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

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 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.