CM lifting of abelian varieties

Ching-Li Chai

Princeton University, March 26, 2007

1 Formulation

(w. B. Conrad & F. Oort) p = a prime number, q = p^{a}, 𝔽 := \overline{𝔽_{p}} Let B be an abelian variety of dimension g > 0 over 𝔽_{q} which is isotypic over 𝔽_{q}. Consider statements (CML), (I), (NI), (IR) for B.

(CML) B admits a CM lifting to characteristic 0, i.e. ∃ a local domain (R, 𝔪), R ↩ ℤ, R/𝔪 ⟶^{∼} 𝔽_{q}, ∃ an abelian scheme B over R and

B ×_{Spec R} Spec(R/𝔪) ⟶^{∼} B

∃ L ↪ End_{R}(B) ⊗_{ℤ} ℚ, where Lis a CM field with

[L : ℚ] = 2 dim(B)

Note. (CML) is a geometric statement, equivalent to: B ×_{Spec 𝔽_{q}} Spec 𝔽 can be CM lifted to char. 0.

(I) B admits a CM-lifting up to isogeny:

∀ CM field L ↪^{α} End^{0}(B), [L : ℚ] = dim(B)

∃ an isogeny A ⟶^{β} B s.t. (A, L) can be CM lifted to char. 0, i.e.

∃ a noetherian local domain (R, 𝔪), R ↩ ℤ,

R/𝔪 ⟶^{∼} 𝔽_{q},

∃ an abelian scheme A over R

∃ a subring O ⊂ O_{L}, [O_{L} : O] < ∞

∃ an abelian scheme A over R and an isomorphism

ϵ: A ×_{Spec R} Spec 𝔽_{q} ⟶^{∼} A

∃ an embedding O ↪ End_{R}(A) which is compatible with α, β and ϵ.

(NI) B admits a CM-lifting to a normal domains up to isogeny: ∀ L ↪^{α} End^{0}(B), L a CM field,

[L : ℚ] = 2 dim(B)

∃ an isogeny A ⟶^{α} B ∃ a noetherian normal local domain (R, 𝔪), R ↩ ℤ,

R/𝔪 ⟶^{∼} 𝔽_{q},

∃ an abelian scheme A over R

∃ a subring O ⊂ O_{L}, [O_{L} : O] < ∞

∃ an abelian scheme A over R and an isomorphism

ϵ: A ×_{Spec R} Spec 𝔽_{q} ⟶^{∼} A

∃ an embedding O ↪ End_{R}(A) which is compatible with α, β and ϵ.

The following consequence of classical CM theory is well-known.

Theorem. (IR) B admits a CM-lifting to normal domains up to isogeny after a finite residue field extension: For any CM field L↪ End^{0}(B), ∃ a finite extension field κ ⊃ 𝔽_{q} such that (NI) holds for

(B×_{Spec 𝔽_{q}}Spec κ, L, α)

2 A 2-dimensional example

Suppose that p ≡ 2, 3 (mod 5). Let C be a ℤ[ζ_{5}]-linear abelian surface over 𝔽_{p^{2}} such that Fr_{C} = pζ_{5}; C exists by Honda-Tate. We have End^{0}(C) = ℚ(ζ_{5}), C is supersingular with a-number 2.

(1) (NI) fails for C: Suppose that (R, 𝔪) is a normal local domain with R/𝔪 = 𝔽_{p^{2}}, and C_{/R} is a CM lift of C with CM type Φ. Then the reflex field of Φ is ℚ(ζ_{5}) because Φ ∩ \overline{Φ} = ∅ and the complex conjugation is the unique element of order 2 in Gal(ℚ(ζ_{5})/ℚ) ≅ ℤ/4ℤ.

R ⊃ ℤ[ζ_{5}] ∵ R is normal, so R/𝔪 ⊃ 𝔽_{p^{4}}.

(2) It is not possible to lift C and the ℤ[ζ_{5}]-action to char. 0: Suppose (C_{/R}, α : ℤ[ζ_{5}] → End_{R}(C)) is such a lift, R ⊃ ℤ. Then the CM type

Φ ⊂ Hom_{ring}(ℚ(ζ_{5}), \overline{frac(R)}) for (C, α)

corresponds to the character of the ℤ[ζ_{5}]-action on Lie(C) in char. p. So Φ is stable under action the unique element of order 2 in

Gal(ℚ(ζ_{5})/ℚ) ≅ Gal(ℤ[ζ_{5}]/p, 𝔽_{p}) ≅ ℤ/4ℤ;

i.e. Φ is stable under the complex conjugation, a contradiction.

(3) (CML) holds for C: ∃ a ℤ[ζ_{5}]-linear isogeny

δ : A → C ×_{Spec 𝔽_{p^{2}}} Spec 𝔽_{p^{4}}, s.t.

(a) Ker(δ) ≅ α_{p}, A has a-number 1, and Ker(δ) is the unique subgroup of A[p] of order p.

(b) The subset Φ′ ⊂ Hom_{ring}(ℤ[ζ_{5}], W (𝔽_{p^{4}})

corresponding to the character of the ℤ[ζ_{5}]-action on Lie(A_{0}) is a CM type for ℚ(ζ_{5}).

Let A= the ℤ[ζ_{5}]-linear lift of A over W (𝔽_{p^{4}}) Let G= a finite flat subgroup of order p of A[p]×_{Spec W (𝔽_{p4})}Spec O_{K} for some finite extension field K of frac(W (𝔽_{p^{4}})). Then

(A×_{Spec W (𝔽_{p^{4}})}Spec O_{K}) /G

is a CM lift of A over O_{K}.

A higher dimensional example. Let p be an inert prime for ℚ(ζ_{17}), i.e. (\frac{−1}{17}) = −1. Let A be an ℤ[ζ_{17}]-linear abelian variety over 𝔽_{p^{2}} with Fr_{A} = pζ_{17}. Then as before,

• (NI) fails for A,

• A does not admit a ℤ[ζ_{17}]-linear lift to char. 0,

• ∃ 𝔽_{q}-isogeny A → B s.t. A can be CM lifted to char. 0.

One needs to generalize the “trick” use for the 2-dimensional example C to produce a CM lifting of A. This turns out to be the key to the question (I).

3 Results

1. (CML) is false. Will generalize Oort’s 1992 example and explain an obstruction to (CML), from the size of fields of definition of B[p^{∞}].

2. There is an obstruction to (NI), from the size of the residue fields above p of reflex fields of CM type of L. This is the only obstruction. (proved 2 years ago)

3. CM lifting up to isogeny (I) is true.

Residual reflex condition: If (NI) holds for a g-dimensional abelian variety B over 𝔽_{q} and a CM subfield L ⊆ End_{𝔽_{q}}^{0}(B) with [L : ℚ] = 2g, then ∃ a p-adic CM type Φ ⊆ Hom_{ring}(L, \overline{ℚ}_{p}) s.t.

(i) The slopes of B are given in terms of (L, Φ) by the Shimura-Taniyama formula

\frac{ord_{v}(Fr_{B,q})}{ord_{v}(q)} = \frac{#\{ϕ∈Φ : ϕ induces v on L\}}{[L_{v}:ℚ_{p}]}

∀ place v of L above p.

(ii) Let E(L, Φ) ⊆ \overline{ℚ}_{p} be the reflex field, w = the induced p-adic place of E(L, Φ). The residue field κ_{w} of O_{E(L,Φ),w} can be realized as a subfield of 𝔽_{q}.

THM 1. (F. Oort, B. Conrad, CLC) Let B be an abelian variety over 𝔽_{q}. Assume that (B, L, Φ, 𝔽_{q}) satisfies the residual reflex condition. Then there exists

• a finite extension E′/E(L, Φ) inside of \overline{ℚ}_{p},

• an abelian variety A over E′ with good reduction at the p-adic place w′ on E′ induced by \overline{ℚ}_{p},

• an inclusion L ↪ End_{E′}^{0}(A) with associated p-adic CM-type Φ

s.t. the reduction of A at w′ is L-linearly isogenous to B over an isom. κ_{w′} ≃ 𝔽_{q}.

4 Proof of Thm. 1 about (NI)

• L is a CM field with [L : ℚ] = 2g

• Φ is a CM-type for L

• δ is the reflex field of (L, Φ)

• N_{Φ} is the reflex norm associated to (L, Φ)

PROP 2. Let F/E be a finite extension of E, and let ϵ : 𝔸_{F}^{×} → L^{×} be an algebraic Hecke character whose algebraic part ϵ_{alg} is equal to N_{Φ} ∘ Nm_{F/E}. Then ∃ a g-dimensional O_{L}-linear abelian variety A over F with CM type Φ, unique up to F -isogeny, s.t. ϵ is the algebraic Hecke character attached to (A, L). ϵ : 𝔸_{F}^{×} → L^{×} an algebraic Hecke character, v: a finite place of F s.t. ϵ is unramified at v E ⊂ F, w= the place of E below v THM 3. Suppose that ϵ_{alg} factors through Nm_{F/E} : \underline{F}^{×} → \underline{E}^{×}, and an element β ∈ L^{×} satisfies β^{[κ_{v}:κ_{w}]} = ϵ(π_{v}). Then ∃

• Ẽ/E, [Ẽ : E] < ∞

• a place w̃ of Ẽ above w,

• an algebraic Hecke character δ : 𝔸× ; Ẽ → L^{×} s.t.

(i) [κw̃ : κ_{w}] = 1. (ii) δ_{alg} = χ_{alg} ∘ Nm_{Ẽ/E}.

(iii)δ is unramified at w̃, and δ(πw̃) = β.

5 Why (CML) fails

Large fields of definition of A[p^{∞}] obstruct CM lifts.

PROP 4. Let X be a p-divisible group over 𝔽_{q}, and let X_{(0,1)} be the maximal p-divisible subgroup of X with no etale or multiplicative part. Assume that

dim(X_{(0,1)}) ≠ 1 ≠ codim(X_{(0,1)}), i.e. X_{(0,1)} is

not a Lubin-Tate formal group or the dual of a Lubin-Tate formal group. Then ∃ a constant c s.t. ∀ finite field k, there exists an isogeny Y → X with the property that if Y can be defined over a finite field k_{1}, then [k_{1}k : k_{1}] | c.

(“∀k, you can produce Y s.t. every field of definition of Y almost contains k”, using a varying family of α_{p}’s in

α_{p} × α_{p}.)

PROP 5. There exists a finite field κ = κ(g) such that if A is a g-dim. abelian variety over a finite field 𝔽_{q} which admits a CM lift to char. 0, then the p-divisible group A[p^{∞}]×_{Spec 𝔽_{q}}Spec 𝔽 over 𝔽 can be defined over κ.

Rmk. A similar statement holds for p-divisible groups—replace dim(A) by the height of p-divisible groups.

• Prop. 5 says: every CM liftable p-divisible group can be defined over a small finite field.

• Prop. 4 ⟹ ∀ abel. var. B over 𝔽 with p-corank ≥ 2, ∀r ∈ ℕ, ∃ an isogeny A → B s.t. A[p^{∞}] cannot be defined over 𝔽_{p^{s}} with s ≤ r.

6 Proof of (I)

L_{0} := max. tot. real subfield of L, Σ_{L_{0},p} := places of L_{0} above p. May assume that O_{L} operates on B. Want: an O_{L}-linear isogeny A → B over 𝔽_{q} such that A admits a CM lift.

Step 1. (localization by Serre-Tate) Need to construct

• an O_{L,p}-linear isogeny β: Y → B[p^{∞}] over 𝔽_{q}, where O_{L,p} = O_{L}⊗ℤ_{p}.

• an O′-linear lifting Y of Y_{𝔽} to char. 0, O′ ⊂ O_{L,p} an order, s.t. the action of L on Lie(Y)⊗ℚ corresponds to a CM type of L.

Places v ∈ Σ_{L_{0},p} give idempotents in O_{L_{0},p} and product decomposition of β.

Step 2. (classification of p-divisible groups over 𝔽 and weak Galois descent) E = a finite extension of ℚ_{p} I = Hom_{ring}(κ_{E}, 𝔽), a torsor under Gal(κ_{E}/𝔽_{p}) R_{𝔽}(O_{E}) = Grothen. group of (O_{E} ⊗ 𝔽)-modules of finite type; it is ≅ ⊕_{i∈I}ℤ and compatible with the Gal(𝔽/𝔽_{p})-actions.

(a) (C.-F. Yu) An element

ξ = (n_{i})_{i∈I} ∈ ⊕_{i∈I} ≅ R_{𝔽}(O_{E})

is the Lie type of an O_{E}-linear p-divisible group Z of CM type (i.e. ht(Z) = [E : ℚ_{p}]) if and only if

0 ≤ n_{i} ≤ e(E/ℚ_{p}) ∀i ∈ I .

(Then dim(Z) = ∑_{i∈I} n_{i} =: dim(ξ).)

(b) (C.-F. Yu) Any O_{E}-linear p-divisible group Z of CM type over 𝔽 is determined up to isomorphism by its Lie type, the class in R_{𝔽}(O_{E}) of the O_{E}-module Lie(Z).

(c) If Z and the O_{E}-action on Z are defined over a finite field κ, then the Lie type of Z is κ-rational, i.e. it is fixed by Gal(𝔽/κ).

(d) If Z is an O_{E}-linear p-divisible group of CM type over a finite field κ, and ξ is a κ-rational Lie type with dim(ξ) = dim(Z), then ∃ an O_{E}-linear isogeny Y → Z over κ s.t. ξ = Lie type of Y . ι = the complex conjugation for L/L_{0} Step 3. (the self-duality condition) Defn. For v ∈ Σ_{L_{0},p}, write O_{L,v} := O_{L}⊗_{O_{L0}}O_{v}. The Lie type δ_{v} of an O_{L,v}-linear p-divisible group Z of CM type over 𝔽 is self-dual if δ_{v} + ιδ_{v} is equal to the class of O_{L,v}⊗𝔽 in R_{𝔽}(O_{L,v}).

(C.F. Yu) The Lie type δ_{v} of Z as above is self-dual ⟺ ∃ an O_{L,v}-linear lift Z of Z to char. 0 whose CM type Φ_{v} satisfies Φ_{v} ∩ ιΦ_{v} = ∅.

Note. If there exists a self-dual 𝔽_{q}-rational Lie type in R_{𝔽_{q}}(O_{L,v}), then ∃ an 𝔽_{q}-isogeny Y_{v} → B[v^{∞}] with lie(Y_{v}) = δ_{v} by Step 2 (d), and Y_{v} has a good O_{L,v}-linear CM lift by Step 3.

Step 4. (good and bad places in Σ_{L_{0},p})

(1) ∃ a self-dual 𝔽_{q}-rational Lie type in R_{𝔽_{q}}(O_{L,v}) unless the following holds:

• v does not split in L;

• e(L_{w}/ℚ_{p}) is odd, f(L_{w}/Qp) ≡ 0 (mod 4),

w = the place of L above v.

• # Gal(κ_{w}/(𝔽_{q} ∩ κ_{w})) is even.

Places v ∈ Σ_{L_{0},p} satisfying the above properties will be called “bad”.

(2) If v is a bad place, and w lies above p, then ∃ an O_{w}-linear 𝔽_{q}-isogeny Y_{w} → B[w^{∞}] and an isom.

Y_{w} ≅ O_{w} ⊗_{O_{w}^{ur}} Y_{w,0} ,

where

• O_{w}^{ur} ⊂ O_{w} is the maximal étale ℤ_{p}-subalgebra,

• Y_{w,0} is an O_{w}^{ur}-linear p-divisible group over 𝔽_{q} whose Lie type corresponds to a

Gal(κ_{w}/𝔽_{p^{2}})-orbit in I_{v} ≅ Hom(κ_{w}, 𝔽).

Note. By Steps 3 & 4, to complete the proof of (I) it suffices to construct, ∀ bad place v ∈ Σ_{L_{0},p}, a lifting Y_{w} of Y_{w,0} to a mixed char. d.v.r. (R, 𝔪) and an embedding α_{w} : E_{w}^{ur} → End^{0}(Y_{w}) s.t. the CM type Ψ_{w} of Y_{w} satisfies Ψ_{w} ∩ ιΨ_{w} = ∅.

Let E = L_{w}^{rm}, κ_{E} ≅ 𝔽_{p^{4m}},

Y = Y_{w,0}, w = the place of L above a bad place v. Step 5. (set up for CM lifting at bad places) Identify I = Hom(κ_{E}, 𝔽) with ℤ/4mℤ. May assume: Lie type of Y is {2, 4, …, 4m}.

(1) Use Dieudonné theory or Step 3 (a) & (b) to construct an O_{E}-linear isogeny

γ : X → Y ×_{Spec κ}Spec 𝔽

characterized by

• the Lie type of X is

{1, 3, …, 2m − 1, 2m + 2, 2m + 4, …, 4m},

which is self-dual (∵ ι ↔ translation by 2m),

• Ker(γ) ≅ α_{p}^{m}

K/frac(W (𝔽)): tame extn. of degree p^{4m} − 1 ζ = a generator of κ×_{E}

(2) Let X be the O_{E}-linear lift of X over W (𝔽); X [p] is of (p, …, p) type with action by κ_{E}, its closed fiber contains Ker(γ).

(3) X [p](K) contains a non-zero point ξ.

(4) For a = 0, 1, …, m − 1, let ξ_{(a)} := [ζ^{a}](ξ).

Define G := closure in X [p]×_{Spec W (𝔽)}Spec O_{K} of the subgroup generated by ξ_{0}, …, ξ_{(m−a)}.

Step 6. (compute schematic closure) Use either Raynaud’s (p, …, p) or integral p-adic Hodge theory (Breuil, Kisin, Zink, Vasiu-Zink) to show

G×_{Spec W (𝔽)}Spec 𝔽 = Ker(γ)×_{Spec W (𝔽)}Spec O_{K}

Then X /G is a desired CM lifting of Y . Q.E.D.

Note. This proof produces an 𝔽_{q}-isogeny A → B and a (ℤ+pO_{L})-linear lift A.

Open question. Let A be an O_{L}-linear abelian variety over 𝔽, L a CM field, [L : ℚ] = 2 dim(A). Does (CML) hold for A?