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
Rϕ with Rϕ 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 184.108.40.206.
220.127.116.11. Remark. By Theorem 18.104.22.168, for any (A, L) as in Proposition 22.214.171.124,
A has a unique simple factor C (in the sense of Definition 126.96.36.199). By Proposition
188.8.131.52, C is a CM abelian variety with complex multiplication by the CM field
(see Proposition 184.108.40.206(1)). Since L is canonically identified with
the center of
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
220.127.116.11) 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
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.
18.104.22.168. 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.
22.214.171.124. 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.