Categories of abelian varieties over finite fields I. Abelian varieties over $\mathbb{F}_p$

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by  Tommaso Giorgio Centeleghe

Categories of abelian varieties over finite fields I
Abelian varieties over F
p
TOMMASO GIORGIO CENTELEGHE AND JAKOB STIX
Abstract — We assign functorially a Z-lattice with semisimple Frobenius action to
eachabelianvarietyoverF . Thisestablishesanequivalenceofcategoriesthatdescribes
p √
abelian varieties over F avoiding p as an eigenvalue of Frobenius in terms of simple
p
commutative algebra. The result extends the isomorphism classification of Waterhouse
5 and Deligne’s equivalence for ordinary abelian varieties.
1
0 Contents
2
1. Introduction 1
n
2. OntheubiquityofGorensteinringsamongminimalcentralorders 5
a
3. Remarksonreflexivemodules 11
J
4. Abelianvarietieswithminimalendomorphismalgebra 13
1
5. Constructionoftheanti-equivalence 15
1
6. PropertiesofthefunctorT 20

7. Ambiguityandcomparison 22
]
References 29
T
N
.
h
t 1. Introduction
a
m 1.1. Let p be a prime number, F an algebraic closure of the prime field F with p elements,
p p
[ and F ⊂ F the subfield with q elements, where q = pe is a power of p. The category
q p

1 AV
q
v
6 of abelian varieties over F is an additive category where for all objects A, B the abelian
q
4 groups HomF (A,B) are free of finite rank. Even though the main result of this paper concerns
4 q
abelian varieties over the prime field F , the general theme of our work is describing suitable
2 p
0 subcategories C of AVq by means of lattices T(A) functorially attached to abelian varieties A of
. C. In contrast to the characteristic zero case, if we insist that
1
0
rkZ(T(A)) = 2dim(A) (1.1)
5
1 then it is not possible to construct T(A) on the whole category AV (see Section §1.6). However,
q
:
v if we take C to be the full subcategory
i AVord
X q
r of ordinary abelian varieties, Deligne has shown that a functor A (cid:55)→ T(A) satisfying (1.1) exists
a and gives an equivalence between AVord and the category of finite free Z-modules T equipped
q
with a linear map F : T → T satisfying a list of axioms easy to state (cf. [De69] §7).
Inspired by a result of Waterhouse (cf. [Wa69], Theorem 6.1), in the present work we show
that a description in the style of Deligne can in fact be obtained, when q = p, for a considerably
larger subcategory C of AV , which excludes only a single isogeny class of simple objects of AV
p p
from occuring as an isogeny factor (see Theorem 1). Deligne’s method is an elegant application
of Serre–Tate theory of canonical liftings of ordinary abelian varieties, whereas our method,
closer to that used by Waterhouse, does not involve lifting abelian varieties to characteristic
Date: January 13, 2015.
The first author was supported by a project jointly funded by the DFG Priority Program SPP 1489 and the
Luxembourg FNR.
1
2 TOMMASOGIORGIOCENTELEGHEANDJAKOB STIX
zero. Even if the main result of this paper generalizes the q = p case of Deligne’s theorem, it is
unlikely that a proof generalizing Deligne’s lifting strategy be possible.
1.2. A Weil q–number π is an algebraic integer, lying in some unspecified field of characteristic
zero, such that for any embedding ι : Q(π) (cid:44)→ C we have
|ι(π)| = q1/2,
where |−| is the ordinary absolute value of C. Two Weil q–numbers π and π(cid:48) are conjugate to
each other if there exists an isomorphism Q(π) −→∼ Q(π(cid:48)) carrying π to π(cid:48), in which case we write
π ∼ π(cid:48). We will denote by
W
q
the set of conjugacy classes of Weil q–numbers. A Weil q–number is either totally real or totally
imaginary, hence it makes sense to speak of a non-real element of W .
q
Let A be an object of AV , denote by π : A → A the Frobenius isogeny of A relative to F .
q A q
If A is Fq–simple then EndFq(A)⊗Q is a division ring, and a well known result of Weil says
that π is a Weil q–number inside the number field Q(π ). Let
A A
(cid:89)
A ∼ Aei (1.2)
i
1≤i≤r
bethedecompositionofAuptoF –isogenyintopowersofsimple,pairwisenon–isogenousfactors
q
A . The Weil support of A is defined as the subset
i
w(A) = {π ,...,π } ⊆ W
A1 Ar q
given by the conjugacy classes of the Weil numbers π attached to the simple factors A . By
Ai i
Honda–Tate theory, the conjugacy classes of the π are pairwise distinct, moreover any class in
Ai
W arises as π , for some F –simple abelian variety A, uniquely determined up to F –isogeny
q A q q
(cf. [Ta68], Théorème 1).
1.3. Consider now the case q = p. Using Honda–Tate theory it is easy to see that for a simple
√
object A of AVp the ring EndFp(A) is commutative if and only if πA (cid:54)∼ p, i.e., if and only if the
Frobenius isogeny π : A → A defines a non-real Weil p–number (cf. [Wa69], Theorem 6.1). Let
A
AVcom
p
be the full subcategory of AV given by all objects A such that w(A) does not contain the
√ p
conjugacy class of p. Equivalently, AVcom is the largest full subcategory of AV closed under
p p
taking cokernels containing all simple objects whose endomorphism ring is commutative. Since
√
the Weil p–number p is associated to an F –isogeny class of simple, supersingular abelian
p
surfaces (cf. [Ta68], exemple (b) p. 97), we have a natural inclusion AVord ⊂ AVcom.
p p
The main result of this paper, proven at the end of Section §5.3, is the following.
Theorem 1. There is an ind-representable contravariant functor
A (cid:55)→ (T(A),F)
which induces an anti-equivalence between AVcom and the category of pairs (T,F) given by a
p
finite, free Z-module T and an endomorphism F : T → T satisfying the following properties.
(i) F ⊗Q is semisimple, and its eigenvalues are non-real Weil p–numbers.
(ii) There exists a linear map V : T → T such that FV = p.
Moreover, the lattice T(A) has rank 2dim(A) for all A in AVcom, and F is equal to T(π ).
p A
In order to prove the theorem we consider in Section §2 a family of Gorenstein rings
R = Z[F,V]/(FV −p,h (F,V))
w w
indexed by the finite subsets w ⊆ W , where h (F,V) is a certain symmetric polynomial built
p w
out of the minimal polynomials over Q of the elements of w. An object (T,F) in the target
Abelian varieties over Fp 3
category of the functor T(−) of Theorem 1 is nothing but an R -module, for w ⊂ W large
w p
enough, that is free of finite rank as a Z-module. In this translation, the linear map F : T → T
is given by the action of the image of F in R , and the relation h (F,V) in R encodes
w w w
precisely that F ⊗ Q acts semisimply and with eigenvalues given by Weil p-numbers lying in
w (see Sections §2.4, §2.5 and §3.2). Thanks to the Gorenstein property, these R -modules
w
are precisely the reflexive R -modules, the category that they form will be denoted by (see
w
Section §3)
Refl(R ).
w
For v ⊆ w, the corresponding rings are linked by natural surjective maps pr : R −→ R .
v,w w v
We denote by Rcom the pro-system (R ,pr ) with w ⊆ W ranging over the finite subsets
p √ w v,w p
avoiding the conjugacy class of p. We further set
Refl(Rcom) = lim Refl(R ).
p −→ w
√
w⊆Wp\{ p}
We refer to Section §3.2 for details.
In this language, Theorem 1 can be stated as saying that
T : AVcom → Refl(Rcom)
p p
is an anti–equivalence of categories. While this formulation of the main result is closer to the
perspective we adopted in its proof, the more concrete statement we chose to give above allows
an immediate comparison to Deligne’s result in [De69].
1.4. The rings R studied in Section §2 are in fact defined for any finite subset w ⊆ W . They
w q
appear naturally in connection to abelian varieties, in that for any A in AV the natural map
q
Z[F,V]/(FV −q) −→ EndF (A) (1.3)
q
sendingF toπ andV totheVerschiebungisogenyq/π inducesanidentificationbetweenR
A A w(A)
and the subring Z[πA,q/πA] of EndFq(A), which has finite index in the center (see Section 2.1).
The rings R have been already considered in [Wa69] and [Ho95] for example, however, to
w
our best knowledge, our observation that they are almost1 always Gorenstein remained so far
unnoticed (see Theorem 11).
AnRcom-linearstructureonAVcom canbededucedfromthemap(1.3)(seeSection§2.3). The
p p
requirement that F = T(π ) precisely means that the functor T(−) is an Rcom-linear functor
A p
(see Section §3.2).
1.5. The proof of the theorem consists of two steps. First, for any finite subset w ⊆ W not
√ p
containing the conjugacy class of p, we construct a certain abelian variety A isogenous to the
w
product of all simple objects attached to the elements of w via Honda–Tate theory. The object
A is chosen in its isogeny class with the smallest possible endomorphism ring, i.e., such that
w
the natural map
Rw → EndFp(Aw)
is an isomorphism (see Proposition 21). In order to show the existence of such an A , which
w
already appears in [Wa69] Theorem 6.1 if w consists of a single element, the assumption q = p
plays an important role. Exploiting the Gorenstein property of R , in Theorem 25 we show that
w
the functor HomFp(−,Aw) gives a contravariant equivalence
∼
HomFp(−,Aw) : AVw −→ Refl(Rw)
where AV is the full subcategory of AV given by all abelian varieties A with w(A) ⊆ w.
w p
1When [F :F ]=e is even we must require that the set w either contains both or none of the two rational
q p
Weilq–numbers±qe/2. Inparticular,R isGorensteinforacofinalsubsystemofthefinitesubsetsw⊆W with
w q
respect to inclusion.
4 TOMMASOGIORGIOCENTELEGHEANDJAKOB STIX
The second step consists in showing that the abelian varieties A previously constructed can
w
be chosen in such a way that the functors HomFp(−,Aw) interpolate well, and define a functor
on AVcom. More precisely we show the existence of an ind-system
p
A = (A ,ϕ ), (1.4)
w w,v
√
indexed by finite subsets w ⊆ W not containing the conjugacy class of p, such that the
p
corresponding direct limit of finite free Z–modules
T(A) = l−i→mHomFp(A,Aw)
w
stabilizes. The contravariant functor T(−) ind-represented by A will produce the required anti–
equivalence.
1.6. As Serre has observed, it is not possible to functorially construct a lattice T(A) satisfying
the expected rkZ(T(A)) = 2dim(A) on the category of abelian varieties over Fp. This is due
to the existence of objects like supersingular elliptic curves E over F . As is well known, the
p
division ring End (E)⊗Q is a non-split quaternion algebra over Q and has no 2-dimensional
F
p
Q–linear representation that can serve as T(E)⊗Q. That just described is the same obstruction
that prevents the existence of a Weil–cohomology for varieties over finite fields with rational
coefficients.
Using the same argument one can show the non-existence of a lattice T(A) as above on the
category AV , where q is a square. When q is not a square, the correct instance of Serre’s
q
observation, which prevents Theorem 1 from extending to all of AV , is given by the isogeny
p
class of F -simple, supersingular abelian surfaces associated via Honda–Tate theory to the real,
q √
non rational, Weil q–number q. The endomorphism ring of any such surface A is an order of
√ √
a quaternion algebra over Q( q) = Q( p) which is ramified at the two real places (cf. [Wa69]
p. 528). It follows that EndF (A) ⊗ R (cid:39) H × H is a product of two copies of the Hamilton
q
quaternions H. Thus it admits no faithful representation on a 4-dimensional real vector space,
such as T(A)⊗R would give rise to.
1.7. The dual abelian variety establishes an anti–equivalence A (cid:55)→ At of AV which preserves
q
Weil supports and has the effect of switching the roles of Frobenius and Verschiebung endomor-
phisms relative to F . This is to say that
q
(π )t = q/π
A At
as isogenies from At to itself. On the module side, we define a covariant involution of Refl(Rcom)
p
denoted by M (cid:55)→ Mτ which interchanges the roles of F and V, i.e., such that
(T,F)τ = (T,p/F).
UsingthesetwodualitieswecanexhibitacovariantversionofthefunctorT(−)ofTheorem1.
More precisely, define
T (A) = T(At)τ
∗
asthepairgivenbytheZ–moduleT(At)equippedwiththelinearmapp/T(π ). Inthenotation
At
as pairs T (A) takes the form
∗
(T(At),p/T(π )) = (T(At),T((π )t)) = (T (A),T (π )).
At A ∗ ∗ A
The functor T (−) gives a covariant, Rcom–linear equivalence
∗ p
T : AVcom → Refl(Rcom) (1.5)
∗ p p
whichispro-representedbythesystemAt = (At ,ϕt )dualto(1.4). InthedefinitionofT (−)
w w(cid:48),w ∗
it is necessary to apply the involution τ to T(At) in order to guarantee that T be Rcom–linear.
∗ p
In Section §7.4 we compare T (−) restricted to AVord with Deligne’s functor from [De69] §7
∗ p
thatwedenotebyT (−). Thecomparisonmakesuseofacompatiblepro-systemofprojective
Del,p
Abelian varieties over Fp 5
R -modules M of rank 1 for all finite subsets w ⊆ W consisting only of conjugacy classes of
w w p
ordinary Weil p–numbers. Proposition 44 then describes, for all abelian varieties A over F with
p
w(A) ⊆ w, a natural isomorphism
∼
T (A)⊗ M −→ T (A).
Del,p Rw w ∗
Furthermore, by choosing a suitable ind-representing system A = (A ,ϕ ) we may assume
w v,w
that M = R for all w, i.e., the anti-equivalence of Theorem 1 may be chosen to extend in its
w w
covariant version Deligne’s equivalence, see Proposition 45 for details.
1.8. Finally, we indicate how to recover the (cid:96)-adic Tate module T (A), for a prime (cid:96) (cid:54)= p,
(cid:96)
and the contravariant Dieudonné module T (A) (cf. [Wa69] §1.2) from the module T(A). This
p
involves working with the formal Tate module T (A) and the formal Dieudonné module T (A)
(cid:96) p
of the direct system A, respectively defined as the direct limit of T (A ) and the inverse limit of
(cid:96) w
the T (A ), with transition maps obtained via functoriality of T and T . More concretely we
p w (cid:96) p
have natural isomorphisms
T (A) (cid:39) Hom (T(A)⊗Z ,T (A)),
(cid:96) R(cid:96) (cid:96) (cid:96)
T (A) (cid:39) (T(A)⊗Z )⊗ˆ T (A),
p p Rp p
see Proposition 27 and 28 for notation and proofs. In this respect the functor T(−) can be
interpreted as an integral lifting of the Dieudonné module functor T (−).
p
In a forthcoming paper we will apply the method used here to study certain categories of
abelian varieties over a finite field which is larger that F . Therefore, although Theorem 1 deals
p
with abelian varieties over F , we only restrict to the case q = p when it becomes necessary.
p
Acknowledgments. The authors would like to thank Gebhard Böckle for stimulating discus-
sions and for his suggestion of the symmetric polynomial h (F,V). We thank Filippo Nuccio
π
for valuable comments on an earlier version of the manuscript, and Hendrik Lenstra for his
interesting observations on local complete intersections. Special thanks go to Brian Conrad and
Frans Oort for their attentive reading of a preliminary version of our work, and for the prompt
and interesting feed back they gave us. Finally we thank the anonymous referees for their quick
work in reviewing the paper.
2. On the ubiquity of Gorenstein rings among minimal central orders
2.1. Minimal central orders. Let w ⊆ W be any finite set of conjugacy classes of Weil q–
q
numbers. Choose Weil q–numbers π ,...,π representing the elements of w, and consider the
1 r
ring homomorphism
(cid:89)
Z[F,V]/(FV −q) −→ Q(π ) (2.1)
i
1≤i≤r
sending F to (π ,...,π ) and V to (q/π ,...,q/π ).
1 r 1 r
Definition 2. The minimal central order R is the quotient
w
Z[F,V]/(FV −q) → R (2.2)
w
by the kernel of the homomorphism (2.1). The image of F in R will be denoted by F , and
w w
the image of V by V .
w
The construction of the ring R is independent of the chosen Weil q–numbers in their respec-
w
tive conjugacy classes. When w consists of a single conjugacy class of a Weil number π, the ring
R , isomorphic to the order of Q(π) generated by π and q/π, will sometimes be denoted sim-
{π}
ply by R . Since the representatives π ,...,π are pairwise non–conjugate, there is a canonical
π 1 r
finite index inclusion
(cid:89)
R ⊆ R ,
w π
π∈w
6 TOMMASOGIORGIOCENTELEGHEANDJAKOB STIX
in particular
(cid:89)
R ⊗Q = Q(π). (2.3)
w
π∈w
Moreover, for finite subsets v ⊆ w ⊆ W we have a natural surjection
q
pr : R −→ R .
v,w w v
Our main goal in this section is showing that, under a mild assumption on w, the ring R
w
is a one dimensional Gorenstein ring. This will be proved in Section 2.5, where we obtain a
description of R by identifying the relations between the generators F and V.
w
Example 3. The equality of closed subschemes
Spec(R ) = (cid:91) Spec(R ) ⊆ Spec(cid:0)Z[F,V]/(FV −q)(cid:1)
w π
π∈w
shows that the spectrum of R is obtained by glueing the spectra of the rings R along their
w π
various intersections inside Spec(cid:0)Z[F,V]/(FV −q)(cid:1). This roughly means that it is the congru-
ences between Weil q-numbers who are responsible for R differing from the product of the R
w π
for all π ∈ w.
(cid:81)
We measure in a special situation the deviation of R from being isomorphic to R .
w π∈w π
Let π for i = 1,2 be quadratic Weil q-numbers with minimal polynomial
i
x2−β x+q,
i
where β ∈ Z, and set ∆ = β −β . Since q/π = β −π we have
i 1 2 i i i
R = Z[π ] (cid:39) Z[x]/(x2−β x+q),
πi i i
moreover the subring R ⊆ Z[π ]×Z[π ] is generated as a Z-algebra by
w 1 2
(0,∆),(π ,π ) ∈ Z[π ]×Z[π ],
1 2 1 2
since it is generated by (π ,π ) and (β −π ,β −π ). Because β ≡ β modulo ∆, there are
1 2 1 1 2 2 1 2
isomorphisms of quotients
Z[π ]/∆Z[π ] (cid:39) Z[π ]/∆Z[π ] =: R
1 1 2 2 0
and R becomes the fibre product
w
R = Z[π ]× Z[π ],
w 1 R0 2
which is an order of index ∆2 in the product R ×R . The congruences between π and π
are encoded by the closed subscheme of Spec(cid:0)Z[πF1,V]/π(F2 V −q)(cid:1) given by 1 2
Spec(R ) = Spec(R )∩Spec(R ).
0 π1 π2
Note that the minimal polynomials x2−β +q yield Weil q-numbers if and only if
i
β2 < 4q.
i
In particular, by letting q range over the powers of the prime p, the Weil q-numbers π may be
i
chosen so as to have ∆ divisible by an arbitrary integer.
Abelian varieties over Fp 7
2.2. Connection to abelian varieties. We proceed to link R to abelian varieties over F .
w q
Any such A has two distinguished isogenies given by the Frobenius π and the Verschiebung
A
q/πA relative to Fq. The Q–algebra EndFq(A) ⊗ Q is semi–simple, and its center is equal to
the sub–algebra Q(π ) generated by π (cf. [Ta66], Theorem 2). It follows that any isogeny
A A
decomposition of A, as in (1.2), induces the isomorphism
(cid:89)
Q(π ) (cid:39) Q(π ), (2.4)
A Ai
πAi∈w(A)
sending π to (π ,...,π ), where π ,...,π are the Weil q–numbers defined by the simple
A A1 Ar A1 Ar
factors of A, and w(A) is the Weil support of A as defined in the introduction.
From (2.4) we deduce that the ring homomorphism
rA : Z[F,V]/(FV −q) −→ EndFq(A)
sending F to π and V to q/π gives an identification between R and the image of r ,
A A w(A) A
namely the subring
Z[π ,q/π ]
A A
which sits inside the center of EndFq(A) with finite index. In this way we see that Rw(A) plays
the role of a lower bound for the center of EndF (A). This justifies the terminology we chose in
q
its definition.
Remark 4. One can raise the question of whether there exists an abelian variety A with Weil
support w such that the natural map Rw → EndFp(A) induced by rA gives an isomorphism
between Rw and the center of EndFp(A). In Proposition 21 below, generalizing a result of
Waterhouse, we obtain a partial result in this direction.
2.3. Linear structures over minimal central orders. For a finite subset w ⊆ W the full
q
subcategory
AV ⊆ AV
w q
consists of all abelian varieties A such that w(A) ⊆ w or, equivalently, such that r factors
A
through the quotient Z[F,V]/(FV −q) → R . Since for any morphism f : A → B in AV and
w q
any η ∈ Z[F,V]/(FV −q) the diagram
f (cid:47)(cid:47)
A B (2.5)
rA(η) (cid:15)(cid:15) (cid:15)(cid:15) rB(η)
f (cid:47)(cid:47)
A B
is commutative, as follows from the naturality of the Frobenius and Verschiebung isogenies, we
deduce an R –linear structure on the category AV . Furthermore, for finite subsets v ⊆ w the
w w
R –linear structure on AV induced by the fully faithful inclusion AV ⊆ AV is compatible, via
w v v w
the surjection pr , with the R –linear structure on AV .
v,w v v
Remark 5. If W ⊆ W is now any subset, denote by R the projective system (R ,pr ) as w
q W w w,v
ranges through all finite subsets of W, and by
AV
W
the full subcategory of AV whose objects are all abelian varieties A with w(A) ⊆ W. We will
q
treat AV as the direct 2-limit of the categories AV , for w a finite subsets of W. The collection
W w
of R –linear structures on the subcategories AV ⊆ AV , which are linked by the compatibility
w w W
conditions described above, form what we will refer to as an R –linear structure on AV .
W W
8 TOMMASOGIORGIOCENTELEGHEANDJAKOB STIX
2.4. The symmetric polynomial. Let π be a Weil q–number. If Q(π) has a real place then
π2 = q, so that Q(π) is totally real, and [Q(π) : Q] is either 2 or 1 according to whether the
degree e = [F : F ] is odd or even, respectively. In the first case there is only one conjugacy
q p
class of real Weil q–numbers, in the second one there are two of them, given by the rational
integers qe/2 and −qe/2. In the general case where π is not real, the field Q(π) is a non-real CM
field, with complex conjugation induced by π (cid:55)→ q/π.
The degree 2d = [Q(π) : Q] is even, except for the two rational Weil q–numbers occurring for
e even. Set
P (x) = x2d+a x2d−1+...+a x+a ∈ Z[x]
π 2d−1 1 0
for the normalized minimal polynomial of π over Q, and accept that d = 1/2 in case π ∈ Z. The
polynomial P (x) depends only on the conjugacy class of π. The following lemma is well known
π
(cf. [Ho95], Prop. 3.4).
Lemma 6. Let π be a non-real Weil q–number. For r ≥ 0, we have a = qra .
d−r d+r
Proof. We can arrange the roots α ,...,α of P (x) so that α and α are complex con-
1 2d π i 2d+1−i
jugates of each other, which is to say α α = q. For a subset I ⊆ {1,...,2d} we set
i 2d+1−i
Ic = {1,...,2d}\I, and I = {i ; 2d+1−i ∈ I}, and moreover we use the multiindex notation
αI = (cid:81) α . Then with summation over subsets of {1,...,2d} we compute
i∈I i
2d
(−1)d+ra = (cid:88) αI = (cid:0)(cid:89)α (cid:1)· (cid:88) 1
d−r i αIc
|I|=d+r i=1 |I|=d+r
= qr · (cid:88) qd−r = qr · (cid:88) αJ = qr(−1)d−ra ,
αJ d+r
|J|=d−r |J|=d−r
and this proves the lemma. (cid:3)
We next construct a symmetric polynomial h (F,V) ∈ Z[F,V]. The idea is to consider the
π
rational function P (F)/Fd ∈ Z[F,q/F] (at least when d ∈ Z), and then formally set V = q/F.
π
Definition 7. We define the symmetric polynomial h (F,V) attached to a Weil q–number
π
π as follows:
(1) If π is a non-real Weil q–number, then we set in Z[F,V]
h (F,V) = Fd+a Fd−1+...+a F +a +a V +...+a Vd−1+Vd.
π 2d−1 d+1 d d+1 2d−1
√
(2) If π = ±pm p is real but not rational, then we set
h (F,V) = F −V ∈ Z[F,V].
π
(3) If π = ±pm is rational, then we set
h (F,V) = F1/2∓V1/2 ∈ Z[F1/2,V1/2].
±pm
The polynomial h (F,V) just defined appears already in [Ho95], §9.
w
Lemma 8. (1) If π is a non-real Weil q–number, then we have h (π,q/π) = 0.
π
(2) If π is a real, but not rational Weil q-number, then h (F,V) = F−V and h (π,q/π) = 0.
π π
(3) If π = ±pm is rational, then h (F,V) · h (F,V) = F − V is again contained in
pm −pm
Z[F,V], and vanishes for F = π and V = q/π.
Proof. Assertion (1) follows from h (π,q/π) = P (π)/πd = 0 which is based on Lemma 6.
π π
Assertion (2) and (3) are trivial. (cid:3)
Definition 9. An ordinary Weil q–number is a Weil q–number π such that exactly half of the
rootsofitsminimalpolynomialP (x)inanalgebraicclosureofQ arep-adicunits. Equivalently,
π p
the isogeny class of abelian varieties over F associated to π by Honda–Tate theory is ordinary.
q
Abelian varieties over Fp 9
If π is ordinary then Q(π) is not real, and precisely half of the roots of the even degree
polynomial P (x) are p-adic units.
π
Lemma 10. Let w ⊆ W be a finite subset of non-real conjugacy classes of Weil q–numbers.
q
Then w consists of ordinary conjugacy classes, if and only if h (0,0) is not divisible by p.
w
(cid:81)
Proof. Let α ,...,α ,q/α ,...,q/α be the roots of P (x). Then
1 d 1 d π∈w π
d
(cid:89)
h (F,V) ≡ (F −(α +q/α )+V) mod (FV −q)
w i i
i=1
so that
d
(cid:89)
h (F,V) ≡ (−1)d (α +q/α ) mod p.
w i i
i=1
This integer is not divisible by p if and only if for all i the algebraic integers α +q/α are p-adic
i i
units. This happens if and only if for all i either α or q/α are p-adic units, hence if w consists
i i
of ordinary conjugacy classes. (cid:3)
2.5. Structure of the minimal central orders. In what follows we will define for a finite
subset w ⊆ W the degree of w by
q
(cid:88)
deg(w) = rkZ(Rw) = [Q(π) : Q].
π∈w
So w is of even degree if and only if w either contains none or both rational Weil q–numbers
±qe/2, which only show up when e = [F : F ] is even. Extending this notion, we will say that
q p
an arbitrary subset W ⊆ W is of even degree if either none or both rational conjugacy classes
q
of Weil q-numbers belong to W.
If w ⊆ W is any finite subset we set
q
(cid:89)
h (F,V) = h (F,V),
w π
π∈w
which is contained in Z[F,V] as soon as w is of even degree.
Theorem 11. Let w ⊆ W be a finite set of Weil q–numbers of even degree.
q
(1) We have R = Z[F,V]/(FV −q,h (F,V)).
w w
(2) The ring R is a 1-dimensional complete intersection, in particular it is a Gorenstein ring.
w
Proof. The ring R is reduced as it injects into a product of number fields. Moreover, R is a
w w
finite Z-algebra, because it is generated by F and V that satisfy integral relations in R . Thus
w
R is free of finite rank as a Z-module and of Krull dimension 1. More precisely, by (2.3) we
w
have
(cid:88)
rkZ(Rw) = [Q(π) : Q] =: 2D
π∈w
The ring Z[F,V]/(FV − q) is a normal ring with at most one rational singularity in p =
(F,V,p). Hence, h (F,V) is a non-zero divisor in Z[F,V]/(FV −q) and it remains to show (1)
w
to conclude the proof of (2).
We now show assertion (1). By Lemma 8 the evaluation of h (F,V) in R vanishes for all
w π
π ∈ w. Hence we obtain a surjection
ϕ : S = Z[F,V]/(FV −q,h (F,V)) (cid:16) R .
w w
We are done if we can show that S is generated by 2D elements as a Z-module.
By construction, h (F,V) is a product of polynomials of the form
w
f (F)+g (V)
π π
10 TOMMASOGIORGIOCENTELEGHEANDJAKOB STIX
with f ,g ∈ Z[X] monic (or −g monic). The degrees are deg(f ) = deg(g ) = [Q(π) : Q]/2
π π π π π
if π is non-rational, and 1 if π is rational. Having a representative of the form f(F)+g(V) for
monic polynomials f,g (or −g) of the same degree is preserved under taking products:
(cid:0) (cid:1)(cid:0) (cid:1)
f (F)+g (V) f (F)+g (V) = f f (F)+g g (V)+lower degree terms in F,V,
1 1 2 2 1 2 1 2
where the mixed terms are of lower degree, because FV = q necessarily leads to cancellations.
Hence the same holds for the product: h (F,V) = f(V)+g(V) with deg(f) = deg(g) = D.
w
In particular
FD,FD−1,...,F,1,V,...,VD−1
generate S as a Z-module. (cid:3)
Part(1)ofTheorem11hasalreadybeenobservedbyHowe,atleastforwordinary(cf. [Ho95],
Prop. 9.1). On the other hand the Gorensteinness of R , so crucial in the present work, seems
w
to have remained unnoticed so far.
Since Theorem 1 deals with abelian varieties over F , our main concern in this paper are
p
the commutative algebra properties of R for finite subsets of W . Here Theorem 11 covers all
w p
cases. In order to complete the picture we answer what happens if w ⊆ W contains exactly one
q
rational conjugacy class of Weil q–numbers.
√
Theorem 12. Let q be the square of a positive or negative integer q ∈ Z. Let v ⊆ W be
√ q
a finite set containing no rational conjugacy class, and set w = v ∪{ q}. Then the following
holds.
√ √
(1) We have R = Z[F,V]/(FV −q,h (F,V)(F − q),h (F,V)(V − q)).
w v v
(2) The ring R is Gorenstein if and only if all conjugacy classes of Weil q-numbers in v are
w
ordinary.
Proof. ReasoningasinLemma8, weseethatthedefiningquotientmapZ[F,V]/(FV −q) → R
w
factors as a surjective map
√ √
S = Z[F,V]/(FV −q,h (F,V)(F − q),h (F,V)(V − q)) (cid:16) R .
v v w
As in Theorem 11 the ring R as a Z-module is free of rank
w
(cid:88)
rkZ(Rw) = 1+ [Q(π) : Q] =: 2D+1.
π∈v
It is easy to see that S is generated as a Z-module by
FD,FD−1,...,F,1,V,...,VD.
This shows assertion (1) as above.
For assertion (2) we first note that after inverting one of the elements p, F or V the three
relations can be reduced to two relations, so that outside of (p,F,V) the ring R is a local
w
complete intersection and hence Gorenstein. It remains to discuss the local ring in p = (p,F,V).
There is a unique polynomial h ∈ Z[X] such that
h (F,V) = h(F)−h(0)+h(V) ∈ Z[F,V],
v
and for this h we have h(0) = h (0,0). Since Z is regular (hence Gorenstein) and R is a flat
v w
Z-algebra, it follows from [Ma89] Theorem 23.4 that R is Gorenstein in p if and only if
w
R /pR = F [F,V]/(FV,h(F)F,h(V)V)
w w p
is Gorenstein in p¯ = (F,V). The ring R /pR is Artinian, hence of dimension 0, so that by
w w
[Ma89] Theorem 18.1 the ring (R /pR ) is Gorenstein if and only if
w w ¯p
1 = dimFpHom(κ(p¯),Rw/pRw).
The space of homomorphisms has the same dimension as the socle, i.e., the maximal submodule
annihilated by (F,V). The socle is the intersection of the kernels of F and V as F -linear maps
p