78 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

By Proposition 1.5.4.1 (applied over the algebraically closed base field Q) there

is a CM abelian variety B over Q with complex multiplication by L and CM type

Φ (viewed as valued in Q). The abelian variety BK over K admits complex multi-

plication by L with associated CM type Φ (viewed as valued in K), so by applying

Proposition 1.5.4.1 over K we see that BK is L-linearly isogenous to A over K (as

these two abelian varieties over K are endowed with complex multiplication by L

yielding the same CM type Φ on L). Fix such an isogeny f : BK → A. By Example

1.7.1.1 (applied to K/Q), the finite kernel of f descends to a finite subgroup of B.

The quotient of B by this descent of ker(f) is a descent A0 of A = BK /ker(f) to

an abelian variety over Q.

By Lemma 1.2.1.2, End(A0) → End(A) is an isomorphism. Thus, we may

assume the base field K is an algebraic closure Q of Q. Express Q as a direct limit

of number fields to see that the Q-group A descends to an abelian variety over a

number field inside Q. The same direct limit argument as used at the start of the

proof of Proposition 1.2.6.1 then shows that we can choose the descent of A to a

number field so that all elements of End(A) descend as well.

Theorem 1.7.2.1 can be formulated with a general ground field K of charac-

teristic 0, but the nature of the descent becomes a bit more subtle. Namely, if A

is an abelian variety over a field K of characteristic 0 and if A admits suﬃciently

many complex multiplications, then there is a finite extension K /K such that AK

descends (along with its entire endomorphism algebra) to an abelian variety over a

number field contained in K . In this formulation it is crucial to introduce the finite

extension K /K, even if we just wish to descend the abelian variety (and not any

specific endomorphisms). This is illustrated by quadratic twists of elliptic curves:

1.7.2.2. Example. Consider a CM elliptic curve over C and extend scalars to

K = C(t). Let E be the quadratic twist of this scalar extension by a quadratic

extension K /K, so E is a CM elliptic curve over K whose -adic representation

for Gal(Ks/K) is non-trivial. No member of the isogeny class of E over K can be

defined over C (let alone over Q), as all members of the isogeny class have non-

trivial action by Gal(Ks/K) for their -adic representations. Of course, if we pass

up to K then the effect of quadratic twisting goes away and there is no obstruction

to descent to Q.

Over number fields, CM abelian varieties extend to abelian schemes over the

entire ring of integers at the cost of a finite extension of the ground field. This is

an application of the semi-stable reduction theorem for abelian varieties (see [109,

Thm. 6]), and we record it here for later reference:

1.7.2.3. Theorem. Every CM abelian variety over a number field has potentially

good reduction at all places.

Since every abelian variety over an algebraic closure of Fp descends to a finite

field and hence has suﬃciently many complex multiplications (by Corollary 1.6.2.5),

a first guess for an analogue of Theorem 1.7.2.1 in positive characteristic is that CM

abelian varieties over algebraically closed fields K with positive characteristic can

be descended to the algebraic closure of the prime field inside K (or equivalently,

to a finite subfield of K).