ENDOMORPHISMS OF EXCEPTIONAL D-ELLIPTIC SHEAVES MIHRANPAPIKIAN 9 0 0 Abstract. We relate the endomorphism rings of certain D-elliptic sheaves 2 of finite characteristic to hereditary orders in central division algebras over functionfields. n a J 3 2 1. Introduction Theendomorphismringsofabelianvarietieshavelongbeenasubjectofintensive ] T investigation in number theory. One of the earliest results in this area was the N determination by Deuring of the endomorphism rings of elliptic curves over finite . fields. His results were later generalized to higher dimensional abelian varieties by h Honda, Tate and Waterhouse [16]. These results have importantapplications, e.g., t a they play a key role in calculations of local zeta functions of Shimura varieties. m In [3], Drinfeld introduced a certain function field analogue of abelian varieties; [ these objects are now called Drinfeld modules. Denote by F the finite field with q q elements. Let X be a smooth, projective, geometrically connected curve defined 1 v over Fq. Let F = Fq(X) be the function field of X. Fix a place ∞ of F (in 9 Drinfeld’s theory this plays the role of an archimedean place). Let o 6= ∞ be 8 another place of F. Denote by F the residue field at o. In [6], Drinfeld proved o 6 the analogue of Honda-Tate theorem for Drinfeld modules defined over extensions 3 of F . In [9], Gekeler extended Drinfeld’s results, in particular, he proved that for . o 1 a rank-d supersingular Drinfeld module φ over F the endomorphism ring End(φ) 0 o is a maximal order in the central division algebra over F of dimension d2, which is 9 0 ramifiedexactly atoand∞with invariants−1/dand1/d,respectively. Moreover, : thereisabijectionbetweentheisomorphismclassesofrank-dsupersingularDrinfeld v modules over F and the left ideal classes of End(φ) (see [9, Thm. 4.3]). i o X In [13], Laumon, Rapoport and Stuhler introduced the notion of D-elliptic r sheaves, which is a generalization of the notion of Drinfeld modules. (One can a think ofthese objectsasfunction fieldanaloguesofabelianvarieties equippedwith an action of a maximal order in a simple algebra over Q.) In Laumon-Rapoport- Stuhler theory one needs to fix a central simple algebra D over F of dimension d2 which is split at ∞, and a maximal O -order D in D (see §2 for definitions). In X [13, Ch. 9], the authors develop the analogue of Honda-Tate theory for D-elliptic sheavesover F with zero o and pole ∞, assuming D is split at o. The assumption o that D is split at o is not superficial. When D is ramified at o, to obtain a rea- sonable theory of D-elliptic sheaves with zero o and pole ∞, one has to assume at least that D⊗ F is the division algebra with invariant 1/d over F , where F is F o o o the completionofF ato. SuchD-elliptic sheavesplayacrucialroleinthe function 1991 Mathematics Subject Classification. Primary11G09; Secondary11R58,16H05. The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship. 1 2 MIHRANPAPIKIAN fieldanalogueofC˘erednik-DrinfelduniformizationtheorydevelopedbyHausberger [11]. Assume D := D ⊗ F is the d2-dimensional central division algebra with o F o invariant 1/d over F . In this paper we define a subclass of D-elliptic sheaves over o F , which we call exceptional, and which are distinguished by a particularlysimple o relationshipbetweentheactionsofD andtheFrobeniusato;seeDefinition5.1. In o general, exceptional D-elliptic sheaves do not correspond to points on the moduli schemes constructed in [13] or [11], so they are not very natural from moduli- theoretic point of view. Nevertheless, we show that the theory of endomorphism ringsoftheseobjectsissimilartothetheoryofendomorphismringsofsupersingular Drinfeld modules. The main result is the following (see Theorems 5.3 and 5.4): Theorem 1.1. Let E be an exceptional D-elliptic sheaf over F of type f. Then o End(E) is a hereditary O -order in the central division algebra D¯ over F with X invariants 1/d, x=∞; inv (D¯)= 0, x=o; x  inv (D), x6=o,∞.  x This orderis maximalat everyplacex6=o, andatoitis isomorphic toahereditary order of type f. There is a bijection between the set of isomorphism classes of exceptional D-elliptic sheaves over F of type f and the isomorphism classes of o locally free rank-1 right End(E)-modules. The type of an exceptional D-elliptic sheaf is determined by the action of the Frobenius at o, and the type of a hereditary order determines the order up to an isomorphism (see §2). In §6, we use Theorem 1.1 to prove a mass-formula for exceptional D-elliptic sheaves, and discuss a geometric application of this formula. In §7, we explain how the argument in the proof of Theorem 1.1 can be used to proveatheoremaboutendomorphismringsofsupersingularD-ellipticsheavesover F , which implies Gekeler’s result mentioned earlier as a special case (in §7 we o assume that D is split at o). Notation. Unless specified otherwise, the following notation is fixed throughout the article. •kisafixedalgebraicclosureofF andFr :k →kistheautomorphismx7→xq. q q • |X| denotes the set of closed points on X (equiv. the set of places of F). • For x ∈ |X|, O is the completion of O , and F (resp. F ) is the fraction x X,x x x field (resp. the residue field) of O . The degree of x is deg(x) := [F : F ], and x x q q :=qdeg(x) =#F . We fix a uniformizer π of O . x x x x • A := ′ F denotes the adele ring of F, and for a set of places S ⊂|X|, F x∈|X| x AS := ′ F denotes the adele ring outside of S. F x∈|QX|−S x • The zeta-function of X is Q ζ (s)= (1−q−s)−1, s∈C. X x xY∈|X| • For any ring R we denote by R× its subgroup of units. • M denotes the ring of d×d matrices. d ENDOMORPHISMS OF EXCEPTIONAL D-ELLIPTIC SHEAVES 3 2. Orders For the convenience of the reader, we recall some basic definitions and facts concerning orders over Dedekind domains. A standard reference for these topics is [14]. LetRbeaDedekinddomainwithquotientfieldK andletAbe acentralsimple K-algebra. For any finite dimensional K-vector space V, a full R-lattice in V is a finitely generated R-submodule M in V such that K ⊗ M ∼= V. An R-order in R theK-algebraAisasubringΛofA,havingthesameunityelementasA,andsuch that Λ is a full R-lattice in A. A maximal R-order in A is an R-order which is not contained in any other R-order in A. A hereditary R-order in A is an R-order Λ which is a hereditary ring, i.e., every left (equiv. right) ideal of Λ is a projective Λ-module. Maximal orders are hereditary. Being maximal or hereditary are local properties for orders: an R-order Λ in A is maximal (resp. hereditary) if and only if Λ := Λ⊗ R is a maximal (resp. hereditary) R -order in A := A⊗ K for p R p p p K p all prime ideals p(cid:1)R, where R and K are the p-adic completions of R and K. p p Let I be a full R-lattice in A. Define the left order of I O (I)={a∈A | aI ⊆I}. ℓ ItiseasytoseethatO (I)isanR-orderinA. Onesimilarlydefinesthe rightorder ℓ O (I) of I. r AssumeRisacompletediscretevaluationringwithauniformizerπ andfraction field K. Let f = (f ,...,f ) be a d-tuple of non-negative integers such that 0 d−1 d−1f = d. Denote by M (f,R) the subgroup of M (R) consisting of matrices i=0 i d d of the form (m ), where m ranges over all f ×f matrices with entries in R if ij ij i j P i≥j, and over all f ×f matrices with entries in πR if i <j (a block of size 0 is i j assumed to be empty), e.g., if f =(d,0,...,0) then M (f,R)=M (R). d d Theorem 2.1. Let Λ be an R-order in M (K). Λ is maximal if and only if there d is an invertible element u ∈ M (K) such that uΛu−1 = M (R); Λ is hereditary if d d and only if uΛu−1 =M (f,R) for some f (uniquely determined up to permutation d of its entries). Proof. See Theorems 17.3 and 39.14 in [14]. (cid:3) When Λ is a hereditary order as in Theorem 2.1, we shall call f the type of Λ. (This is slightly different from the terminology used in [14, p. 360].) LetAbeacentralsimplealgebraoverF. AnO -orderin Aisacoherentlocally X free sheaf A of O -algebras with generic fibre A. The O -order A is maximal X X (resp. hereditary) if for every open affine U = Spec(R) ⊂ X the set of sections A(U) := Γ(U,A) is a maximal (resp. hereditary) R-order in A. For x ∈ |X| we denote A := A⊗ F and A := A⊗ O , so A is isomorphic to a subring x F x x OX,x x x of A . A is maximal (resp. hereditary) if and only if A is a maximal (resp. x x hereditary) O -order in A . x x Let A be a hereditary O -order. Let I be a coherent sheaf on X which is a X locally free rank-1 right A-module. (The action of A on I extends the action of O .) The generic fibre I ⊗ F is isomorphic to A as an F-vector space. Define X OX a sheaf O (I) on X as follows. For an open affine U ⊂X let ℓ O (I)(U)=O (I(U)). ℓ ℓ 4 MIHRANPAPIKIAN It is easy to see that O (I) is an O -order in A, which is locally isomorphic to A. ℓ X 3. Dieudonn´e modules Let R be a complete discrete valuation ring of positive characteristic p>0 and residue field F . Fix a uniformizer π of R and identify R = F [[π]]. Let K be the q q fraction field of R. Let R = R⊗F k ∼= k[[π]] be the completion of the maximal q unramified extension of R, and K = K⊗F k ∼= k((π)) be the field of fractions of q R. We will denote the canonicalblifting of Fr ∈Gal(k/F ) to Aut(K) by the same q q symbol, so b ∞ ∞ Fr a πi = aqπi, n∈Z. q i i ! i=n i=n X X The following definition and Theorem 3.2 below are due to Drinfeld [8]; see also [12, §2.4]. Definition 3.1. A Dieudonn´e R-module over k is a free R-module of finite rank M endowedwith an injective Fr -linear mapϕ:M →M such that the cokernelof q ϕ is finite dimensional as a k-vectorspace. The rank of (M,ϕ) is the rank ofM as a R-module. A Dieudonn´e K-module over k is a finite dimensional K-vector space N endowed with a bijective Fr -linear map ϕ:N →N. The rank of (N,ϕ) is the q dimension of N as a K-vector space. A morphism of Dieudonn´e R-modules (resp. K-modules) over k is a linear map between the underlying R-modules (resp. K- vectorspaces)whichcommuteswiththeFr -linearmapsϕ. If(M,ϕ)isaDieudonn´e q R-module over k, then (K ⊗ M,K⊗ ϕ) is a Dieudonn´e K-module over k. R R LetK{τ}bethenon-commutativepolynomialringwithcommutationruleτ·a= Fr (a)τ, a∈K. For each pair of integers (r,s) with r≥1 and (r,s)=1, let q N =K{τ}/K{τ}(τr −πs). r,s Then (N ,ϕ ), where ϕ is the left multiplication by τ, is a Dieudonn´e K- r,s r,s r,s module over k of rank r. Theorem3.2. ThecategoryofDieudonn´eK-modulesoverkisK-linearandsemi- simple. Its simple objects are (N ,ϕ ), r,s∈Z, r ≥1, (r,s)=1. The K-algebra r,s r,s of endomorphisms D = End(N ,ϕ ) of such an object is the central division r,s r,s r,s algebra over K with invariant −s/r. This theorem implies that given a Dieudonn´e K-module (N,ϕ), its endomor- phism algebra End(N,ϕ) is a finite dimensional semi-simple K-algebra such that the center of each simple component is K. It is clear that for a Dieudonn´e R- module (M,ϕ), the endomorphism ring End(M,ϕ) is an R-order in End(N,ϕ), where (N,ϕ)=K⊗(M,ϕ). Proposition 3.3. Let (M,ϕ) be a Dieudonn´e R-module over k of rank n. Suppose ϕ(M)=M. Then (M,ϕ)∼=(Rn⊗Fqk,Id⊗FqFrq). Proof. This is proven in [8, Prop. 2.5] using π-divisible groups. An alternative b b argumentis asfollows. Let(N,ϕ) be the associatedDieudonn´eK-module. By[12, Prop. 2.4.6], the assumption of the proposition is equivalent to (N,ϕ)∼=(N ,ϕ )n. 1,0 1,0 ENDOMORPHISMS OF EXCEPTIONAL D-ELLIPTIC SHEAVES 5 Hence (N,ϕ)∼=(Nϕ⊗Fqk,Id⊗FqFrq), where Nϕ = {a ∈ N | ϕ(a) = a} is an n-dimensional K-vector space. Since N =M ⊗K, Mϕ :=M ∩Nϕ is a full Rb-latticebin Nϕ and M =Mϕ⊗F k. (cid:3) q Let D be the d2-dimensionalcentraldivision algebraoverK with invariant1/d. b Let D be the maximal R-order in D. Denote by R = F [[π]] the ring of integers d qd of the degree d unramified extension of K. We can identify D with the R-algebra R [[Π]] of non-commutative formal power series in the indeterminate Π satisfying d the relations Πa=Fr (a)Π for any a∈R , q d Πd =π. Definition 3.4. A Dieudonn´e R-module (M,ϕ) over k is connected if for all large enough positive integers m, ϕm(M) ⊂ πM. This is equivalent to saying that (N ,ϕ ) does not appear in the decomposition of the associated Dieudonn´e K- 1,0 1,0 module into simple factors. A Dieudonn´e D-module over k is a rank-d2 connected Dieudonn´eR-module(M,ϕ)overkequippedwitharightD-actionwhichcommutes withϕ andextends the naturalactionofR. Amorphism ofDieudonn´eD-modules is a morphism of the underlying Dieudonn´e R-modules which commutes with the action of D. For each Dieudonn´e D-module (M,ϕ) over k there is an associated Dieudonn´e D-module (N,ϕ)=K⊗(M,ϕ). A Dieudonn´e D-module (M,ϕ) over k is naturally a right D⊗F k-module. Fix q an embedding F ֒→k. Since F is also embedded in D, we obtain a grading qd qd b M = M , i i∈MZ/dZ whereM ={m∈M |m(λ⊗1)=m(1⊗λqi),λ∈F }. EachM isafreefiniterank i qd i R-module. TheactionofΠ⊗1onM inducesinjectivelinearmapsΠ :M →M , i i i+1 i∈Z/dZ. The compositionb b M −Π→i bM −Π−i−+→1 ···−Π−i−+−d−−→1 M =M i i+1 i+d i isM (π⊗1)=πM . Inparticular,allM havethesamerankoverR,whichmustbe i i i d,sincetherankofM isd2. Sincedim (coker(π))=d2,wehavedim (coker(Π))= k k d. b Similarly, ϕ induces injective Fr -linear maps q M −ϕ→i M −ϕ−i−+→1 ···−ϕ−i−+−d−−→1 M =M . i i+1 i+d i Let f := dim (coker(ϕ )), so d−1f = dim (coker(ϕ)). We call the ordered i k i i=0 i k d-tuple f =(f ,...,f ) the type of M. 0 d−1 P Definition3.5. (cf. [15],[10])ADieudonn´eD-module(M,ϕ)overkisexceptional if Im(ϕ)=Im(Π). (M,ϕ) is special if f =1 for all i. (M,ϕ) is superspecial if it is i special and exceptional. Note that (M,ϕ) being exceptional is equivalent to Im(ϕ ) = Im(Π ) for all i i i ∈ Z/dZ, i.e., every index of M is critical in the terminology of [10, Def. II.1.3]. In particular, if (M,ϕ) is exceptional of type f then d−1f =d. i=0 i P 6 MIHRANPAPIKIAN Proposition 3.6. Let (M,ϕ) be an exceptional Dieudonn´e D-module over k of type f. Then End (M,ϕ) ∼= M (f,R). In particular, End (M,ϕ) is a hereditary D d D R-order in End (N,ϕ)∼=M (K). D d Proof. Using the injections Π , we can identify all M ⊗ K with the same d- i i dimensionalK-vectorspaceV. ThenΠ inducesabijectivelinearmapV →V,and i ϕ induces a bijective Fr -linear map. Consider Π−1 ◦ϕ : V → V as a bijective i q i i Fr -linear map. Since (M,ϕ) is exceptional, Im(Π ) = Im(ϕ ) for all i ∈ Z/dZ. q i i Hence Π−1◦ϕ is bijective on M . By Proposition 3.3, there are d full R-lattices i i i Λ in Kd, i ∈ Z/dZ, such that M = Λ ⊗k. Since the action of D commutes with i i i the actionofϕ, wehaveϕ ◦Π =Π ◦ϕ . Thisimplies thatthe identifications i+1 i i+1 i Mi =Λi⊗k can be made compatibly sobthat (3.1) Λ −π→0 Λ −π→1 ···−π−d−−→2 Λ −π−d−−→1 Λ , b 0 1 d−1 0 where π ’s are injections, Π =π ⊗k, ϕ =π ⊗Fr . The inclusions π satisfy i i i i i q i π ◦π ◦···◦π =π, i i+1 i+d−1 b b so each coker(πi) has no nilpotents and dimFq(coker(πi))=fi. Now it is easy to see that giving an endomorphism of (M,ϕ) commuting with the action of D is equivalent to giving an endomorphism g of Kd which preserves the flag of lattices (3.1). Such endomorphisms form an R-algebra isomorphic to M (f,R). That this is a hereditary order in M (K) follows from Theorem 2.1. (cid:3) d d Every Dieudonn´e D-module over k satisfies the properties in [10, p. 20], hence corresponds to a formal D-module of height d2. It is instructive to give explicit examples of such formal modules. What follows below is motivated by [10, I.4.2]. TheunderlyingformalgroupisisomorphictoGd ,whereG =Spf(k[[t]])isthe a,k a,k formal additive group. Denote by τ the Frobenius isogeny of G corresponding a,k to t7→tq. To give a formal D-module essentiallybamounts tobgiving an embedding Φ:F [[Π]]=D ֒→End(Gd )∼=M (k{{τ}}),b qd a,k d where k{{τ}} is the non-commutative ring of formal power series in τ satisfying τa=aqτ for a∈k. Now let (M,ϕ) be an exceptional Dieudonn´e D-module of type f. Being excep- tional, i.e., Im(Π)=Im(ϕ), translates into Φ(Π)=τ ·Id. The type translates into Φ(λ)=diag(χ (λ)) , λ∈F , ij 0≤j≤d−1,1≤i≤fj qd where χ (λ) = λqj if f 6= 0 and is omitted from diag(·) otherwise. For example, ij j if d=3 and f =(2,0,1) then τ 0 0 λ 0 0 Φ(Π)= 0 τ 0 and Φ(λ)= 0 λ 0 . 0 0 τ 0 0 λq2     The endomorphism ring End (M,ϕ) is isomorphic to the opposite algebra of the D centralizerof Φ(D) in M (k{{τ}}). One cancheck asin [10, I.4.2]thatthis central- d izer is isomorphic to M (f,R)opp. d ENDOMORPHISMS OF EXCEPTIONAL D-ELLIPTIC SHEAVES 7 4. D-elliptic sheaves InthissectionwerecallthedefinitionofD-ellipticsheavesoffinitecharacteristic and their basic properties as given in [13, Ch. 9]. Fix a closed point ∞ ∈ |X|. Let D be a central simple algebra over F of dimension d2. Assume D is split at ∞, i.e., D⊗ F ∼= M (F ). Fix a maximal F ∞ d ∞ O -order D in D. Denote by Ram ⊂ |X| the set of places where D is ramified; X hence for all x 6∈ Ram the couple (D ,D ) is isomorphic to (M (F ),M (O )). x x d x d x Fix another closed point o ∈ |X|−∞, and an embedding F ֒→ k. Let z be the o morphism determined by these choices z :Spec(k)→Spec(F )֒→X. o Definition 4.1. A D-elliptic sheaf of characteristic o over k is a sequence E = (Ei,ji,ti)i∈Z, where Ei is a locally-free OX⊗Fqk-module of rankd2, equipped with a right action of D which extends the O -action, and X j :E ֒→E i i i+1 t :τE :=(Id ⊗Fr )∗E ֒→E i i X q i i+1 are injective D-linear homomorphisms. Moreover, for each i ∈ Z the following conditions hold: (1) The diagram E ji //E i i+1 OO OO ti−1 ti τE τji−1 // τE i−1 i commutes; (2) E =E ⊗ O (∞), and the inclusion i+d·deg(∞) i OX X E −j→i E −j−i+−→1 ···→E =E ⊗ O (∞) i i+1 i+d·deg(∞) i OX X is induced by O ֒→O (∞); X X (3) dim H0(X ⊗k,cokerj )=d; k i (4) E /t (τE )=z H , where H is a d-dimensional k-vector space. i i−1 i−1 ∗ i i When o6∈Ram, the previous definition is exactly the one found in [13, p. 260]. When o ∈ Ram this definition is not restrictive enough, but for our purposes it is adequate to take it as a starting point. Definition 4.2. LetDESbe thecategorywhoseobjectsaretheD-ellipticsheaves of characteristic o over k, and a morphism between two objects in this category ψ =(ψi)i∈Z :E′ =(Ei′,ji′,t′i)i∈Z →E′′ =(Ei′′,ji′′,t′i′)i∈Z isasequenceofsheafmorphismsψ :E′ →E′′ whicharecompatiblewiththeaction i i i of D and commute with the morphisms j and t : i i ψ ◦j′ =j′′◦ψ and ψ ◦t′ =t′′◦τψ . i+1 i i i i i−1 i i−1 Denote by Hom(E′,E′′) the set of all morphisms E′ → E′′, and let End(E) = Hom(E,E). 8 MIHRANPAPIKIAN Definition 4.3 ([8]). Aϕ-space overk isafinite dimensionalF⊗F k-vectorspace q N equipped with a bijective F ⊗Fq Frq-linear map ϕ : N → N. A morphism α betweentwoϕ-spaces(N′,ϕ′)and(N′′,ϕ′′)isaF⊗F k-linearmapN′ −→α N′′ such q that ϕ′′◦α=α◦ϕ′. LetE∈DES. DenoteN =H0(Spec(F⊗Fqk),E0). ThisisafreeD⊗Fqk-module of rank 1. The ti’s induce a bijective F ⊗FqFrq-linear map ϕ:N →N, compatible withtheactionofD ontheright,sotoEonecanattachaϕ-spaceoverk equipped with an action of D, which commutes with ϕ. This action induces an F-algebra homomorphism ι:Dopp →End(N,ϕ). We denote by End (N,ϕ) the F-algebra of endomorphisms of (N,ϕ) which com- D mute with the action of D. The triple (N,ϕ,ι) is called the generic fibre of E ([13, Def. 9.2]). It is independent of the choice of E since the sheavesE are isomorphic 0 i over (X −∞)⊗k via j’s. For x∈|X|, denote Mx :=H0(Spec(Ox⊗k),E0). This is a free Ox⊗Fqk-module of rank d2 with a right action of D . Let N = F ⊗ M . The t ’s induce x x x Ox x i a bijective Fx⊗FqFrq-linear map ϕx : Nxb→ Nx, compatible with tbhe action of D . The pair (N ,ϕ ) is the Dieudonn´e module of E at x. The F -algebra of x x x x endomorphismbs of (Nx,ϕx) which commute with the action of Dx will be denoted by End (N ,ϕ ). Note that (N ,ϕ )=(F ⊗ N,F ⊗ ϕ). Dx x x x x x F x F As easily follows from definitions, the lattices M have the following properties x (see [13, Lem. 9.3]): b b (M1) If x=∞, then M ⊂ϕ (M ) ∞ ∞ ∞ dim (ϕ (M )/M )=d k ∞ ∞ ∞ ϕd·deg(∞)(M )=π−1M . ∞ ∞ ∞ ∞ (M2) If x=o, then π M ⊂ϕ (M )⊂M o o o o o andtheFo⊗Fqk-moduleMo/ϕo(Mo)isoflengthdandissupportedonthe connected component of Spec(Fo⊗Fq k) which is the image of z. (M3) If x6=o,∞, then ϕ (M )=M . x x x (M4) Some basis of N generates M in N for all but finitely many x∈|X|. x x Definition 4.4. Let DMod be the category whose objects are the pairs ((N,ϕ,ι),(M ) ) x x∈|X| where (N,ϕ) is a ϕ-space of rank d2 over F ⊗k, ι : Dopp → End(N,ϕ) is an F- algebra homomorphism, and (M ) is a collection of D -lattices in (N ,ϕ )= x x∈|X| x x x (F ⊗ N,F ⊗ ϕ) which satisfy (M1)-(M4). A morphism α between two such x F x F objects b bα:((N′,ϕ′,ι′),(Mx′)x∈|X|)→((N′′,ϕ′′,ι′′),(Mx′′)x∈|X|) is a morphism of the ϕ-spaces α:(N′,ϕ′)→(N′′,ϕ′′) such that ι′′◦α=α◦ι′ and α⊗F (M′)⊂M′′ x x x for all x∈|X|. b ENDOMORPHISMS OF EXCEPTIONAL D-ELLIPTIC SHEAVES 9 Proposition 4.5. The functor DES → DMod which associates to a D-elliptic sheaf of characteristic o over k its generic fibre along with the lattices M in its x Dieudonn´e modules is an equivalence of categories. Proof. From the description of a locally free sheaf on a curve through lattices in its generic fibre, it follows that the functor in question is fully faithful. Now let ((N,ϕ,ι),(Mx)x∈|X|)∈DModandi∈Z. Define a sheafEi on X⊗Fqk as follows. Let U ⊂X be an open affine. If ∞6∈|U|, then let Ei(U ⊗Fq k):= (N ∩Mx), x∈\|U| where the inner intersections are taken in N , and the outer in N. If ∞∈|U|, let x Ei(U ⊗Fq k):= (N ∩Mx) ϕi∞(M∞)∩N . x∈|\U|−∞ \(cid:0) (cid:1)   Thanks to (M4), E is a locally-free O -module of rank d2. The inclusions i X⊗Fqk ϕi (M ) ⊂ ϕi+1(M ) induce inclusions j : E ֒→ E . The action of ϕ on N ∞ ∞ ∞ ∞ i i i+1 induces homomorphisms t : τE → E . The action of D on (N,ϕ) and D on i i i+1 x M , defines an action of D on E compatible with j and t . Finally, the conditions x i i i (M1)-(M3) ensure that (Ei,ji,ti)i∈Z ∈ DES. Hence our functor is essentially surjective. (cid:3) Let x ∈ |X| and r := [Fx :Fq]. Since Nx is a free Fx⊗Fqk-module, by fixing an embedding F → k, we obtain two actions of F on N . These actions induce a x x x grading b N = N , x x,i i∈MZ/rZ where N = {a ∈ N | (λqi⊗1)a = (1⊗λ)a, λ ∈ F }. Now ϕ maps N x,i x x x x,i bijectively into Nx,i+1, and Nx,0 is an Fx⊗Fxk-vector space. Hence (Nx,0,ϕrx) is a Dieudonn´e Fx-module over k inbthe sense ofb§3. We can recover (Nx,ϕx) uniquely from (Nx,0,ϕrx) since as Fx⊗Fqk-vector spbace b Nx ∼= Nx,0 i∈MZ/rZ with the action of ϕ given by x (a ,a ,...,a )7→(ϕr(a ),a ,...,a ). 0 1 r−1 x r−1 0 r−2 Finally, since the action of D commutes with ϕ , (N ,ϕr) is a Dieudonn´e D - x x x,0 x x module over k and End (N ,ϕ ) = End (N ,ϕr). Similar argument applies Dx x x Dx x,0 x also to the lattices M (x6=∞) and produces Dieudonn´e D -modules over k. x x 5. Endomorphism rings Let D be as in §4. Assume D is a division algebra such that D is the d2- o dimensional central division algebra over F with invariant 1/d. In this case D is o o 10 MIHRANPAPIKIAN the unique maximal order of D which we identify with F(d)[[Π ]]. Here F(d) is the o o o o degree d extension of F and o Π a=Frdeg(o)(a)Π , o q o Πd =π . o o Definition 5.1. Let E∈DES. We say that E is exceptional if ϕdeg(o)(M )=M ·Π . o o o o Clearly E is exceptional if and only if the Dieudonn´e D -module (M ,ϕdeg(o)) o o,0 o associated to (M ,ϕ ) is exceptional in the sense of Definition 3.5. The type of an o o exceptionalEisthetypeof(M ,ϕdeg(o)). Similarly,wesaythatEisspecial (resp. o,0 o deg(o) superspecial) if (M ,ϕ ) is special (resp. superspecial). o,0 o Remark 5.2. Exceptional D-elliptic sheaves do not correspond to points on the moduli schemes constructed in [11], unless they are superspecial. Let D¯ be the central division algebra over F with invariants 1/d, x=∞; (5.1) inv (D¯)= 0, x=o; x  inv (D), x6=o,∞.  x Theorem 5.3. If E is exceptional of type f, then End(E) has a natural structure  of a hereditary O -order in D¯. This order is maximal at every x∈|X|−o, and at X o it is isomorphic to M (f,O ). d o Proof. Let ((N,ϕ,ι),(M ) ) ∈ DMod be the object attached to E by Propo- x x∈|X| sition 4.5. Giving an endomorphism of E is equivalent to giving ψ ∈End(N,ϕ,ι)=End (N,ϕ) D such that ψ⊗ F ∈End(N ,ϕ ) preserves the lattice M for all x∈|X|. F x x x x Let (F,Π) be the ϕ-pair of (N,ϕ); see [13, App. A] for the definition. Since E is exceptionabl e e ϕd·deg(∞)(M )=π−1M , ∞ ∞ ∞ ∞ ϕd·deg(o)(M )=π M , o o o o ϕ (M )=M , if x6=o,∞. x x x Let h be the class number of X. The divisor h(deg(∞)o−deg(o)∞) is principal, so from the previous equalities ϕdh ∈ F. By construction of (F,Π), this implies F =F and Π∈F has valuations e e 1/ddeg(o), x=o; (e5.2) e ord (Π)= −1/ddeg(∞), x=∞; x  0, x6=o,∞.  e Since (N,ϕ) is isotypical [13, Lem. 9.6], (5.2) and [13, Thm. A.6] imply that  A=End(N,ϕ) is the cental simple algebra over F of dimension d4 with invariants inv (A) = −1/d, inv (A) = 1/d, inv (A) = 0, x 6= o,∞. End (N,ϕ) is exactly o ∞ x D the centralizer of ι(Dopp) in End(N,ϕ). By the double centralizer theorem [14, Cor. 7.14] End (N,ϕ)⊗ Dopp ∼=A. D F