The isomorphism classes of abelian varieties of
CM-type
Chia-Fu Yu1
Abstract
We prove that any abelian variety with CM by OL of characteristic p is defined
over a finite field, where OL is the ring of integers of the CM field L. This
general-izes a theorem of Grothendieck on isogeny classes of CM abelian varieties. We also provide a direct proof of the Grothendieck theorem, which does not require several ingredients based on Weil’s foundation as the original proof does. A description of the isomorphism classes is given. We analyze the reduction map modulo p for the abelian varieties concerned and solve the lifting and algebraization problem.
2000 Mathematics Subject Classification: 11G15, 14K22
§1. Introduction
(1.1) A basic result of the theory of complex multiplication says that any abelian variety A of CM-type over C is defined over a number field. By an abelian variety of CM-type we understand the same terminology that an abelian variety admits sufficiently many complex multiplication (see (2.1)(a)). The well-known proof of this result is using the fine mod-uli scheme. Let O be intersection of the CM field L with End(A). Choose an O-linear polarization λ : A → At, say of degree d2. This can be done by complex uniformization
L ⊗ R/a → A(C) and choose a Riemann form TrL/Q(ξx¯y) with an appropriate element ξ
in L with respect to the CM type. Then one shows that the set of isomorphism classes of polarized abelian O-varieties of fixed polarization degree with a suitable level structure is finite. This is achieved by counting the equivalence classes of lattices a in L and all compat-ible complex structures on L ⊗ R. As the fine moduli scheme is of finite type over Q, all the points in this moduli space are defined over a number field.
(1.2) This argument does not stand anymore in the case of characteristic p. A theorem of Grothendieck ([M1, p. 220], also see [O, Thm. 1.1]) says that any abelian variety A of CM-type over a field k of characteristic p is isogenous, over a finite field extension of k, to an abelian variety A1 that is defined over a finite field. The statement “up to isogeny”
is necessary for the ground field descending down to a finite field. It is well-known (eg. in [LO]) that any supersingular abelian variety is of CM-type, but not all of them are defined over finite fields. Let (A, λ) be a generic principally polarized supersingular abelian variety of dimension g > 1 over k. The endomorphism algebra End0k¯(A) is isomorphic
to Mg(Dp), where Dp is the definite quaternion algebra over Q of discriminant p. Let
L be a CM field of dimension 2g in End0¯k(A) stable under the Rosati involution and let
O := L ∩ End¯k(A). The argument fails because the moduli space of polarized abelian
O-varieties is not finite over Fp. The order O above is not the ring of integers and this might
be the main reason that the family is not discrete. A natural question arose is that if End(A) ⊃ OL or End(A) ⊗ Zp ⊃ OL⊗ Zp, is A defined over a finite field? In this note we
prove the following
(1.3) Theorem Let A be an abelian variety over a field k of characteristic p such that End(A) ⊗ Zp ⊃ OL⊗ Zp, where OL is the ring of integers of a CM field L of degree 2 dim A.
Then A¯k is defined over a finite field.
(1.4) Theorem (Grothendieck, Oort) Up to isogeny and over a finite field extension, any abelian variety of CM-type over k is defined over a finite field.
(1.5) We recall the ingredients of the original proof of Theorem 1.4 given in [O]. The proof in loc. cit. is divided into two steps:
(1) Show that any abelian variety of CM-type over a field K with a discrete valuation v has potentially good reduction.
(2) When the abelian variety A has good reduction at v and the ground field K is finitely generated over the perfect residue field k = k(v), then A is isogenous to the K/k-trace of A over K (see [L, VIII.3] or [O, p.404])
(1) is achieved by carefully analyzing the Galois representation on the Tate module and using the results of Serre-Tate [ST] on the criteria of good reduction and potentially good reduction. The existence of N´eron models is needed. (1) is also a consequence of Grothendieck’s semi-table reduction theorem. The proof of (2) applies the Lang-N´eron generalization of the Mordell-Weil theorem [LN].
The existence of N´eron models and Lang-N´eron’s theorem were established under the old fashion foundation of Weil. Therefore, it is desirable to have a simple proof of Theorem 1.4 not depending on the old foundation.
(1.6) The idea of our proof is using the fine moduli scheme again (1.1). The logical order is the following:
1. Show that any abelian variety with CM is isogenous to abelian variety with an OL
-action and with a separable OL-linear polarization.
2. Show that a fine moduli scheme of polarized abelian OL-varieties of dimesnion g and
of a prime-to-p polarization degree is finite. Hence Theorem 1.4 follows.
3. After having established Theorem 1.4, we can count the isomorphism classes of polar-ized abelian OL-varieties of given “local conditions”. Then we can show that the fine
moduli scheme of polarized abelian OL-varieties of dimesnion g and of fixed polarization
degree is finite. Thus Theorem 1.3 follows.
(1.7) In the proof of our main theorem, we classify the isomorphism classes of p-divisible groups of CM-type (over an algebraically closed field of characteristic p). The classification of the isomorphism classes of the Lie algebras of these p-divisible groups gives us some invariant (called Lie type in section 4). In section 5, we compute the Lie type of abelian varieties which are arising from abelian varieties of CM-type (L, Φ) of characteristic 0. With this information, we give a necessary and sufficient condition for a polarized abelian OL
-varieties in characteristic p to be lifted to that of characteristic 0. In this note, schemes are all locally noetherian.
Acknowledgments. Part of this work was done during my stay in National Center for Theoretic Sciences in Taiwan. I would like to thank the Center for the kind hospitality and the excellent working condition.
§2. Terminology and preliminary
(2.1) Terminology. (a) Let k be a field of any characteristic. Recall that an abelian variety A is of CM-type over k, or an abelian variety admits smCM (sufficiently many complex multiplication) over k if there exists a commutative subalgebra L of E := End0k(A) such that [L : Q] = 2 dim A. An equivalent definition is that A is isogenous to Q Ani
i over k with
each factor Ai k-simple such that the division algebra End0k(Ai) contains a field Li of degree
2 dim Ai.
After a finite extension of k, we may and do assume that all endomorphisms of A are defined over k. In this case, if chark = 0 then Li = End0(Ai) is a CM-field and End0(A) =
Q Mni(Li). When chark = p, End
0
(Ai) is a division algebra over Q that admits a positive
involution. Note that a division algebra with a positive involution may contain a non-CM maximal commutative subfield. A priori, the maximal commutative subalgebra L in E may not be a product of CM fields. However, we can always find one CM maximal commutative subalgebra in E which is stable under a Rosati involution. See a proof in (2.2) below. Hence we may assume that L is a CM subalgebra.
(b) Let B be a finite dimensional semi-simple algebra over Q with a positive involution ∗. Let OB be an order of B stable under the involution ∗. Recall that a polarized abelian
OB-variety is a triple (A, λ, ι) where A is an abelian variety, ι : OB → End(A) is a ring
monomorphism and λ : A → Atis a polarization satisfying the compatible condition λι(b∗) = ι(b)tλ for all b ∈ OB.
Let A be an abelian variety up to isogeny with ι : B → End(A). Then dual abelian variety At admits a natural action by B by ιt(b) := ι(b∗)t. The compatible condition above
is saying that the polarization λ : A → At is O
B-linear.
(c) Let Bp be a finite dimensional semi-simple algebra over Qp with an involution ∗.
p-divisible Op-group is a triple (G, λ, ι) where G is a p-divisible group, ι : Op → End(G) is
a ring monomorphism and λ : G → Gt is a quasi-polarization (i.e. λt = −λ) such that
λι(b∗) = ι(b)tλ for all b ∈ O
p. For convenience, we also introduce the term “quasi-polarized
Dieudonn´e Op-modules” for the associated covariant Dieudonn´e module to a quasi-polarized
p-divisible Op-group over a perfect field k of characteristic p. It is a Dieudonn´e module M
over k equipped with a W := W (k)valued nondegenerate alternating pair h , i and W -linear action by O satisfying the usual condition hax, yi = hx, a∗yi and hF x, yi = hx, V yiσ
for all a ∈ Op and x, y ∈ M .
(2.2) Lemma Let B be a finite-dimensional simple algebra over Q with a positive involution ∗. Then there exists a ∗-invariant maximal subfield of B.
Proof. Let K be the center of B. Let L1 = K(α) be a maximal subfield of B. If L1 is
∗-invariant, then we are done. Suppose not. Set β := α + α∗ and L
2 := K(β). Note that
β 6∈ K, otherwise α∗ would be in L1. As L2 is ∗-invariant, the commudant C of L1 in B is
also stable under the involution. Replacing B by C, K by L1, we reduce the degree [B : K].
By induction, the proof is completed.
Let A be an abelian variety of CM-type over k and λ be a polarization on A. Write End0(A) = ⊕Bi, where Bi = End0(Ai) is simple Q-algebra with a positive involution ∗
induced by λ. Each factor Bi contains a ∗-invariant subfield Li of dimension 2 dim Ai. It
follows that Li is a CM field and End0(A) contains ⊕Li, a CM algebra of degree 2 dim A.
Furthermore, we can choose Ai in the isogeny class so that OLi ⊂ End(Ai).
(2.3) Proposition Let A be an abelian B-variety up to isogeny over k. Then there exists a B-linear polarization λ : A → At.
This is Lemma 9.2 in Kottwitz [K2]. The proof only uses semi-simple algebras with positive involutions and the knowledge on the polarization cone of the abelian variety. We refer the reader to [K2, sec. 2 and 9] for the well-written proof.
§3. The proof of the theorems
(3.1) In the rest of this note, k will denote an algebraically closed field of characteristic p > 0. Let L be a CM field of degree 2g and L0 be the maximal totally real subfield of L.
We denote by x 7→ ¯x the nontrivial element of Gal(L/L0). Let (A, λ, ι) be a polarized abelian
OL-variety of dimension g and let H be the attached p-divisible group. The decomposition
OL0 ⊗ Zp = ⊕v|pOv gives rise to the decomposition for the p-divisible group H = ⊕v|pHv.
Each Hv is a quasi-polarized p-divisible OLv-group of height 2gv, where gv := [L0,v : Qp]. If v is inert or ramified in L, then Lw (w|v) has degree 2gv over Qp. By [K1, Sect. 3], Hv
corresponding decomposition. Then Hw (resp. Hw¯) is a p-divisible group of height gv, hence
it has single slope, which is dim Lie(Hw)/gv (resp. dim Lie(Hw¯)/gv). Note that the constant
´
etale group Qp/Zp has dimension 0 and single slope 0.
Let ev and fv denote the ramification index and residue degree of v respectively. Let πv
be a uniformizer of Ov. Write dw = dim Hw and dw¯ = dim Hw¯.
(3.2) We will find for each v a separably quasi-polarized p-divisible OLv-group (H
0 v, λ 0 v, ι 0 v)
such that there is an OLv-linear isogeny φ : H
0
v → Hv with φ∗λv = λ0va for some a ∈ L×v.
Then by a theorem of Tate, there is a separably polarized abelian OL-variety (A0, λ0, ι0) and
an OL-linear isogeny φ : A0 → A such that φ∗λ = λ0a for some a ∈ L×.
(3.3) We shall simplify the notations (in (3.3)-(3.4)) by replacing L and L0 by their
com-pletion at v. The Qp-algebra L is either a quadratic field extension of the field L0 or is
Lw× Lw¯ with Lw = Lw¯ = L0. Let e and f be the ramification index and residue degree of
L0/Qp respectively. Let (H0, λ0, ι0) be a quasi-polarized p-divisible OL-group. Then there is
an OL-linear (quasi-)polarization λ0 of minimal degree such that any OL-linear polarization
is of the form λ0a for some a ∈ OL. It follows from (λ0a)t = −(λ0a) that a ∈ OL0. If λ1 is another OL-linear polarization of minimal degree, then λ1 = λ0a for some a ∈ O×L0. Let
b ∈ EndOL(H
0)×, then the pull-back b∗λ
0 is λ0(¯bb). Therefore, all OL-linear polarizations (of
minimal degree) on H0 are classified by O×L
0/NL/L0(O
× L).
If two p-divisible OL-group H1 and H2 are isogenous, then we can exhibit an OL-linear
isogeny by the Noether-Skolem theorem. Indeed, given an isogeny φ : H1 → H2, we get a
ring monomorphism ι0 : L → End0(H1) given by ι0(a) = φ−1ι2(a)φ. By the Noether-Skolem
theorem, there is an element c ∈ End(H1) such that ι1(a) = c−1ι0(a)c for all a ∈ L. Replacing
φ by φc, we obtain an OL-linear isogeny.
Choose a uniformizer π of L0, and a uniformizer πw of L with πw2 = π if L/L0 is ramified.
Let Onr be the maximal unramified extension of Zp in OL0. We identify the set of embed-dings Hom(Onr, W (k)) = {σi} with Z/fZ. The Lie algebra of H has a decomposition into
eigenspace under the action of Onr:
Lie(H) = ⊕i∈Z/f ZLie(H)i.
(3.4) Lemma (a) The isomorphism classes of p-divisible OL-groups H over k are classified
by the isomorphism classes of Lie(H) as OL⊗ k-modules.
(b) Any OL-linear polarization of minimal degree is separable if and only if dimk(Lie(H)i)
are the same for all i ∈ Z/f Z.
The proof will be given in Sect. 4. We shall also write down an p-divisible group with the property (b). Hence we have proved
(3.5) Proposition Let notations be as in (3.1). For any polarized abelian OL-variety
(A, λ, ι), there is a separable polarized abelian OL-variety (A0, λ0, ι0) and an OL-linear isogeny
(3.6) Let L, v, dw be the notations in (3.1). Let N ≥ 3 and d be positive integers with
(N, pd) = 1. Let Fq be a finite field containing the residue fields of L over p and N
-th roots of unitary. Denote by W (Fq) the ring of Witt vectors over Fq. We choose a
trivialization ζN : Z/N Z ' µN over W (Fq) and fix a perfect truncated skew-Hermitian form
ψ0 : N1OL/OL× N1OL/OL → Z/NZ. We consider the moduli problem over W (Fq) which
associates a W (Fq)-scheme S the isomorphism classes of the data (A, λ, ι, η)S, where
• (A, λ, ι) is a polarized abelian OL-scheme of relative dimension g over S with
polariza-tion degree d2 such that Lie(A)w is a locally free OS-module of rank dw for all v = w ¯w
split in L, • η : 1
NOL/OL ' A[N ](S) is an OL-linear isomorphism such that the pull of the Weil
pairing is ψ0 via the trivialization ζN.
The moduli problem is representable by a separated scheme of finite type over W (Fq) by the
standard result [M2]. We denote the fine moduli scheme by M.
(3.7) Proposition Let notations as before. Then MFq is finite over Fq provided (d, p) = 1. It suffices to show that dimMFq is zero. One can check that this moduli space is not smooth, hence tangent space calculation does not compute the dimension. However, one can compute the dimension easily by Rapoport-Zink’s local models [RZ] (also due to Deligne-Pappas [DP]). For our purpose, we formulate a special case. The reader is referred to [RZ] for generalization.
Let D := (K, K0, R, ψ) be a datum as follows:
• K is a finite dimensional commutative semi-simple Qp-algebra with an involution a 7→
¯
a. and K0 is the subalgebra of fixed elements by the involution.
• ψ is a Qp-valued non-degenerate skew-Hermitian form on K,
• R be a self-dual OK-lattice in K, where OK is the maximal order of K.
We may take R = OK and ψ a perfect pairing on OK in the datum. For each datum D,
we associate a projective scheme N[D] over Zp: For any Zp-scheme S, the set N[D](S) of
S-points consists of locally free OS-submodules F of R ⊗ZpOS which is maximally isotropic and stable under the OK-action.
After a finite ´etale base change, the local modelN[D] becomes a product of simpler local models N[K0, K00, ψ0, R0], where K00 is a field totally ramified over Qp and K0 is either a field
ramified over K00 or K0 = K00 × K0
0. We may replace D by D
0. In the product case, the
local model N[D] has a closed immersion into another local model N0[K0, R0] ×N0[K0, R0],
where N0[K0, R0] the scheme that parameterizes locally free OS-submodules of R0⊗Zp OS. The immersion simply sends F = F1⊕ F2 to (F1, F2). In the ramified case, N[D] is a closed
When S = SpecQp, we have N0[K, R](Qp) ' N0[Qp, Zp](Qp)[K:Qp]. When S = Speck, an
object inN0[K, R](k) is a submodule of k[T ]/Te, which is of the form Td·k[T ]/Te. This shows
that N := N[D] is quasi-finite over Zp. Therefore, N is finite over Zp and has 0-dimensional
fibres.
Put K = L ⊗ Qp and suitable ψ, R in D. By [RZ, Prop. 3.33, p. 92] (also see [DP, p. 60]),
for any geometric point x of M there is a point y such that M∧x ' N∧
y This shows that
dimMFq is zero. Proposition 3.7 is proved.
By Proposition 3.5 and 3.7, we proved Theorem 1.4. The rest of this section is to prove (3.8) Theorem The moduli scheme MFq is finite over Fq.
(3.9) Proof of Theorem 1.3. Let A be an abelian variety over k such that End(A) ⊃ OL⊗Z(p).
Then there is a prime-to-p isogeny φ : A0 → A with End(A) ⊃ OL. By Proposition 2.3, we
equip A0 with the additional structures and get a point in M(k). By Theorem 3.8, we have M(k) = M(Fp), therefore A0 is defined over Fp. The kernel ker φ is a finite subgroup scheme
of A0[N ] ⊗
Fp k over k for some (N, p) = 1. As A
0[N ]
Fp is a constant group, ker φ is defined over Fp and hence the same for A. We finish the proof.
(3.10) Lemma Let (Ai, λi, ιi) ∈M(k) for i = 1, 2. Then
(a) (A1, ι1) is isogenous to (A2, ι2) over k.
(b) The map ϕp : HomOL(A1, A2) ⊗ Zp → HomOL(A1[p
∞], A
2[p∞]) over k is an
isomor-phism
Proof. (a) By Theorem 1.4, we may assume that k = ¯Fp as the statement is up to isogeny.
Suppose Ai are defined over a finite field k0 and let πi be their Frobenius endomorphism.
With the isomorphism OL ' EndOL(Ai), we embed Q[πi] into L. The normalized p-adic valuations w(πi) are dw/g, 1/2, 1/2 according to v is split, inert, or ramified in L. Then
π1/π2 is a root of unity in L and we have πn1 = π2n for some n. By the Honda-Tate theory, Ai
are isogenous (over the degree n extension of k0). We can adjust the isogeny to be OL-linear
by the Noether-Skolem Theorem.
(b) As the map ϕp is co-torsion free, we can replace Zp by Qp. If the source of ϕp is
non-empty, then both the source and the target are isomorphic to L ⊗ Qp. It follows from
(a) that the source of ϕp is non-empty. This completes the proof.
(3.11) Proof of Theorem 3.8. We want to show that the setM(k) is finite. Let M1(k) be the
set of isomorphism classes of objects inM(k) forgetting the level structure. Then we have a finite-to-one map M(k) → M1(k) given by the forgetful morphism. Therefore, it suffices to
show that the set M1(k) is finite.
Let Xpbe the set of isomorphism classes of rank 1 free OL⊗ ˆZ(p)-module Vp together with a ˆZ(p)-valued non-degenerate skew-Hermitian form h , i. Let Xp be the set of isomorphism
map
T = Tp× Tp :M1(k) → Xp× Xp
by sending an object (A, λ, ι) to the p-divisible group A[p∞] and the prime-to-p Tate module Q
`6=pT`(A) with the additional structure.
Let x = (Ax, λx, ιx) ∈ M1(k) and put Λx := T−1(T (x)). Let Gx be the group scheme of
automorphisms of x over SpecZ which is defined as follows: for any commutative ring R, the group of R-points of Gx is given by
Gx(R) := { g ∈ (EndOL(Ax) ⊗ R)
×| g0
g = 1 },
where g 7→ g0 is the Rosati involution induced from λx. The next proposition shows that
Λx is finite. As the image of T is finite, one shows that M1(k) is finite. This completes the
proof.
(3.12) Proposition Let Λx and Gx be as above. Then there is a bijection between Λx and
Gx(Q)\Gx(Af)/Gx(ˆZ).
Proof. We first note that Gx(Q)\Gx(Af)/Gx(ˆZ) classifies the isomorphism classes of
polarized abelian OL-varieties (A, λ, ι) over k such that
(i) (A[`∞], λ, ι) ' (Ax[`∞], λx, ιx) for all `, including p.
(ii) There is an OL-linear quasi-isogeny φ : A → Ax such that φ∗λx = λ.
A proof of this result can be found in [Y1, Thm. 10.5]. Therefore, we need to show that for any (A, λ, ι) ∈ Λx, the condition (ii) is automatic.
It follows from Lemma 3.10 that there is an OL-linear quasi-isogeny φ : A → Ax such
that φ∗λx = λa for some totally positive element a ∈ L×0. The element a is changed up to an
element in NL/L0(L
×) if a different φ is chosen. As (i) is satisfied, the induced quasi-isogeny
φ` can be adjusted to be an isomorphism, hence a ∈ L0⊗ Q` is a local norm for all primes `.
It follows from positivity that a is also a local norm at the infinite place. By the local-global principle for the norm equation of the cyclic extension L/L0, a is a global norm. Therefore
we can exhibit a morphism φ satisfying (ii).
(3.13) Remark (1) Theorem 3.8 can be strengthened that M is finite over W (Fq). We
have proved the quasi-finiteness. The properness is equivalent to that all objects in M have potentially good reduction. As the residue fields in the valuation criterion are finite in this case, one can apply a result of Serre-Tate that if the `-adic Galois representation T`(AQp) is
abelian, then A has potentially good reduction, see [ST, Thm. 2 (i)]. The existence of N´eron models is required in this argument.
(2) The preimage f−1(Λx) of Λx under the forgetful map f : M(k) → M1(k) can be
identified with Gx(Q)\Gx(Af)/Gx(Zp) × KNp, where K p
N is the kernel of the projection
§4. Classification of p-divisible groups of CM type
In this section we classify the p-divisible groups with the additional structures over an al-gebraically closed field k of characteristic p. We are in the local situation and let notations L, L0, π, e, f be as in (3.3) and write O for OL0. Let O
nr (resp. Onr
L) denote the maximal
unramified extension of Zp in O (resp. OL). There are three cases:
(split) L = L0⊕ L0 and H = Hw⊕ Hw¯.
(inert) L is the unramified quadratic extension of L0.
(ramified) L is a ramified quadratic extension of L0.
As pointed out before, we can classify objects (H, λ, ι) with the polarization degree minimal. In this case, all OL-linear polarizations on (H, ι) are classified by O×/NL/L0(O
×
L). Hence we
only need to classify the pairs (H, ι). Let M be its covariant Dieudonn´e module. We treat them by cases as notations are not uniform.
(split) We have M = Mw ⊕ Mw¯ with each factor a Dieudonn´e O-module. Write O ⊗ W =
⊕Wiinto eigenspaces with respect to the action of Onrby embeddings Hom(Onr, W ) = Z/f Z.
Then we have the decomposition for the Dieudonn´e modules Mw = ⊕i∈Z/f ZMwi, Mw¯ = ⊕i∈Z/f ZMwi¯.
and also for the Lie algebras
Lie(Mw) = ⊕i∈Z/f ZLie(Mw)i, Lie(Mw¯) = ⊕i∈Z/f ZLie(Mw¯)i.
If Lie(Mw) ' ⊕i∈Z/f Zk[π]/(πe i w), Lie(M ¯ w) ' ⊕i∈Z/f Zk[π]/(πe i ¯ w) as O ⊗ k-modules for some ei
w, eiw¯, we put lie(M ) := (eiw, eiw¯)i∈Z/f Z, called the Lie type. The
invariant lie(M ) determines the isomorphism classes of O ⊗ k-modules Lie(M ). Note that we have P ei
w = dw and P eiw¯ = dw¯ (3.1).
First, we have Vf(M
w) = πdw(Mw). Then Mw is generated by the “unital sections” of
the operator π−dwVf. So we can choose a Wi-generator xi
w for Mwi such that V xi+1w = πe
i wxi
w
for all i ∈ Z/f Z. Similarly we can find a Wi-generator xiw¯ for Mwi¯. This shows that if lie(N ) = lie(M ) then N and M are isomorphic as Dieudonn´e OL-modules.
Let D−1 be the inverse of the different of O over Zp with exponent −δ. If h , i is a minimal
degree OL-linear polarization on M . Then there is a unique pairing ( , ) on M with values
in O ⊗ W such that hx, yi = TrO⊗W/Wπ−δ(x, y) for all x, y ∈ M . Put ui = (xiw, xiw¯) and it
follows from (V x, V y)σ = p(x, y) that πeiw+ewi¯(ui)σ = pui+1. The pairing h , i is perfect if and only if ui are all units. This happens exactly when eiw + eiw¯ = e for all i ∈ Z/f Z.
(inert) Write Hom(Onr
L, W ) = {τi} ' Z/2fZ in a way that σ(τi) = τi+1and τi = τi+f. Then
we can write OL ⊗ W = ⊕(Wi ⊕ Wi) into eigenspaces with respect to the action of OnrL.
Similarly we have the decomposition for the Dieudonn´e module and the Lie algebra: M = ⊕i∈Z/f Z(Mi⊕ Mi), Lie(M ) = ⊕
where Mi and Mi are the corresponding eigenspaces of M for τ
i and τi and the same for
Lie(M )i and Lie(M )i. Note that Mi ⊕ Mi is a free Wi⊕ Wi-module of rank one. Write
Lie(M )i ' k[π]/(πei
), Lie(M )i ' k[π]/(πe¯i ) and put lie(M ) = (ei, ¯ei)
0≤i<f.
In this case, we have V2f(M ) = πef(M ) and there is a generator xi and ¯xi for Mi and
Mi respectively such that V xi+1 = πei
xi and V ¯xi+1= πe¯i ¯
xi. It follows that the isomorphism
classes of Dieudonn´e OL-modules are uniquely determined by the Lie types lie(M ).
The last step is the same as the split case except that we lift the pairing h , i to ( , ) with values in OL⊗ W here. Put ui = (xi, ¯xi) and we obtain πe
i+¯ei
(ui)σ = pui+1. From this we conclude that the pairing h , i is perfect if and only if ei+ ¯ei = e for all 0 ≤ i < f .
(ramified) Write OL⊗ W = ⊕Wi into eigenspaces with respect to the action of OLnr = O nr
by the embeddings Hom(Onr, W ) ' Z/f Z. Let π
w be a uniformizer of L and denote by πw
again the projection of πw ⊗ 1 in Wi. We have the decomposition
M = ⊕i∈Z/f ZMi, Lie(M ) = ⊕i∈Z/f ZLie(M )i.
and put lie(M ) = (ei)
i∈Z/f Z if Lie(M ) ' ⊕i∈Z/f Zk[πw]/(πe
i
w) for some ei.
As Vf(M ) = πef
w(M ), we can find a generator xi for Mi such that V xi+1 = πe
i
wxi. It
follows that the isomorphism classes of Dieudonn´e OL-modules are uniquely determined by
the Lie types lie(M ).
By the same way treated in the inert case, we show that an OL-linear polarization (of
minimal degree) is separable if and only if ei = e for all i ∈ Z/f Z. This proves Lemma 3.4.
§5. Reduction
In the previous section, we showed that the set of the isomorphism classes of polarized abelian OL-varieties of characteristic p can be decomposed as a disjoint union of double
coset spaces. The union is indexed by the isomorphism classes of underlying Dieudonn´e and Tate modules. On the other hand, there is also the similar decomposition in characteristic 0. As these abelian varieties have good reduction, the reduction map should identify these double coset spaces for characteristic 0 and p. In this section, we study this reduction map. (5.1) Let notations be as in (3.1). We fix an algebraic closure Qp of Qp and fix an embedding
Q ⊂ Qp. Let Φ be a CM type of L and put Σ := Hom(L, C) and Σ0 := Hom(L0, C). We
identify the sets of embeddings by the inclusions Q ⊂ C and Q ⊂ Qp
Hom(L, C) = Hom(L, Q) = Hom(L, Qp).
For a prime v of L0 and a prime w of L over p, we put
Σ0,v := Hom(L0,v, Qp), Σv := Hom(Lv, Qp), Σw := Hom(Lw, Qp).
Clearly, we have Σv = ( Σw∪ Σw¯ if v = w ¯w splits in L, Σw otherwise, and Σ = ∪wΣw = ∪vΣv, Σ0 = ∪vΣ0,v.
We put Φw = Φ ∩ Σw and Φv = Φ ∩ Σv. If v = w ¯w splits in L, then Φv = Φw ∪ Φw¯ and
Φw∩ Φw¯ = ∅, Φw∪ Φw¯ = Σw. If v is inert or ramified in L, then Φw is a CM type of Σw. In
this case, 2 · | Φw| = | Σw|.
(5.2) Let A be an abelian OL-variety of dimension g over Q. Then A extends an abelian
OL-scheme eA over a localization Op of some number field k at the chosen prime p (5.1).
Reduction modulo p and base change to Fp, we get an abelian OL-variety Ap over Fp. It
follows from the uniqueness of N´eron model that Ap is uniquely determined. Hence one gets
the reduction map Red :M(Q) → M(Fp) (3.6). This is nothing but the composition of
M(Q) ∼
→M(Qp) = M(OQ
p) →M(Fp).
If A has CM type Φ, then it follows from the canonical isomorphism
Lie( eA/Op) ⊗Op Fp = Lie(Ap) (1)
that the Lie type lie(Ap) (in Sect. 4) is determined by Φ, and we denote it by lie(Φ).
There-fore, the reduction map Red refines a map
Red :MΦ(Q) → Mlie(Φ)(Fp), (2)
where MΦ(Q) (resp. Mlie(Φ)(Fp)) is the subset of M(Q) (resp. of M(Fp)) consists of objects
with CM type Φ (resp. with Lie type lie(Φ)).
(5.3) Let H be the p-divisible group of Ap and write H = ⊕Hv. As A admits a prime-to-p
OL-linear polarization, each local factor Hv satisfies the condition of Lemma 3.4 (b).
Re-garded as an OL0⊗Zk-module, [Lie(Ap)] is [OL0⊗Zk] in the Grothendieck group K0(OL0⊗Zk).
In other words, Lie(Ap) satisfies the Kottwitz condition defined from abelian varieties with
RM by L0 [K2, Sect. 5]. More precisely, if we write, for each i ∈ Z/fvZ = Hom(OLnr0,v, W ),
lie(Hv)i = (eiw, eiw¯) if v = w ¯w splits in L, (ei, ¯ei) if v is inert in L, ei if v is ramified in L, then the condition is that
(∗) eiw+ eiw¯ = ev, ei+ ¯ei = ev, or ei = ev, respectively, for all i.
We will call a Lie type good if it satisfies this condition.
To describe lie(Φ), we first note that the Lie type is determined by the dimensions of eigenspaces. Namely, we have
dim Lie(Hw)i = eiw, dim Lie(Hw¯)i = eiw¯, (split)
dim Lie(Hv)i = ei, dim Lie(Hv)i = ¯ei, (inert)
dim Lie(Hv)i = ei, (ramified).
It follows from the identification (1) that
dim Lie(Hw)i = | Φiw|, dim Lie(Hw¯)i = | Φiw¯|, (split)
dim Lie(Hv)i = | Φiv|, dim Lie(Hv)i = | Φ¯i|, (inert)
dim Lie(Hv)i = | Φiv|, (ramified),
where
(split) Φi
w (resp. Φiw¯) is the preimage in Φw (resp. Φw¯) of i under the restriction map
Hom(L, Qp) → Hom(Lnrw, W ) = Z/fvZ (resp. Hom(L, Qp) → Hom(Lnrw¯, W )).
(inert) Φiv (resp. Φ¯iv) is the preimage of i, for 0 ≤ i < fv, (resp. of ¯i = i + fv) in Φv under
the restriction map Hom(L, Qp) → Hom(Lnrv , W ) = Z/2fvZ.
(ramified) Φi
v is the preimage of i in Φv under the restriction map Hom(L, Qp) →
Hom(Lnr
0,v, W ) = Z/fvZ.
We conclude that lie(Φ) = (lie(Φv))v, where
lie(Φv) = (| Φi w|, | Φiw¯| )i∈Z/fvZ if v = w ¯w splits in L, (| Φiv|, | Φ¯i v| )i∈Z/fvZ if v is inert in L, (| Φi v| )i∈Z/fvZ if v is ramified in L.
(5.4) It is clear that the map Φ 7→ lie(Φ) is not injective in general. This map is surjective from the set of CM types to that of good Lie types. To see this, we first note that the sets Σi
w∪ Σiw¯, Σiv∪ Σ ¯i
v, and Σiv are stable under the complex conjugate, where the notations are
similarly defined as Φ∗•’s above. Choose a subset of each component of embeddings
Φi w ⊂ Σiw, (split) Φiv ⊂ Σi v, (inert) Φiv ⊂ Σi v, (ramified) such that | Φi
w| = eiw, | Φiv| = eiv and Φiv is a CM type of Σiv respectively. Put Φiw¯ := Σiw¯r Φiw
and Φ¯i v := Σ
¯i
v r Φiv. Then the union of such Φ ∗
•’s is a desired CM type Φ.
We will prove
(5.5) Proposition The reduction map Red in (5.2) (2) is bijective.
(5.6) Corollary A polarized abelian OL-variety (A0, λ0, ι0) of dimension g over Fp can be
lifted to that over Q if and only if the underlying abelian OL0-variety satisfies the Kottwitz condition defined by abelian varieties with real multiplication by OL0.
(5.7) Let x = (Ae xe, λxe, ιxe ∈ MΦ(Q) and x = (Ap, λp, ιp) be its reduction in Mlie(Φ)(Fp).
Define the group schemes G
e
x and Gx over SpecZ as in (3.11). We identify Gex with Gx by
the canonical isomorphism EndOL(A) ' EndOL(Ap). Let Xp and X
p be the isomorphism classes of p-divisible groups and prime-to-p Tate
modules in question defined in (3.11). Let eXp be the set of isomorphism classes of
quasi-polarized p-divisible OL⊗ Zp-groups of height 2g over OQp. We have natural maps
T = Tp× Tp : Mlie(Φ)(Fp) → Xp × Xp
e
T = eTp× Tp : MΦ(Q) → eXp× Xp
(3)
and put Λx := T−1(T (x)) and Λex := eT −1( eT (
e
x)). As objects in MΦ(Q) are isogenous, the
proof of Proposition 3.12 shows that there is a natural bijection Λex ' Gx(Q)\Gx(Af)/Gx(Zp) × K
p
N (4)
via the identification Gex = Gx. The natural bijections between Λ• and the double coset are
compatible with the reduction map, as the construction of the bijection uses the isogeny. Therefore, the map Red : Λ
e
x → Λx induces the identification on the double coset (4). This
shows the injectivity of the reduction map Red.
For the surjectivity part, let x0 = (A0, λ0, ι0) be a point in Mlie(Φ)(Fp). We first show
that there is a polarized abelian OL-variety y = (A, λ, ι) over Q with CM type Φ such that
Tp(Red(y)) = Tp(x0). We know that the isomorphism classes of the p-divisible groups H in
question are determined by the Lie types and the polarization classes of the components in the decomposition H = ⊕Hv, (3.3) (3.4) and Sect. 4. Choose a prime-to-p polarized abelian
OL-variety (A, λ0, ι) of CM type Φ. Then for each prime v of L0 over p there is av in OL0,v such that λ0vav = λ0,v. By the weak approximation, there is a totally positive element a in OL
close to av for all v|p. Put λ := λ0a, and then (A, λ, ι) is the desired one. Therefore we get a
p-divisible group (A[p∞], λ[p∞], ι ⊗ Zp) with the additional structure over a ring Op of p-adic
integers such that its reduction is isomorphic to (A0[p∞], λ0[p∞], ι0⊗ Zp). By the Serre-Tate
theorem, x0 lifts to an object in MΦ(Op). This finishes the proof of Proposition 5.5.
(5.8) Corollary Any quasi-polarized p-divisible OL⊗ Zp-group (H, λ, ι) of height 2g over k
is attached from a polarized abelian OL-variety of dimension g over k.
Proof. Choose an OL-linear isogeny φ : (H0, ι0) → (H, ι) such that H0 has good Lie type.
Take the pull-back polarization λ0 := φ∗λ. Then there is a polarized abelian OL-variety
(A1, λ1, ι1) over k which induces (H0, λ0, ι0), see (5.7). By a theorem of Tate, there is a
polarized abelian OL-variety (A0, λ0, ι0) whose attached p-divisible group is isomorphic to
(H, λ, ι).
(5.9) Remark (1) The special lifting result Corollary 5.6 holds for p = 2, which is limited for general lifting results as in [Y2] by the deformation theory of Grothendieck-Messing and
Fontaine’s theory. However, we do not have control on the ramification of the base ring to lift to.
(2) In characteristic 0, two abelian OL-varieties are OL-linear isogenous if and only if
their CM types are the same. However, in characteristic p, the Lie type is not an isogeny invariant. Lemma 3.10 (a) gives the necessary and sufficient condition when two abelian OL-varieties in characteristic p are OL-linear isogenous.
(3) It has been known for a while that the naive Rapoport-Zink local model [RZ] is not flat in some ramified cases. This was first found by G. Pappas [P]. One can also use the degeneracy of the Lie types under the reduction map to explain this.
Let OK be the ring of integers of a quadratic imaginary field K in which p is ramified.
Consider the moduli spaces of 4-fold polarized abelian OK-varieties with signature (3, 1) and
(2, 2) using the naive moduli problem [RZ]. Say they are M(3,1) and M(2,2). For a point
(A, λ, ι) either inM(3,1)(Fp) or inM(2,2)(Fp), the Lie algebra Lie(A) defines the same element
in the Grothendieck group K0(OK ⊗Z Fp). Therefore, the special fibres of both moduli
spaces are the same, which has dimension at least 4. However, the generic fibre ofM(3,1) has
dimension 3, which violates the flatness.
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National Center for Theoretical Sciences Mathematics Division
National Tsing-Hua University Tsingchu, Taiwan, 30043, R.O.C. Email: chiafu@math.cts.nthu.edu.tw
Department of Mathematics Columbia University
New York, NY 10027
