68 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
subalgebra R for which A with its OL-action over C descends to an abelian scheme
A over R equipped with an OL-action.
By localization of R, we can arrange that the tangent space Lie(A ) is finite
and free as an R-module, and by increasing R to contain the integer ring of the
Galois closure of L in C we can arrange that the OL-action on Lie(A ) decomposes
as
ϕ∈Φ
with equal to R having action by c OL through multiplication
by ϕ(c) Q R. For any maximal ideal m of R, the natural map of Q-algebras
Q R/m is an isomorphism. Thus, passing to the fiber of A at a closed point of
Spec(R) gives a pair (A, j) over Q with CM type Φ.
This proposition has an important consequence for descending the field of defi-
nition of a CM abelian variety in characteristic 0, as we will see in Theorem 1.7.2.1.
1.5.4.2. Remark. By Theorem 1.3.1.1, for any (A, L) as in Proposition 1.5.4.1,
A has a unique simple factor C (in the sense of Definition 1.2.1.5). By Proposition
1.3.2.1, C is a CM abelian variety with complex multiplication by the CM field
L :=
End0(C)
(see Proposition 1.3.6.4(1)). Since L is canonically identified with
the center of
End0(A),
it naturally embeds into L. Hence, there is a K-valued CM
type Ψ on L arising from C, and the pair (L , Ψ) is determined by (L, Φ) since
A with its complex multiplication by L is determined up to L-linear isogeny by Φ
(and L = L if and only if A = C, which is to say that A is simple). It is therefore
natural to seek an intrinsic recipe to directly construct (L , Ψ) from (L, Φ), and in
particular to characterize in terms of Φ whether or not A is simple.
The criterion is this: among the CM fields in L from which Φ is obtained by full
preimage under restriction, (L , Ψ) is the unique such pair with [L : Q] minimal
and Φ the full preimage of Ψ. Indeed, since the CM type is Q-valued (Remark
1.5.2.2) and the base field K is algebraically closed, it suffices to treat the case
K = C. In this case the desired recipe is established in the complex-analytic theory
(see [82, §22, Rem. (1)]).
1.5.5. Positive characteristic. Assume char(K) 0, and let A be an abelian
variety over K of dimension g 0 admitting an action by an order O in a CM field
L of degree 2g over Q. There is no action by L = Q ⊗Z O on the tangent space
T := Lie(A) of A at the origin since L is a Q-algebra and T is a K-vector space.
Thus, there is not a good notion of CM type on L associated to the embedding of
L into
End0(A).
More specifically, for the CM order O := L End(A) in L, T has
a K-linear action by O/(p) and there is generally no constraint on this action akin
to the eigenspace decomposition considered in characteristic 0 (as in Lemma 1.5.2).
The lack of such a constraint occurs for a couple of reasons, as we now explain.
1.5.5.1. Example. If p divides the discriminant of O over Z or pOL is not prime
in OL then O/pO fails to be a field. In such cases, the K-linear O/(p)-action on T
admits no notion of eigenspace decomposition that closely resembles the situation
in characteristic 0.
1.5.5.2. Example. Suppose that O has discriminant not divisible by p (so O(p) =
OL,(p)) and that p is totally inert in L. In such cases κ := O/(p) is a finite field of
degree 2g over Fp and Aut(L/Q) injects into Gal(κ/Fp), so the canonical complex
conjugation on L induces a non-trivial involution on κ.
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