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