Endomorphisms of exceptional $D$-elliptic sheaves

0.24 MB·English

by  Mihran Papikian

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