1.5. CM TYPES 65
1.5. CM types
Let A be an isotypic abelian variety of dimension g 0 over a field K such
that A admits sufficiently many complex multiplications. By Theorem 1.3.4, we
may and do choose a CM field L
with degree 2g. (Conversely, by
Theorem, the existence of such an L forces A to be isotypic.) It turns out
that the L-linear isogeny class of A is encoded in terms of a rather simple discrete
invariant when char(K) = 0 and K is algebraically closed. We wish to review the
basic features of this invariant, called the CM type, and to discuss some useful
replacements for it in positive characteristic.
The order O = L End(A) in L acts on A over K, and is called the CM order.
It acts K-linearly on the tangent space T = Lie(A) at the origin, so if char(K) = 0
then L = O ⊗Z Q acts K-linearly on T , whereas if char(K) = p 0 then O/(p)
acts K-linearly on T . In particular, if char(K) = 0 then T is an L ⊗Q K-module
whose isomorphism class is an invariant of the L-linear isogeny class of A over K;
nothing of the sort is true when char(K) = p 0.
1.5.1. Characteristic 0. We now focus on the case char(K) = 0. Let K /K
be an algebraically closed extension. Since L ⊗Q K Ki for finite (separable)
extensions Ki/K, any L ⊗Q K-module M canonically decomposes as Mi for
a Ki-vector space Mi. Thus, if dimK M is finite then the isomorphism class of
M is determined by the numbers dimKi Mi, which in turn are determined by the
isomorphism class of the L ⊗Q K -module M ⊗K K .
The K -algebra L ⊗Q K has a very simple form: it is
where ϕ ranges
through all field embeddings L K and denotes K viewed as an L-algebra
via ϕ. Hence, any L ⊗Q K -module M decomposes into a corresponding product
of eigenspaces over K on which L acts through ϕ. We conclude that for
an L ⊗Q K-module M with finite K-dimension, the isomorphism class of M is
determined by the numbers dimK (M ⊗K K as ϕ varies through Hom(L, K ).
On the set Hom(L, K ) = Hom(L, Q) (with Q the algebraic closure of Q in K )
there is a natural involution defined by precomposition with the intrinsic complex
conjugation ι of the CM field L (i.e., the non-trivial automorphism of L over its
maximal totally real subfield L+). This decomposes the set Hom(L, K ) of size
2g into g “conjugate pairs” of embeddings. In the special case K = C we can
also compute the involution on Hom(L, K ) by using composition with complex
conjugation on K = C.
An especially interesting example is the L ⊗Q K -module M = T ⊗K K with
T = Lie(A) for a CM abelian variety A over K with complex multiplication by L.
There is a non-trivial constraint on the eigenspaces (T ⊗K K for the L-action
on T ⊗K K (with ϕ varying through the embeddings ϕ : L K ):
1.5.2. Lemma. When char(K) = 0, each eigenspace (T ⊗K K is at most 1-
dimensional over K . If Φ denotes the set of g distinct embeddings ϕ : L K
for which there is a ϕ-eigenline in T ⊗K K then Φ contains no “conjugate pairs”.
That is, we have a disjoint union decomposition Hom(L, K ) = Φ ι).
Proof. By considerations with direct limits (as in the proof of Proposition, we may and do first arrange that K is finitely generated over Q. The
choice of algebraically closed extension K /K does not matter, so we can replace
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