By Proposition (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 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 (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, 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 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 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: 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: 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,
a first guess for an analogue of Theorem 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).
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