} such that

ϕ_{∞}^{d deg(∞)} = ϖ_{∞}, ϕ_{𝔭}^{d deg(𝔭)} = ϖ_{𝔭}.

Now fix a level I ∈ Div(X), I ≥ 0. We put

k(∞)_{⋆} = \{ \begin{matrix}k(∞) & if ∞ ∉ |I|, \\ k(∞)_{d} & if ∞ ∈ |I|, \end{matrix} k(𝔭)_{♯} = \{ \begin{matrix}k(𝔭) & if 𝔭 ∉ |I|, \\ k(𝔭)_{d} & if 𝔭 ∈ |I|. \end{matrix}

There exists a canonical map W(A, I, ∞) → W(A, B, I, D) induced by C_{I}(A^{∞}×B_{∞})[θ] → C_{I}(A^{∞}×B_{∞})[θ_{∞}, θ_{𝔭}] given by θ ↦ θ_{∞}^{−1}. Using the diagram with exact rows

\xymatrix{0 \ar[r] & C_{I}(A^{∞} × B_{∞})_{0} \ar[r] \ar[d] & W(A, I, ∞) \ar[r] \ar[d] & G(k(∞)_{⋆}/𝔽_{q}) \ar[d] \\ 0 \ar[r] & C_{I}(A^{∞} × B_{∞})_{0} \ar[r] & W(A, B, I, D) \ar[r] & G(k(∞)_{⋆}/𝔽_{q}) × G(k(𝔭)_{♯}/𝔽_{q})}

it is easy to see that W(A, I, ∞) → W(A, B, I, D) is injective and is equal to the kernel of the canonical projection W(A, B, I, D) → G(k(𝔭)_{♯}/𝔽_{q}) (compare 4.19). Recall that C_{I}(A^{∞} × B_{∞})_{0} ≅ Pic_{I,0}(A) if ∞ does not divide I. In the following we will consider W(A, I, ∞) as a subgroup of W(A, B, I, D). From Proposition 4.20 we deduce that SE_{A,B,I}^{D} is a W(A, I, ∞)-torsor over Spec k(𝔭)_{♯}. By 4.17 (iv) there exist ξ_{∞}, ξ_{𝔭} ∈ C_{I}(A^{∞} × B_{∞})_{0} such that

θ_{∞}^{[k(∞)_{⋆}:𝔽_{q}]} = ξ_{∞}, θ_{𝔭}^{[k(𝔭)_{♯}:𝔽_{q}]} = ξ_{𝔭}

ξ_{∞} and ξ_{𝔭} are given as follows. Let Π_{∞} ∈ B_{∞} (resp. Π_{𝔭} ∈ A_{𝔭}) be a generator of the radical of B_{∞} (resp. of A_{𝔭}). If ∞ ∉ |I| (resp. ∞ ∈ |I|) then ξ_{∞} denotes the class in C_{I}(A^{∞} × B_{∞}) of the idele ({1}_{x≠∞}, Π_{∞}^{−1}) (resp. ({1}_{x≠∞}, ϖ_{∞}^{−1})) in C_{I}(A^{∞} × B_{∞}). If 𝔭 ∉ |I| (resp. 𝔭 ∈ |I|) then ξ_{𝔭} denotes the class in C_{I}(A^{∞} × B_{∞}) of the idele ({1}_{x≠𝔭}, Π_{𝔭}) (resp. ({1}_{x≠𝔭}, ϖ_{𝔭})) in C_{I}(A^{∞} × B_{∞}). The tensor product

⊗ : Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} ⟶ Eℓℓ_{B,I}^{𝔭}, (E, L) ↦ E ⊗ L  (33)

is a morphism of Spec k(𝔭)_{⋆}-stacks having the following properties:

(i) The morphism (33) is compatible with deg_{A} and deg_{B}, i.e. for m, n ∈ \frac{1}{d}ℤ it induces a morphism

Eℓℓ_{A,I,m}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I,n}^{D} ⟶ Eℓℓ_{B,I,m+n}^{𝔭}.

(ii) The morphism of stacks

Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} ⟶ Eℓℓ_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯}} SE_{A,B,I}^{D}, (E, L) ↦ (E ⊗ L, L)

is an isomorphism with quasi-inverse

Eℓℓ_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯}} SE_{A,B,I}^{D} ⟶ Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D}, (E, L) ↦ (E ⊗ L^{−1}, L).

(iii) The following diagram commutes

\xymatrix{Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar[dd]^{θ⊗θ_{∞}} \ar@{-->}[dr]^{⊗} \\  & _{Eℓℓ{∞ \atop B,I}} \\ Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar@{-->}[ur]^{⊗}}

(iv) For ξ ∈ C_{I}(A^{∞} × B_{∞}) the following diagram commutes

\xymatrix{Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar[dd]^{ξ⊗ξ^{−1}} \ar@{-->}[dr]^{⊗} \\  & _{Eℓℓ{∞ \atop B,I}} \\ Eℓℓ_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar@{-->}[ur]^{⊗}}

.

To define (33) let S ∈ Sch /𝔽_{q} and let ∞_{S} : S → X, 𝔭_{S} : S → X be morphisms in Sch /𝔽_{q} which factor through ∞ → X and 𝔭 → X respectively. Let E = (E, ∞_{S}, t) be an A-elliptic sheaf over S with zero z : S → X and let L = (L, Φ) be an invertible A-B-bimodule of slope \frac{1}{d}(∞_{S} − 𝔭_{S}). Define

E ⊗ L: = (E ⊗_{A} L, 𝔭_{S}, t ⊗_{A} Φ).

Note that E ⊗_{A} L(−\frac{1}{d}𝔭_{S}) = E(−\frac{1}{d}∞_{S}) ⊗_{A} L(\frac{1}{d}(∞_{S} − 𝔭_{S})). One easily checks that t ⊗_{A} Φ is an injective B ⊠ O_{S}-linear homomorphism with Coker(t ⊗_{A} Φ) ≅ Coker(t) ⊗_{A} L. It follows from 3.8 (c) that E ⊗ L is a B-elliptic sheaf with pole 𝔭 and zero z. Thus we have defined (33) if I = 0. When considering additionally level-I-structure, it is enough to treat separately the three cases ∞, 𝔭 ∉ |I|, |I| = {∞} and |I| = {𝔭}. In the first case if E carries a level-I-structure α and L a level-I-structure β then one defines a level-I-structure α • β on E ⊗ L as in (21).

Suppose now I = n∞, n > 0 and let E = (E, ∞_{S}, t), L = (L, Φ, ∞_{S}, 𝔭_{S}) be as above. Let (α, λ), (μ, β) be level-I-structures on E and L respectively such that λ = μ : S → Spec k(∞)_{d} lifts ∞_{S}. Thus

α : (id_{I} ×λ)^{∗}(M_{I}) ⟶^{≅} E|_{I×S}, β : B_{I} ⊠ O_{S} ⟶^{≅} (id_{I} ×μ)^{∗}(M_{I}) ⊗_{A} L|_{I×S}.

Let α • β be the composition

α • β : B_{I} ⊠ O_{S} ⟶^{β} (id_{I} ×μ)^{∗}(M_{I}) ⊗_{A} L|_{I×S} ⟶^{α⊗id} (E ⊗_{A} L)|_{I×S}

Finally let I = n𝔭, n > 0 and let α and (μ, β) be level-I-structures on E and L. In this case level-I-structures on E and L are given by

α : A_{I} ⊠ O_{S} ⟶ E|_{I×S}, β : (id_{I} ×μ)^{∗}(N_{I}) ⟶ L|_{I×S}

where μ : S → Spec k(𝔭)_{d} is a lift of 𝔭_{S}. We set

α • β : (id_{I} ×μ)^{∗}(N_{I}) ⟶^{β} L|_{I×S} = A_{I} ⊗_{A} L|_{I×S} ⟶^{α⊗id} (E ⊗_{A} L)|_{I×S}.

In both cases one easily checks that α • β defines a level-I-structure on E ⊗ L. Thus we have defined (33). The straight forward but tedious verification of the properties (i)–(iv) will be left to the reader. Recall that Ell_{A,I}^{∞}, Ell_{B,I}^{𝔭} and SE_{A,B,I}^{D} denote the coarse moduli spaces of Eℓℓ_{A,I,0}^{∞}, Eℓℓ_{B,I,0}^{𝔭} and SE_{A,B,I,0}^{D} respectively (these are fine moduli spaces if I ≠ 0). By (i)–(iv), (33) induces an W(A, I, ∞)-equivariant isomorphism of 𝔽_{q}-schemes

Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} ⟶ Ell_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯}} SE_{A,B,I}^{D} .  (34)

Here the (free) action of the finite group W(A, I, ∞) on the right is given by id ⊗ξ, ξ ∈ W(A, I, ∞) ↪ W(A, B, I, D) whereas on the left it is given by ξ^{−1} ⊗ ξ. Consequently by passing to quotients under the action and using the fact that SE_{A,B,I}^{D} /k(𝔭)_{♯} is a W(A, I, ∞)-torsor we obtain:

Theorem 4.24. The isomorphism (34) induces an isomorphism of k(𝔭)_{♯}- schemes

(Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D})/W(A, I, ∞) ≅ Ell_{B,I}^{𝔭}

We shall give now another formulation of this result. Note that

Θ_{D}^{[k(𝔭)_{♯}:𝔽_{q}]} = θ_{∞}^{[k(𝔭)_{♯}:𝔽_{q}]}ξ_{𝔭} = Frob_{SE_{A,B,I}^{D} /k(𝔭)_{♯}}

In particular Θ_{D}^{−[k(𝔭)_{♯}:𝔽_{q}]} lies in W(A, I, ∞) and is equal to = θ^{[k(𝔭)_{♯}:𝔽_{q}]}ξ_{𝔭}^{−1}. The fact that (34) is in particular Θ_{D}^{[k(𝔭)_{♯}:𝔽_{q}]}-equivariant implies that the following diagram commutes

\xymatrix{Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar[r]^{(34)} \ar[d]^{θ^{[k(𝔭)_{♯}:𝔽q]}ξ_{𝔭}^{−1}⊗Frob_{SE /k(𝔭)_{♯}}} & Ell_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯}} SE_{A,B,I}^{D} \ar[d]^{id ⊗ Frob_{SE /k(𝔭)_{♯}}} \\ Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆}} SE_{A,B,I}^{D} \ar[r]^{(34)} & Ell_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯}} SE_{A,B,I}^{D}}

Documenta Mathematica · Extra Volume Suslin (2010) 595–654

Fix L ∈ SE_{A,B,I}^{D}(\overline{𝔽}_{q}). Its poles correspond to 𝔽_{q}-embeddings λ : k(∞)_{⋆} → \overline{𝔽}_{q}, μ : k(𝔭)_{♯} → \overline{𝔽}_{q}. By taking base change of the above diagram with respect to the morphism Spec \overline{𝔽}_{q} → SE_{A,B,I}^{D} corresponding to L we obtain:

Theorem 4.25. Let m = [k(𝔭)_{♯} : 𝔽_{q}]. Thus m = deg(𝔭) if 𝔭 ∉ |I| and m = d deg(𝔭) otherwise. The isomorphism · ⊗ L : Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆},λ}\overline{𝔽}_{q} → Ell_{B,I}^{𝔭} ⊗_{k(𝔭)_{♯},μ}\overline{𝔽}_{q} induces an isomorphism of k(𝔭)_{♯}-schemes

(Ell_{A,I}^{∞} ⊗_{k(∞)_{⋆},λ}\overline{𝔽}_{q})/ < θ^{m}ξ_{𝔭}^{−1} ⊗ Frob_{q}^{m} > ≅ Ell_{B,I}^{𝔭} .

Remark 4.26. A pair (λ, μ) ∈ Hom_{𝔽_{q}}(k(∞)_{⋆}, \overline{𝔽}_{q}) × Hom_{𝔽_{q}}(k(𝔭)_{♯}, \overline{𝔽}_{q}) will be called admissible for (A, B, I) if there exists L ∈ SE_{A,B,I}^{D}(\overline{𝔽}_{q}) with poles λ and μ. The surjectivity of the homomorphism W(A, B, I, D) → G(k(∞)_{⋆}/𝔽_{q}) implies that for all λ there exists a μ such that (λ, μ) is admissible.

4.6 Application to uniformization

Let A be locally principal O_{X}-order of rank d^{2} with generic fiber A such that e_{∞}(A) = d. Let I ∈ Div(X) denote an effective divisor. For a closed point x ∈ |X| − |I| we denote by Êll_{A,I}^{∞}/ Spf(O_{x}) the formal completion of Ell_{A,I}^{∞} along the fiber at x of the characteristic morphism Ell_{A,I}^{∞} → X −I. Also for an arbitrary x ∈ |X| we let Ell_{A,I}^{∞,an} /F_{x} denote the rigid analytic space associated to Ell_{A,I}^{∞} ×_{X} Spec F_{x}. There exists two types of uniformization of Ell_{A,I}^{∞}, i.e. explicite descriptions of Êll_{A,I}^{∞}/ Spf(O_{∞}) and Ell_{A,I}^{∞,an} /F_{x} as (finite unions of) certain quotients of Drinfeld’s symmetric spaces and its coverings. These are called uniformization at the pole and Cherednik-Drinfeld uniformization. The first concerns the point x = ∞ (under the assumption inv_{∞} A = 0) whereas the second the points 𝔭 ∈ |X| − \{∞\} with inv_{𝔭} A = \frac{1}{d}. By using Theorem 4.25 we show that the two types of uniformization are equivalent (see Proposition 4.28 below). In order to introduce the quotients of symmetric spaces appearing in the uniformization results below we have to introduce more notation. Fix a closed point x ∈ X. We denote by Ô_{x}^{nr} the completion of the strict henselisation of O_{x} and by F̂_{x}^{nr} its function field. For each positive integer m we denote by F_{x,m} the unramified extension of degree m of F_{x} in F̂_{x}^{nr} and let O_{x,m} be its ring of integers. Note that the projection O_{x,m} → k(x)_{m} has a canonical section, i.e. k(x)_{m} ⊆ F_{x,m}. Similarly \overline{k(x)} ⊂ Ô_{x}^{nr}. Denote by D_{x} the central division algebra over F_{x} with invariant \frac{1}{d} and let D_{x} be the maximal order in D_{x}. We also fix a uniformizer ϖ_{x} ∈ O_{x} and an element Π_{x} ∈ D_{x} with Π_{x}^{d} = ϖ_{x}. Let σ denote the automorphism on O_{x,m} and Ô_{x}^{nr} which induces the Frob_{q} on the residue fields. Let Ω_{x}^{d} be Drinfeld’s (d − 1)-dimensional symmetric space over F_{x} and Ω̂_{x}^{d}/ Spf(O_{x}) its canonical formal model (see e.g. [Ge]). The rigid analytic variety Ω_{x}^{d} parametrizes certain formal groups. The formal scheme Ω̂_{x}^{d} is equipped with a canonical GL_{d}(F_{x})-action.

We define an action of GL_{d}(F_{x}) on Ω̂_{x}^{d} ˆ⊗_{O_{x}}O_{x,m} = Ω̂_{x}^{d} ⊗_{k(x)} k(x)_{m} and Ω̂_{x}^{d} ˆ⊗_{O_{x}} Ô_{x}^{nr} = Ω̂_{x}^{d} ⊗_{k(x)} \overline{k(x)} by letting g ∈ GL_{d}(F_{x}) act canonically on Ω̂_{x}^{d} and by σ^{−v_{x}(det(g))} on O_{x,m} and O_{x} respectively. There exists a tower … Σ_{n+1,x}^{d} → Σ_{n,x}^{d} → … → Σ_{1,x}^{d} → Σ_{0,x}^{d} = Ω_{x}^{d} ⊗_{F_{x}} F_{x,d} of finite étale Galois coverings ([Ge], IV.1). Each Σ_{n,x}^{d} carries a GL_{d}(F_{x})/ϖ_{x}^{ℤ}- and D_{x}^{∗}/ϖ_{x}^{ℤ}-action and the covering maps Σ_{n+1,x}^{d} → Σ_{n,x}^{d} are equivariant. Finally for n ≥ 0 we equip Σ_{n,x}^{d} ⊗_{F_{x,d}} F̂_{x}^{nr} = Σ_{n,x}^{d} ⊗_{k(x)_{d}} \overline{k(x)} with a GL_{d}(F_{x})- and D_{x}-action by letting g ∈ GL_{d}(F_{x}) (or inD_{x}) act canonically on the first factor and by σ^{−v_{x}(Nrd(g)} on the second factor.

Rigid analytic Drinfeld-Stuhler varieties. Suppose that inv_{∞} A = 0 and fix an isomorphism A_{∞} ≅ M_{d}(F_{∞}). We write I = n∞+I^{∞} with ∞ ∉ |I^{∞}|. Assume first that n = 0. We define

Ŝh_{A,I}^{∞}: = A^{∗}\ (A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Ω̂_{∞}^{d}) .

This is formal scheme over Spf(O_{∞}). Next assume ∞ ∈ |I| and write I = n∞ + I^{∞} with ∞ ∉ |I^{∞}|. Then we define

Sh_{A,I}^{∞}: = A^{∗}\ (A^{∗}(𝔸^{∞})/U_{I^{∞}}(A^{∞}) × Σ_{n,∞}^{d}) .

This is rigid analytic space over F_{∞}. There exists a canonical right action of the group C_{I}(A^{∞} × D_{∞}) on Ŝh_{A,I}^{∞} and Sh_{A,I}^{∞} which is defined as follows. Let a = ({a_{x}}_{x≠∞}, d_{∞}) ∈ (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞}) and assume first n = 0. Then the right action

of the class [a] ∈ C_{I}(A^{∞} × D_{∞}) of a on Ŝh_{A,I}^{∞} is given by right multiplication by {a_{x}}_{x≠∞} on A^{∗}(𝔸^{∞})/U_{I}(A^{∞}). Now assume that n > 0. Then [a] acts on Sh_{A,I}^{∞} by right multiplication of {a_{x}}_{x≠∞} on A^{∗}(𝔸^{∞})/U_{I^{∞}}(A^{∞}) and letting d_{∞}^{−1} act on Σ_{n,∞}^{d}. There are canonical morphism

pole : Ŝh_{A,I}^{∞} → Spec k(∞)  if n = 0,  (35)
pole : Sh_{A,I}^{∞} → Spec k(∞)_{d}  if n > 0.  (36)

which we are going to defined now. Denote by l = l_{∞} : A^{∗}(𝔸^{∞}) → ℤ the composite

l_{∞} : A^{∗}(𝔸^{∞}) ⟶^{Nrd} F^{∗}(𝔸^{∞}) ⟶^{div} ⨁_{x≠∞} ℤx ⟶^{deg} ℤ.

Note that for a ∈ A^{∗} ⊂ A^{∗}(𝔸^{∞}) we have l_{∞}(a) = − deg(∞)v_{∞}(Nrd(a)). First assume n = 0. Let

A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Ω̂_{∞}^{d} ⟶ Spec k(∞)  (37)

be given on the component ηU_{I}(A^{∞}) × Ω̂_{∞}^{d} by

Ω̂_{∞}^{d} ⟶ Spec k(∞) ⟶^{Frob_{q}^{−l(η)}} Spec k(∞).  (38)

Clearly, (38) factors through Ŝh_{A,I}^{∞}, hence it induces (35). Now suppose n > 0. Since k(∞)_{d} ⊆ F_{∞,d}, we get a map Σ_{n,∞}^{d} → Spec F_{∞,d} → Spec k(∞)_{d}. Note that for g ∈ GL_{d}(F_{∞}) the diagram

Σ_{n,∞}^{d}  ⟶^{g·}  Σ_{n,∞}^{d}
↓  ↓
Spec k(∞)_{d} ⟶^{Frob_{q}^{−v∞(det(g))}} Spec k(∞)_{d}

(39)

commutes. We define

A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Σ_{n,∞}^{d} → Spec k(∞)_{d}  (40)

on the component corresponding to ηU_{I}(A^{∞}) ∈ A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) by

Σ_{n,∞}^{d} ⟶ Spec k(∞)_{d} ⟶^{Frob_{q}^{−l(η)}} Spec k(∞)_{d}.

The commutativity of (39) implies that (40) factors through Sh_{A,I}^{∞}, i.e. it yields the map (36).

Cherednik-Drinfeld varieties. Let ξ̃ = {ξ̃_{x}}_{x≠∞} ∈ (∏′_{x∈|X|−{∞}}N(A_{x}))

and let ξ ∈ C_{I}(A^{∞} × D_{∞}) be the idele class represented by ({ξ̃_{x}}_{x≠∞}, 1) ∈ (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞}). We assume that ξ is a central element in

C_{I}(A^{∞} × D_{∞}) and that m = −d deg_{A}(ξ) = −l_{∞}(ξ̃) ≠ 0. We define

Ŝh_{A,I,∞}^{ξ}: = A^{∗}\ (A^{∗}(𝔸^{∞})/U_{I}(A^{∞})ξ̃^{ℤ} × Ω̂_{∞}^{d} ⊗_{k(∞)} \overline{k(∞)})  if n = 0,
Sh_{A,I,∞}^{ξ}: = A^{∗}\ (A^{∗}(𝔸^{∞})/U_{I^{∞}}(A^{∞})ξ̃^{ℤ} × Σ_{n,∞}^{d} ⊗_{k(∞)_{d}} \overline{k(∞)}) if n > 0.

As above one defines a right action of C_{I}(A^{∞} × D_{∞}) on Ŝh_{A,I,∞}^{ξ} and Sh_{A,I,∞}^{ξ} by letting a = ({a_{x}}_{x≠∞}, d_{∞}) ∈ (∏′_{x∈|X|−{∞}} N(A_{x})) × N(D_{∞}) act by right multiplication by {a_{x}}_{x≠∞} on A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) and letting d_{∞}^{−1} act on Ω̂_{∞}^{d} ˆ⊗_{O_{∞}} Ô_{∞}^{nr} (if n = 0) and Σ_{n,∞}^{d} ⊗_{F_{∞,d}} F̂_{∞}^{nr} (if n > 0). Note that ξ acts trivially. Let k(ξ) denote the fixed field of Frob_{q}^{m} in \overline{k(∞)}. There are canonical morphisms

Ŝh_{A,I,∞}^{ξ} → Spec k(ξ)  if n = 0,  (41)
Sh_{A,I,∞}^{ξ} → Spec k(ξ)  if n > 0.  (42)

Their definition is similar to the definition of (35) and (35). For example (41) is induced by the maps

η(U_{I}(A^{∞})ξ̃^{ℤ}) × Ω̂_{∞}^{d} ⊗_{k(∞)} \overline{k(∞)} ⟶ Spec k(ξ) ⟶^{Frob_{q}^{−l(η)}} Spec k(ξ).

The rigid analytic varieties Sh_{A,I}^{∞} and Sh_{A,I,∞}^{ξ} are twists of each other. More precisely we have the following result.

Lemma 4.27. (a) There exists a canonical isomorphism of formal schemes over Spf(O_{∞}) (resp. rigid analytic varieties over F_{∞})

Ŝh_{A,I}^{∞} ⊗_{k(∞)}\overline{k(∞)}/ < ξ ⊗ Frob_{q}^{m} > ≅ Ŝh_{A,I,∞}^{ξ} for n = 0, resp. (43)
Sh_{A,I}^{∞} ⊗_{k(∞)_{d}} \overline{k(∞)}/ < ξ ⊗ Frob_{q}^{m} > ≅ Sh_{A,I,∞}^{ξ}  for n > 0. (44)

Here Ŝh_{A,I}^{∞} ⊗_{k(∞)}\overline{k(∞)} (resp. Sh_{A,I}^{∞} ⊗_{k(∞)_{d}} \overline{k(∞)_{d}}) denotes the base change to \overline{k(∞)} of the morphism (35) (resp. (36)).

(b) Let ξ_{∞} ∈ C_{I}(A^{∞} × D_{∞}) be the class of the idele ({1}_{x≠∞}, Π_{∞}^{−1}) (if n = 0) resp. of ({1}_{x≠∞}, ϖ_{∞}^{−1}) (if n > 0). Then we have

Ŝh_{A,I,∞}^{ξ} ⊗_{k(ξ)}\overline{k(ξ)}/ < ξ_{∞} ⊗ Frob_{q}^{deg(∞)} > ≅ Ŝh_{A,I}^{∞} for n = 0, resp. (45)
Sh_{A,I,∞}^{ξ} ⊗_{k(ξ)}\overline{k(ξ)}/ < ξ_{∞} ⊗ Frob_{q}^{d deg(∞)} > ≅ Sh_{A,I}^{∞}  for n > 0. (46)

Proof. We prove only the existence of (43). The other cases are similar and will be left to the reader. For ηU_{I}(A^{∞}) ∈ A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) we denote the base change of the map (38) to \overline{k(∞)} by (ηU_{I}(A^{∞}) × Ω̂_{∞}^{d}) ⊗_{k(∞)} \overline{k(∞)}. Let

(ηU_{I}(A^{∞}) × Ω̂_{∞}^{d}) ⊗_{k(∞)} \overline{k(∞)} ⟶^{id ⊗ Frob_{q}^{l(η)}} Ω̂_{∞}^{d} ⊗_{k(∞)} \overline{k(∞)}  (47)

and let

(A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Ω̂_{∞}^{d})⊗_{k(∞)}\overline{k(∞)} → A^{∗}(𝔸^{∞})/U_{I}(A^{∞})×(Ω̂_{∞}^{d}⊗_{k(∞)}\overline{k(∞)})

(48)

be made up of all the morphisms (45). One easily checks that it is A^{∗}- equivariant and that the following diagram commutes:

(A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Ω̂_{∞}^{d}) ⊗ \overline{k(∞)} ⟶^{(48)} A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × (Ω̂_{∞}^{d} ⊗ \overline{k(∞)})
↓^{·ξ⊗Frob_{q}^{m}}  ↓_{·ξ̃⊗id}
(A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × Ω̂_{∞}^{d}) ⊗ \overline{k(∞)} ⟶^{(48)} A^{∗}(𝔸^{∞})/U_{I}(A^{∞}) × (Ω̂_{∞}^{d} ⊗ \overline{k(∞)})

Hence (48) induces the isomorphism (43). □

Note that, since ξ_{∞} acts trivially on Ŝh_{A,I}^{∞} (resp. Sh_{A,I}^{∞}), the C_{I}(A^{∞} × D_{∞})- action on Ŝh_{A,I}^{∞} (resp. Sh_{A,I}^{∞}) induces a right W(A, I, ∞)-action (by Remark 4.8 (c)). In terms of the latter, Lemma 4.27 (a) can be reformulated as follows:

Ŝh_{A,I}^{∞} ⊗_{k(∞)}\overline{k(∞)}/ < ξθ^{m} ⊗ Frob_{q}^{m} > ≅ Ŝh_{A,I,∞}^{ξ} for n = 0, resp. (49)
Sh_{A,I}^{∞} ⊗_{k(∞)_{d}} \overline{k(∞)}/ < ξθ^{m} ⊗ Frob_{q}^{m} > ≅ Sh_{A,I,∞}^{ξ}  for n > 0. (50)

Uniformisation at the pole. Suppose that inv_{∞} A = 0 and assume first that ∞ does not divide the level I. Then there exists an isomorphism of formal schemes over Spf(O_{∞})

Êll_{A,I}^{∞}/ Spf(O_{∞}) ≅ Ŝh_{A,I}^{∞}  (51)

which is compatible with the W(A, I, ∞)-action and the morphisms pole. Now assume ∞ ∈ |I|. Then it is expected

Ell_{A,I}^{∞,an} / Spec F_{∞} ≅ Sh_{A,I}^{∞} .  (52)

Again, (52) should be compatible with the W(A, I, ∞)-action and the morphisms pole. We say that Ell_{A,I}^{∞} admits uniformization at the pole if (51) (resp. (52)) holds. Suppose that ∞ ∉ |I|. (51) has been proved in ([BS], 4.4) if A is a division algebra or A = M_{d}(F). As in loc. cit. the general case can be easily deduced from ([St], Corollary, p. 531 and Theorem 1, p. 538) or ([Ge], III.3.1.1). If ∞ ∈ |I| then (52) in known in the case of Drinfeld modular varieties (i.e. A = M_{d}(F)) the uniformization (52) is proved in [Dr4].

Cherednik-Drinfeld uniformization: Let 𝔭 ∈ |X| − {∞} and assume that inv_{𝔭} A = \frac{1}{d}. Let B be a locally principal O_{X}-order of rank d^{2} with \underline{Disc}(B) = \underline{Disc}(A) and such that the local invariants of the generic fiber B of B are given by inv_{∞}(B) = inv_{∞}(A) + \frac{1}{d}, inv_{𝔭}(B) = 0 and inv_{x}(B) = inv_{x}(A) for all x ∈ |X| − {∞, 𝔭}. We fix an isomorphism B_{𝔭} ≅ M_{d}(F_{𝔭}) and isomorphisms B_{x} ≅ A_{x} for all x ∈ |X| − {∞, 𝔭}. Using the latter we can identify

the groups C_{I}(A^{∞} × B_{∞}) and C_{I}(B^{𝔭} × A_{𝔭}). Since ξ_{∞} acts trivially on Ŝh_{B,I,𝔭}^{ξ_{∞}} (resp. Sh_{B,I,𝔭}^{ξ_{∞}}) we obtain a right W(A, I, ∞) ≅ C_{I}(B^{𝔭} × A_{𝔭})/ξ_{∞}^{ℤ}-action on

Ŝh_{B,I,𝔭}^{ξ_{∞}} (resp. Sh_{B,I,𝔭}^{ξ_{∞}}). We also fix an isomorphism k(∞)_{⋆} ≅ k(ξ_{∞}) ⊂ \overline{k(𝔭)} such that the pair (k(∞)_{⋆} ≅ k(ξ_{∞}) ↪ \overline{k(𝔭)}, k(𝔭)_{♯} ↪ \overline{k(𝔭)}) is admissible in the sense of Remark 4.26 (k(∞)_{⋆} and k(𝔭)_{♯} are defined as in the last section) and define

pole : Ŝh_{A,I,∞}^{ξ_{∞}} ⟶^{(41)} Spec k(ξ_{∞}) ≅ Spec k(∞)_{⋆}  if n = 0,
pole : Sh_{A,I,∞}^{ξ_{∞}} ⟶^{(42)} Spec k(ξ_{∞}) ≅ Spec k(∞)_{⋆}  if n > 0.

Assume that 𝔭 ∉ |I| (resp. 𝔭 ∈ |I|). Then we expect that there is a canonical isomorphism of formal schemes over Spf(O_{𝔭}) (resp. of rigid analytic spaces over F_{𝔭})

Êll_{A,I}^{∞}/ Spf(O_{𝔭}) ≅ Ŝh_{B,I,𝔭}^{ξ_{∞}}  if 𝔭 ∉ |I|,  (53)
Ell_{A,I}^{∞,an} /F_{𝔭} ≅ Sh_{B,I,∞}^{ξ_{∞}}  if 𝔭 ∈ |I|  (54)

compatible with W(A, I, ∞)-action and the morphisms pole. We say that Ell_{A,I}^{∞} admits Cherednik-Drinfeld uniformization if (53) (resp. (54)) holds. Both (53) and (54) are proved in ([Hau], 8.1 and 8.3) in the case deg(∞) = 1, inv_{∞} A = 0 and ∞ ∉ |I|. Under these assumptions the

formal scheme Ŝh_{B,I,𝔭}^{ξ_{∞}} and the rigid analytic variety Sh_{B,I,∞}^{ξ_{∞}} have the following simpler description

Ŝh_{B,I,𝔭}^{ξ_{∞}} = B^{∗}\ (B^{∗}(𝔸^{𝔭,∞})/U_{I}(B^{∞,𝔭}) × Ω̂_{𝔭}^{d} ⊗_{k(𝔭)} \overline{k(𝔭)})  if n = 0,
Sh_{B,I,𝔭}^{ξ_{∞}} = B^{∗}\ (B^{∗}(𝔸^{𝔭,∞})/U_{I}(B^{∞,𝔭}) × Σ_{n,𝔭}^{d} ⊗_{k(𝔭)_{d}} \overline{k(𝔭)}) if n > 0,

where n denotes now the exact multiple of 𝔭 occuring in I. By combining Theorem 4.25, (49), (50) and Lemma 4.27 (b) we obtain:

Proposition 4.28. Let 𝔭 ∈ |X| − {∞} and let A and B be locally principal O_{X}-orders of rank d^{2} with \underline{Disc}(B) = \underline{Disc}(A) such that the local invariants of the generic fibers A and B are given by inv_{∞}(A) = 0, inv_{∞}(B) = \frac{1}{d}, inv_{𝔭}(A) = \frac{1}{d}, inv_{𝔭}(B) = 0 and inv_{x}(B) = inv_{x}(A) for all x ∈ |X| − \{∞, 𝔭\}. The following conditions are equivalent:

(i) Ell_{A,I}^{∞} admits uniformization at the pole. (ii) Ell_{B,I}^{𝔭} admits Cherednik-Drinfeld uniformization.

By applying 4.28 to the results of [BS] and [Hau] we obtain further cases where Ell_{A,I}^{∞} admits uniformization at the pole or Cherednik-Drinfeld uniformization. For example if inv_{∞}(A) = 0, ∞ ∈ |I| and if there exists a point 𝔭 ∈ |X| − {∞} such that inv_{𝔭}(A) = \frac{1}{d} and deg(𝔭) = \frac{1}{d} then Ell_{A,I}^{∞} admits uniformization at the pole. Conversely Cherednik-Drinfeld uniformization for Ell_{A,I}^{∞} holds whenever if 𝔭 does not divide the level.

5 Appendix

5.1 Commutative subalgebras in semisimple algebras

Let k be a perfect field and A a finite-dimensional semisimple k-algebra. We collect a few facts about maximal separable and commutative subalgebras of A for which we could not find any references. Let Z denote the center of A. By Wedderburns Theorem we have Z ≅ k_{1}×…× k_{r} for some finite separable extensions k_{i}/k. For a finite Z-module M, rank_{Z} M denotes the (not necessarily constant) rank of the corresponding locally free O_{Spec Z}-module.

Lemma 5.1. Let T be a commutative separable k-subalgebra of A. The following conditions are equivalent.

(i) T = Z_{A}(T) = {x ∈ A | tx = xt ∀t ∈ T }. (ii) T is a maximal commutative separable k-subalgebra of A. (iii) T ⊇ Z and (rank_{Z} T)^{2} = rank_{Z} A. Moreover if A = End_{k_{1}}(V_{1}) × … × End_{k_{r}}(V_{r}) where V_{i} a finite-dimensional k_{i}-vector space for i = 1, …, r, then (i) – (iii) are equivalent to (iv) V_{1} ⊕ … ⊕ V_{r} is a free T -module of rank 1.

A commutative separable k-subalgebra T of A satisfying the equivalent conditions (i) – (iii) above will be called a maximal torus of A.

Lemma 5.2. Let T_{1}, T_{2} be two maximal tori of A. Then there exists a finite extension k′/k such that T_{1} ⊗_{k} k′ and T_{2} ⊗_{k} k′ are conjugated in A ⊗_{k} k′.

A finite A-module M is called a generator of Mod_{A} if the functor

Hom_{A}(M, ·) : Mod_{A} ⟶ Mod_{k}

is faithfull. M is called a minimal generator if dim_{k}(M) is minimal. Assume now that A is split, i.e. A = End_{k_{1}}(V_{1}) × … × End_{k_{r}}(V_{r}) as in condition (iv) of Lemma 5.1 and let T be a maximal torus in A. We have

Lemma 5.3. Let M be a finite A-module. The following conditions are equivalent.

(i) M is a minimal generator. (ii) M ≅ V_{1} ⊕ … ⊕ V_{r} (iii) M is a free T -module of rank 1.

5.2 A-elliptic sheaves according to Laumon-Rapoport-Stuhler

The aim of this section is to show that under suitable assumptions on A the moduli stack Eℓℓ_{A}^{∞} defined in section 4.1 is isomorphic to the stack defined in ([LRS], 2.4). Firstly, we establish an equivalence between certain parabolic vector bundles and locally free modules of a hereditary algebra. We use the following notations and assumptions. Let k be a perfect field of cohomological dimension ≤ 1 and let X be a smooth connected curve over k and F is the function field of X. We also fix a closed point ∞ ∈ X. To simplify the notation we assume that deg(∞) = 1 (see Remark 5.11 below for the case deg(∞) > 1). Let A′ be a locally principal O_{X}-order of rank d^{2} with generic fiber A′. We assume that e_{∞}(A′) = 1, i.e. A′_{∞} ≅ M_{d}(O_{∞}). To begin with we introduce the notion of a parabolic A′-modules and parabolic vector bundles with A′-action (compare [Yo]). A filtered object in a category C is a functor C_{⋆} : ℤ → C. Morphisms of filtered objects are natural transformations. Here we regard the ordered set ℤ as a category in the usual way. The set of objects is ℤ and for i, j ∈ ℤ we have

♯(Mor(i, j)) = \{ \begin{matrix}1 & if i ≤ j \\ 0 & otherwise. \end{matrix}

For i ∈ ℤ the morphism C_{i} → C_{i+1} will be denoted by j_{i} = j_{i}^{C}. For a filtered object C_{⋆} in C and n ∈ ℤ the shifted filtered object C[n]_{⋆} is defined as the composite ℤ ⟶^{+n} ℤ → C. A morphism ϕ : C_{⋆} → D_{⋆} of filtered objects induces a morphism ϕ[n] : C[n]_{⋆} → D[n]_{⋆}. Recall that for S ∈ Sch /k we have set _{A′} Mod(S): = _{A′⊠O_{S}} Mod (resp. Mod_{A′}(S): = Mod_{A′⊠O_{S}}).

Definition 5.4. Let S be a k-scheme.

(a) For e ∈ ℤ with e ≥ 1 let PMod_{A′,e}(S) denote the category of pairs (F_{⋆}, ψ_{⋆}) consisting of a filtered Mod_{A′}(S)-object F_{⋆} and an isomorphism ψ_{⋆} : F[e]_{⋆} → F_{⋆}(∞): = F_{⋆} ⊗_{O_{X×S}} (O_{X}(∞) ⊠ O_{S}) such that the restriction of j_{i} : F → F_{i+1} to X − {∞} × S is an isomorphism and such that the following diagram commutes

\xymatrix{ & F_{i+e} \ar[dd]_{ψ_{i}} \\ F_{i} \ar[ur]^{j_{i+e−1}∘...∘j_{i}} \ar[dr]_{id ⊗ι} \\  & F_{i}(∞)}  (55)

where ι : O_{X×S} ↪ O_{X}(∞) ⊠ O_{S} is the inclusion. Morphisms in PMod_{A′,e}(S) are morphisms of filtered objects compatible with the isomorphisms ψ.

(b) Let PCoh_{A′,sp,e}^{r}(S) denote the groupoid of (K_{⋆}, ψ_{⋆}) in PMod_{A′,e}(S) such that K_{i} ∈ Coh_{A′,sp}^{r}(S) and N(K_{⋆}): = N(K_{i}) = N(K_{i+1}) for all i ∈ ℤ.

(c) For e, r ∈ ℤ with e, r ≥ 1 and e | rd. We denote by PVect_{A′,e}^{r}(S) the full subcategory of (F_{⋆}, ψ_{F_{⋆}}) in PMod_{A′,e}(S) such that F_{i} ∈ Vect_{A′}^{r}(S) for all i ∈ ℤ and such that Coker(j_{⋆} : F_{⋆} → F[1]_{⋆}) ∈ PCoh_{A′,sp,e}^{s}(S) with s = \frac{rd}{e}. Similarly one defines _{A′} PMod_{e}(S) and _{A′} PVect_{e}^{r}(S) using left A′ ⊠ O_{S}- modules.

Note that for (F_{⋆}, ψ_{⋆}) in PMod_{A′,e}(S) with F_{i} ∈ Vect_{A′}^{r}(S) for all i ∈ ℤ, the commutativity of diagram (55) implies that j_{i} : F_{i} → F_{i+1} is injective and Coker(j_{i}) is a sheaf on ∞ × S. For A′ = O_{X} we write Mod_{X}, PMod_{X,e}, Vect_{X} etc. for Mod_{O_{X}}, PMod_{O_{X},e}(S), Vect_{O_{X}} etc. Let E_{⋆} ∈ PMod_{A′,e}(S) and F_{⋆} ∈ _{A′} PMod_{e}(S). We are going to define now a tensor product (E_{⋆} ⊗_{A′} F_{⋆})_{⋆}. For i ∈ ℤ we set

T_{i}(E_{⋆}, F_{⋆}): = ⨁_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ}.

For i ∈ ℤ with we define homomorphisms

α_{i} : T_{i}(E_{⋆}, F_{⋆}) ⟶ T_{i+1}(E_{⋆}, F_{⋆}), β_{i} : T_{i}(E_{⋆}, F_{⋆}) ⟶ T_{i+1}(E_{⋆}, F_{⋆})

as the direct sums of the inclusions j_{λ} ⊗ id : E_{λ} ⊗_{A′} F_{μ} → E_{λ+1} ⊗_{A′} F_{μ} (resp. id ⊗j_{μ} : E_{λ} ⊗_{A′} F_{μ} → E_{λ} ⊗_{A′} F_{μ+1}). Also let

γ_{i} : T_{i}(E_{⋆}, F_{⋆}) ⟶ T_{i}(E_{⋆}, F_{⋆})

be the isomorphism given on the summand E_{λ} ⊗_{A′} F_{μ} by

E_{λ} ⊗_{A′} F_{μ} ≅ E_{λ}(∞) ⊗_{A′} F_{μ}(−∞) ⟶^{ψ^{−1}⊗ψ^{−1}} E_{λ+e} ⊗_{A′} F_{μ−e}.

Finally let

δ_{i} : T_{i−1}(E_{⋆}, F_{⋆}) ⊕ T_{i}(E_{⋆}, F_{⋆}) ⟶ T_{i}(E_{⋆}, F_{⋆})

be given on the summand T_{i−1}(E_{⋆}, F_{⋆}) by α_{i−1} − β_{i−1} and by id − γ_{i} on T_{i}(E_{⋆}, F_{⋆}). We define

∑_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ} = Coker(δ_{i})  (56)

There are canonical morphisms

∑_{λ+μ=i−1,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ} ⟶ ∑_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ}.  (57)

The isomorphisms

E_{λ+d} ⊗_{A′} F_{μ} ⟶^{ψ⊗id} (E_{λ} ⊗_{A′} F_{μ})(∞), E_{λ} ⊗_{A′} F_{μ+d} ⟶^{id ⊗ψ} (E_{λ} ⊗_{A′} F_{μ})(∞)

induces an isomorphism

∑_{λ+μ=i+d,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ} ⟶ ( ∑_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ})(∞).  (58)

Definition 5.5. The tensor product (E_{⋆} ⊗_{A′} F_{⋆})_{⋆} ∈ PMod_{X,e}(S) is defined as the collection of O_{X×S}-modules

(E_{⋆} ⊗_{A′} F_{⋆})_{i} = ∑_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ}

(for i ∈ ℤ) together with the maps (57) and (58).

Lemma 5.6. Let E_{⋆} ∈ PVect_{A′,e}^{r}(S) and F_{⋆} ∈ _{A′} PVect_{e}^{r}(S). Then (E_{⋆} ⊗_{A′} F_{⋆})_{⋆} lies in PVect_{X,e}^{rd^{2}}(S). In particular ∑_{λ+μ=i,λ,μ∈ℤ} E_{λ} ⊗_{A′} F_{μ} is a locally free O_{X×S}-module of rank rd^{2} for all i ∈ ℤ.

Proof. For A′ = O_{X} this follows immediately from the fact that a parabolic O_{X×S}-module is a parabolic vector bundle if and only if it is parabolically flat ([Yo], Proposition 3.1). The general case can be deduce from this special case by Morita equivalence. More precisely since the question is local we can replace X by an étale neighbourhood of ∞ and therefore can assume that A′ ≅ M_{d}(O_{X}). Let I be an invertible A′-O_{X}-bimodule and J its inverse. Since (E_{⋆} ⊗_{A′} F_{⋆})_{⋆} = ((E_{⋆} ⊗_{A′} I) ⊗_{O_{X}} (J ⊗_{A′} F_{⋆}))_{⋆} the assertion follows from the case A′ = O_{X}. □

Let A be another locally principal O_{X}-order of rank d^{2} and assume that e_{∞}(A) = d and A|_{U} and A′|_{U} are Morita equivalent where U = X − {∞}. There exists an increasing families of A-A′-bimodules {I_{i} | i ∈ ℤ} and of A′-A-bimodules {J_{i} | i ∈ ℤ} such that (I_{i})|_{U} = (I_{i+1})_{U} =: I_{U} and (J_{i})|_{U} = (J_{i+1})_{U} =: J_{U} for all i ∈ ℤ, I_{U} is an invertible A_{U}-A′_{U}-bimodule with inverse J_{U} and such that {(I_{i})_{∞} | i ∈ ℤ} and {(J_{i})_{∞} | i ∈ ℤ} are as in 2.2. It follows from Corollary 2.12 that I_{i} and J_{j} are locally free A′-modules of rank 1. Also we have

A(\frac{1}{d}∞) ⊗_{A} I_{i} = I_{i+1}, J_{i} ⊗_{A} A(\frac{1}{d}∞) = J_{i+1}

for all i ∈ ℤ.

Proposition 5.7. Put Vect_{A} = Vect_{A}^{1} and PVect_{A′} = PVect_{A′,d}^{1}. The morphisms

· ⊗_{A} I_{⋆} : Vect_{A} ⟶ PVect_{A′}, (· ⊗_{A′} J_{⋆})_{d−1} : PVect_{A′} ⟶ Vect_{A} (59)

given by E ↦ E ⊗_{A} I_{⋆}: = {F ⊗_{A} I_{i} | i ∈ ℤ} and E_{⋆} ↦ ∑_{λ+μ=d−1} E_{λ} ⊗_{A′} J_{μ} are mutually inverse isomorphisms of stacks. Define θ : Vect_{A} → Vect_{A} and θ′ : PVect_{A′} → PVect_{A′} by θ(E) = E(\frac{1}{d}∞) and θ′(E_{⋆}, ψ_{⋆}) = (E[1]_{⋆}, ψ[1]_{⋆}). Then the diagrams

Vect_{A} ⟶^{⊗_{A}I_{⋆}} PVect_{A′}  PVect_{A′} ⟶^{(·⊗_{A′} J_{⋆})_{d−1}} Vect_{A}
↓^{θ}  ↓_{θ′}  ↓_{θ′}  ↓^{θ}
Vect_{A} ⟶^{⊗_{A}I_{⋆}} PVect_{A′}  PVect_{A′} ⟶^{(·⊗_{A′} J_{⋆})_{d−1}} Vect_{A}

(60)

are 2-commutative.

Proof. In view of 5.6 we only have to show that the second morphism is welldefined. By 5.6 and 3.11 we have to prove that for E_{⋆} ∈ PVect_{A′}(S) the quotient

( ∑_{λ+μ=0,λ,μ∈ℤ} E_{λ} ⊗_{A′} J_{μ})/(  ∑_{λ+μ=−1,λ,μ∈ℤ} E_{λ} ⊗_{A′} J_{μ}) ≅ ∑_{λ+μ=0,λ,μ∈ℤ} \overline{E}_{λ} ⊗_{A′} J_{μ}

is a special A-module on S ≅ ∞ × S where \overline{E}_{⋆}: = Coker(E_{⋆}[−1] ↪ E_{⋆}) ∈_{A′} PMod_{d}(S). However this follows from:

Lemma 5.8. The assignement K_{⋆} ↦ ∑_{λ+μ=d−1} K_{λ}⊗_{A′} J_{μ} defines a morphism

(· ⊗_{A′} J_{⋆})_{d−1} : PCoh_{A′,sp,d}^{1} ⟶ Coh_{A,sp}^{1} .

Proof. By Lafforgue’s Lemma ([Laf], I.2.4) (applied to a maximal tori in A) it suffices to consider the case where S = Spec k and k is an algebraically closed field. If N(K_{⋆}) ≠ ∞ then the assertion follows from Remarks 3.8 (b), (c). Now assume N(K_{⋆}) = ∞. Let O = O_{∞}, K = F_{∞}. Since the question is local with respect to the étale topology we can replace X by Spec O where O = O_{∞}. Then A′ ≅ M_{d}(O) and A ≅ End(L_{⋆}) for a lattice chain L_{⋆} of period d in K^{d}. Morita equivalence allows us to replace A′ by O, i.e. we can assume that A′ = O. Then M_{i}: = Γ(Spec O, K_{i}) is a onedimensional k-vector space for all i ∈ ℤ and we have to show that

∑_{i+j=0} M_{i} ⊗_{O} J_{j}  (61)

is a free \overline{T} = T ⊗ k-module of rank 1 where T ≅ O^{d} is any maximal torus in A. If 1 = e_{1} + … + e_{d} is a decomposition of 1 ∈ T into primitive idempotents we obtain a corresponding decompositon of (61) into

∑_{i+j=0} M_{i} ⊗_{O} J_{j}^{(ν)}, ν = 1, …, d

where J_{j}^{(ν)} : = J_{j}e_{ν}. Since J_{j} is free of rank 1 as a T -module, J_{⋆}^{(ν)} is a shifted parabolic line bundle (compare [Yo]) for each ν ∈ {1, …, d}. Therefore (M_{⋆} ⊗_{O} J_{⋆}^{(ν)})_{⋆} ≅ M_{⋆}[m] for some m ∈ ℤ. Consequently

∑_{i+j=0} M_{i} ⊗_{O} J_{j}^{(ν)} ≅ M_{m}

is a onedimensional k-vector space. It follows that (61) is a free T ⊗_{O} R-module of rank 1. □

Now assume that k = 𝔽_{q} and that A′ is a maximal O_{X}-order in a central division algebra A′ of dimension d^{2} with A′_{∞} ≅ M_{d}(F_{∞}). Let us recall the definition of an A′-elliptic sheaf given in ([LRS], 2.2) and ([BS], 4.4.1) (here we do not require deg(∞) = 1).

Definition 5.9. Let S ∈ Sch /𝔽_{q}. An A′-elliptic sheaf E′ = (E_{i}, j_{i}, t_{i})_{i∈ℤ} with pole ∞ in the sense of [LRS] consists of a commutative diagram

… ⟶ E_{i−1} ⟶^{j_{i−1}} E_{i} ⟶^{j_{i}} E_{i+1} ⟶ …
↑^{t_{i−2}}  ↑^{t_{i−1}}  ↑^{t_{i}}
… ⟶ ^{τ} E_{i−2} ⟶^{j_{i−1}} ^{τ} E_{i−1} ⟶^{j_{i−1}} ^{τ} E_{i} ⟶ …

where E_{i} are locally free O_{X×S}-modules of rank d^{2} additionally equipped with a right action of A′ compatible with the O_{X}-action. The maps are injective A′ ⊠ O_{S}-linear homomorphisms. Furthermore the following conditions should hold:

(i) (Periodicity) E_{i+e deg(∞)} = E_{i}(∞): = E_{i} ⊗_{O_{X×S}} (O(∞) ⊠ O_{S}) where the canonical embedding of E_{i} on the right side corresponds on the left to the com-

postion E_{i} ↪^{j} … ↪^{j} E_{i+d′deg(∞)}.

(ii) The quotient sheaf E_{i}/j_{i−1}(E_{i−1}) is a locally free sheaf of rank d on the graph of a morphism ι_{∞,i} : S → X. (iii) There exists a morphism z : S → X − | \underline{Disc}(A′)| – called the zero or characteristic of E′ – such that for all i ∈ ℤ, Coker(t_{i}) is supported on the graph of a morphism z and is a direct image of a locally free O_{S}-module of rank d by Γ_{z} = (z, id_{S}) : S ⟶ X × S.

We first remark that condition (iii) implies that E_{i} is actually a locally free A′ ⊠O_{S}-module. This follows from ([Laf], I.4, proposition 7) or can be deduced from Lemma 3.11 together with ([LRS], 2.6). Secondly condition (i) implies that ι_{∞,i}(S) = {∞} and we have

ι_{∞,i} ∘ Frob_{S} = ι_{∞,i+1}

for all i ∈ ℤ. For that consider the two filtrations of E_{i+1}/t_{i−1}(^{τ} E_{i−1})

0⊆E_{i}/t_{i−1}(^{τ} E_{i−1})⊆E_{i+1}/t_{i−1}(^{τ} E_{i−1}),
0⊆t_{i}(^{τ} E_{i})/t_{i−1}(^{τ} E_{i−1})⊆E_{i+1}/t_{i−1}(^{τ} E_{i−1}).

The first shows that the support of E_{i+1}/t_{i−1}(^{τ} E_{i−1}) is Γ_{z} + Γ_{ι_{∞,i+1}} and the second that it is Γ_{z} + Γ_{ι_{∞,i}∘Frob_{S}}. Suppose again that deg(∞) = 1. Hence the stack PEℓℓ_{A′}^{∞}(S) of A′-elliptic sheaves as defined in 5.9 is isomorphic to the stack of triples E′ = (E_{⋆}, t_{⋆}) where E_{⋆} = (E_{⋆}, ψ_{⋆}) ∈ PVect_{A′}(S) and t_{⋆} : ^{τ} E[−1]_{⋆} → E_{⋆} is a morphism in PVect_{A′}(S) such that (iii) above holds for Coker(t_{⋆}). We show that the isomorphisms (59) yield isomorphisms between PEℓℓ_{A′}^{∞} and Eℓℓ_{A}^{∞}|_{X−| \underline{Disc}(A′)|} = Eℓℓ_{A}^{∞} ×_{X} (X − | \underline{Disc}(A′)|). Define

· ⊗_{A} I_{⋆} : Eℓℓ_{A}^{∞}|_{X−| \underline{Disc}(A′)|}(S) ⟶ PEℓℓ_{A′}^{∞}(S)

by (E, t) ↦ (E ⊗_{A} I_{⋆}, t ⊗_{A} I_{⋆}). The commutativity of the first diagram (60) shows that t ⊗_{A} I_{⋆} is a map ^{τ} E ⊗_{A} I_{⋆}[−1] → E ⊗_{A} I_{⋆}. That E′ has property (iii) above follows from Remark 3.8 (c). Conversely, we define

(· ⊗_{A′} J_{⋆})_{d−1} : PEℓℓ_{A′}^{∞}(S) ⟶ Eℓℓ_{A}^{∞}|_{X−| \underline{Disc}(A′)|}(S)

by (E_{⋆}, t_{⋆}) ↦ ((E_{⋆} ⊗_{A′} J_{⋆})_{d−1}, (t_{⋆} ⊗_{A′} J_{⋆})_{d−1}). Again the commutativity of the second diagram of (60) implies that (t_{⋆} ⊗_{A′} J_{⋆})_{d−1} is a morphism τ (E(−\frac{1}{d}∞_{S})) → E where E = (E_{⋆}⊗_{A′}J_{⋆})_{d−1}. Finally condition (*) of Definition 4.2 follows Lemma 5.8. We deduce from 5.7:

Proposition 5.10. Let S be a k-scheme. The morphisms

· ⊗_{A}I_{⋆} : Eℓℓ_{A}^{∞}|_{X−| \underline{Disc}(A′)|} ⟶ PEℓℓ_{A′}^{∞},  (62)
(· ⊗_{A′} J_{⋆})_{d−1} : PEℓℓ_{A′}^{∞} ⟶ Eℓℓ_{A}^{∞}|_{X−| \underline{Disc}(A′)|}  (63)

are mutually inverse isomorphisms.

Remarks 5.11. (a) In order to extend 5.10 to the case deg(∞) > 1 we have to modify Definition 5.4 (b) as follows. For S ∈ Sch /k let PVect_{A′}(S) denote the category of triples (F_{⋆}, ψ_{F_{⋆}}, ∞_{S}) where ∞_{S} : S → X is a k-morphism which factors through ∞ → X and (F_{⋆}, ψ_{F_{⋆}}) is an element of PMod_{A′,d deg(∞)}(S) such that F_{i} ∈ Vect_{A′}^{1}(S) for all i ∈ ℤ and such that the sheaf Coker(j_{i}) is a locally free sheaf of rank d on the graph of ∞_{S} ∘ Frob_{S}^{i} : S → X. To define isomorphisms similar to (59) we consider increasing families of A⊠k(∞)- A′ ⊠ k(∞)-bimodules {I_{i} | i ∈ ℤ} and A′ ⊠ k(∞)-A ⊠ k(∞)-bimodules {J_{i} | i ∈ ℤ} with the following properties:

(i) A(\frac{1}{d}∞_{i}) ⊗_{A} I_{i} = I_{i+1}, J_{i} ⊗_{A} A(\frac{1}{d}∞_{i}) = J_{i+1} for all i ∈ ℤ. Here ∞_{0} denotes the canonical morphism Spec k(∞) → X and ∞_{i}: = ∞_{0} ∘Frob^{i} : Spec k(∞) → X.

(i) I_{U} = (I_{i})|_{U×_{𝔽q} k(∞)} is an invertible A_{U} ⊠k(∞)-A′_{U} ⊠k(∞)-bimodule with inverse J_{U} = (J_{i})|_{U×_{𝔽q} k(∞)}.

(iii) For all i ∈ ℤ, I_{i} and J_{j} are locally free A′ ⊠ k(∞)-modules of rank 1.

As in 5.7 one defines isomorphisms

· ⊗_{A}I_{⋆} : Vect_{A} ×_{𝔽_{q}}k(∞) ⟶ PVect_{A′},
(· ⊗_{A′} J_{⋆})_{d−1} : PVect_{A′} ⟶ Vect_{A} ×_{𝔽_{q}}k(∞)

which then yield the isomorphisms (62), (63) above.

(b) Let 𝔭 be a closed point of X such that inv_{𝔭}(A′) = \frac{1}{d}. In [Hau], Hausberger constructed a flat proper model of Eℓℓ_{A′}^{∞} over (X −| \underline{Disc}(A′)|)∪\{𝔭\} by extending the definition of the moduli problem 5.9 of Laumon-Rapoport-Stuhler to characteristic 𝔭. By using ([Hau], 2.16) it is easy to see using that his condition spéciale ([Hau], section 3) corresponds to our condition (*).

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Michael Spieß

Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld Germany