76 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Since the qv-Frobenius generates the center of the endomorphism algebra of
any abelian variety over κv, the Skolem-Noether theorem ensures that any two
Q(πv)-embeddings of L into the central simple Q(πv)-algebra
End0(Aκv
) are related
through conjugation by a unit. Hence, there is an isogeny u End(Aκv ) such that
u φ is L-linear. By renaming this as φ, we may arrange that φ is L-linear. Thus,
in Theorem 1.6.5.1 we may choose the CM lift so that the action of a specified
degree-2g CM field L
End0(A)
also lifts.
It is natural to ask for a strengthening of Theorem 1.6.5.1 in which the isogeny
is applied prior to making a residue field extension (to acquire a CM lifting). As
we will record near the end of 1.8, such a stronger form is true and follows from
one of the main results proved later in this book.
1.7. A theorem of Grothendieck and a construction of Serre
1.7.1. Isogenies and fields of definition. Let A be an abelian variety over a
field K and let K1 K be a subfield. We say that A is defined over K1 if there
exists an abelian variety A1 over K1 and an isomorphism f : A (A1)K. We use
similar terminology for a map h : A B between abelian varieties over K (i.e.,
there exists a map h1 : A1 B1 between abelian varieties over K1 such that (h1)K
is identified with h).
For example, suppose K/K1 is a primary extension of fields (i.e., K1 is separably
algebraically closed in K) and consider abelian varieties A and A over K such that
there are isomorphisms f : A (A1)K and f : A (A1)K for abelian varieties
A1 and A1 over K1. By Lemma 1.2.1.2, the pairs (A1,f) and (A1,f ) are unique
up to unique isomorphism and every map A A as abelian varieties over K is
defined over K1 in the sense that it uniquely descends to a map A1 A1 as abelian
varieties over K1. Likewise, by Corollary 1.2.1.4, all abelian subvarieties of A are
defined over K1 (and even uniquely arise from abelian subvarieties of A1). For
general extensions K/K1 such K1-descents may not exist, and when (A1,f) does
exist it is not necessarily unique (up to isomorphism).
1.7.1.1. Example. Assume char(K) = 0 and let K /K be an algebraically closed
extension (a basic example of interest being K = C). We claim that each member
of the isogeny class of AK is defined over the algebraic closure K of K in K (and
hence over a finite extension of K inside K ). To prove this, observe that the kernel
of any isogeny ψ : AK B over K is contained in some torsion subgroup A[n]K ,
and A[n] becomes constant over K (since A[n] is K-´ etale, as char(K) = 0). Hence,
we can descend ker(ψ) to a constant finite subgroup of AK , and the quotient of AK
by this gives a descent of (B, ψ) to K K .
1.7.1.2. Example. When char(K) = p 0, the naive analogue of Example 1.7.1.1
fails. An interesting counterexample is A =
E2
for a supersingular elliptic curve
E over a field K of characteristic p 0. The kernel H of the Frobenius isogeny
E
E(p)
is a local-local K-group of order p, and it is the unique infinitesimal
subgroup of E with order p (as any commutative infinitesimal K-group of order p
has vanishing Frobenius morphism).
The only local-local finite commutative group scheme of order p over K is αp.
For perfect K this is easily proved by a computation with Dieudonn´ e modules (as
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